text:
3 Florida teens in stolen vehicle die in violent crash
ST. PETERSBURG, Fla. (AP) - Florida authorities said three teenagers - one 14-year-old and two 16-year-olds - stole a sport utility vehicle, sped away from officers and died in a fiery, violent crash early Sunday morning.
The 4:30 a.m. crash happened in Pinellas County, in west-central Florida.
Sheriff Bob Gualtieri said in a news conference Sunday morning that a fourth teen in the SUV, who is 14, survived and is hospitalized. He said all the teens all had criminal histories - including one who had gotten out of jail on July 31.
Among the deceased: 16-year-olds Keontae Brown and Dejarae Thomas, and 14-year-old Jimmie Goshey.
Gualtieri said a Ford Explorer and a Chrysler Sebring were stolen from a Clearwater car dealership Thursday. Both vehicles were spotted by deputies in early Sunday morning. Deputies tried to apprehend the drivers of the cars, but didn't initiate pursuits of them. The sheriff's office's pursuit policy generally does not allow deputies to chase stolen cars.
Deputies believed the teens were using the stolen cars to commit burglaries. At one point, they set up a perimeter to try to catch the teens, but a deputy spotted the vehicles outside of the perimeter.
At that point, the sheriff said, the cars were in a "cat and mouse" game with each other, accelerating and slowing down. Gualtieri said that the cars hit speeds 100 mph or more.
The deputy that spotted them did not initiate a high-speed chase, but the Explorer continued at about 100 mph when it hit another vehicle, caught fire and went airborne. The Explorer also hit a billboard pole.
The driver hit by the Explorer suffered minor injuries, the sheriff said.
Officers found the Chrysler Sebring and arrested the two in that car; one is 16 and the other 18.
Gualtieri said the teens involved were being monitored under a program for repeat offenders.
In 2015, police in Pinellas made 499 felony arrests for juvenile auto theft, more than any other county in Florida going back eight years, and more than the most populous counties in America, including Los Angeles.. The Tampa Bay Times did an investigation into teen car thefts in April of 2017; the paper found that every four days, a teen crashes a stolen car in Pinellas County. It also found thatiIn nearly every other county in Florida, most people arrested for auto theft are adults. But in Pinellas County, 62 percent are younger than 18 - the largest rate of juveniles arrested for grand theft auto of any sizable Florida county for at least a decade.
"This needs to stop. It's a deadly game," Gualtieri said Sunday. "These kids have got to stop this."
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Florida delivery driver guilty of killing 75 year old woman, setting fire to her home


The fjords and small fishing villages of the east coast were calling, following our adventures along the southern coast of Iceland. At the beginning of our third day on our drive around Iceland's Ring Road, we couldn't believe how much we'd already seen.
The kilometres were low today (about 250 km), but some of the driving proved to be a bit more rugged than the previous days. The eastern fjords are a key attraction on this side of the country. Similar to the western coastline, the shoreline here is a mass of peninsulas jutting out into the ocean and forming fjords in between. To get into many of these fjord areas, you must divert off the Ring Road, and onto the smaller roads.
Doing this will add time to your journey as the roads are often of a lesser quality than the main route. The reward for doing so, however, will be seeing the beauty of these small towns that would otherwise be missed. With so many harbours in these fjord areas, it became a natural fit for the fishing industry to set up along the coast.
With approximately 2,200 inhabitants, Höfn feels like a large town. It is, when you consider most of the small towns we've driven through have very low populations, some are barely populated at all. Höfn is a fishing village and has grown from its early origins as a result of expanding the seafood it catches. Now known as the lobster capital of Iceland, it brings in Norwegian and Icelandic lobster along with their fish catch.
With the colourful boats sitting on the harbour, all under the watchful eye of the Vatnajökull Glacier is it any wonder this is a must-see town.
It has several restaurants and small museums where you can absorb some of the local history.
Dating back to the 1500s, Djúpivogur is another cute fishing village, home to only around 220 people.
The house below is Langabúð, one of the oldest trading houses in the town. This building, dating back to 1790. now houses a folk museum, art museum and a cafe.
As we continued in a northerly direction, the icy landscapes gave way to strange looking mountains. With green vegetation growing up the sides, they stop at a point, keeping the tops of these mountains undressed and bare. There's not a tree to be seen anywhere around here.
Iceland is full of waterfalls. Whilst there are plenty on the southern coast, especially in the Golden Circle, they can be found almost everywhere else too. Sometimes, a hike is needed to get to them as they are located further inland, but many, like the Fossardalur waterfall, are only metres off the side of the Ring Road.
Having our own vehicle means we are free to stop at places like these whenever we want to. There were some days I felt like I was constantly getting in and out of the campervan. There's an area to park and you can walk right out to the edge to get a great view. The real beauty of this 122-metre high waterfall is the bright blue waters of the lagoon it empties into below.
Lagarfljot, a large lake in the east of Iceland has a connection with a similarly large lake in Scotland. Since the 14th century, legend has it that a giant serpentine has made itself a home in this 38-kilometre long lake. With its water originating from the Vatnajökull Glacier, it is the centrepiece of an extraordinary part of the countryside, well-known for hiking and other outdoor adventure activities. Being so close to the glacier, it also goes without saying that there are many more waterfalls around too.
It's a pleasant drive up and around the lake and offers so many opportunities to just take in the ever-changing landscapes.
Egilsstadir and its neighbouring town of Fellabaer found across the Lagarfljot River are two of the larger towns in this area, and have formed the base of the trade hub here. If you are driving this is a perfect time to stop and stock up on food and fuel as many of the basic services are available here. It is also the gateway to the fjord town of Seydisfjordur, our next stop.
If it wasn't for the terrible one road in, one road out experience, this town would be inundated with people all the time. In one way, that's great as it keeps the numbers down, but even me, who is used to driving some very difficult roads, was ready to tap out on this one. No doubt, it had everything to do with us driving through the mountains during a total white-out. Given that this happened in summer, I would truly hate to witness it in winter. If nothing else, it gives you a chance to test out your Iceland driving skills.
Seyðisfjörður is a beautiful town located right out on the eastern coast, on a fjord of the same name. In the mid-1800's it became known as a trading centre, and many Norwegian fishermen came to call this their home. During World War Two, the town also became a base for the Allies.
A large industrial port also sees ferries come across the ocean from Europe during summer. If you are planning on bringing your own motorhome, this is the port in which you do so.
Whilst the cloud up on the mountain didn't amuse me much, once we were down in the town, my thoughts switched from it being treacherous to more aligned with a fairytale. Here, those clouds that made it impossible to see up there, now descended on this town enveloping it in its soft, fluffy arms.
There is an official campground here but we also found ourselves a signed camping park just out of town and within walking distance of the main centre.
The street on the other side of the cafe is the best in town, with bright stripes on the cobbled roadway leading down to the sky-blue church. Known simply as the "Blue Church", this building has been more mobile than any other church I know of. Starting its life on a farm, it was later moved to another area in 1882. In 1894 however, it was blown down from its hilltop position by a storm and was relocated once more to lower ground.
It remained there until 1920 when it was removed once again, and repositioned in Seyðisfjörður. Even here it wasn't safe, with fire ripping through parts of the church in 1989. Today, it stands strong in the centre of town and is easily the most beautiful building here.
There are many points of interest in this eastern region. The best way to see them is to self-drive and take it slow.


Wow … that's wild. He looks QUITE hot in that last pic … It was worth all the effort!
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text:
10 Ways to Become Indispensable at Work
Tips for maximizing your value in the workplace
By Seth Michael Donsky
Economists can debate the finer points of whether or not we are in a recession, but there's no doubt that our current economy is one centered on cost-cutting. With labor costs comprising 85 to 90 percent of a business's operating costs, there's good reason that job security is a top concern in the workforce. We asked leading career experts for insider tips on how to maximize your value on the job. To learn their advice for making yourself indispensable in today's workplace, read on.
What's the biggest difference between an employee who's indispensable and one who's not? "The number-one quality of an indispensable employee is a willingness to go above and beyond," says
Honor your commitments
"Always do what you say you're going to do," Low-Kramen says. "If you're responsible for something, people should be able to bank on its getting done." Luke Kreinberg, a career coach with San Francisco's
"Develop an entrepreneurial spirit," Kreinberg says. "Spot opportunities and execute them." Make it a habit to do more than you're asked. To make a good impression on your colleagues, make an effort to volunteer for additional assignment
You've heard this advice time and again, and for good reason: Attitude is everything. "Smile," advises Jim Penny, a partner at Heidrick & Struggles, one of the world's largest executive search firms. "Be positive and help out, if it's at
Build internal relationships
It's not just about what you know, it's about who you know. "Try to take on a project or role that both strengthens your skills and lets executives outside your immediate area see you," advises Penny. Help out with special
If people ask you to do things that you don't think are your responsibility, do them anyway. "It seems so obvious," says Penny Burns, senior agent with Randstad Staffing Services, "but never say 'That's not my job.' Do that and you'll ne
Understand the politics
Pay careful attention to the dynamics of your office. Who's respected? Who's connected? Who's taken seriously? (Who's not?) Try to get a feel for—and adopt—the culture of the fast-trackers around you. Show up early, stay late and ask
Know your market
A valuable employee is well-informed about trends in her company's industry. Know how your company is different from its competitors—this allows you to make important contributions to the strategic conversations that are typically only
In today's technological environment, it's not enough to be indispensable now—you need to take measures to ensure that you'll be indispensable in the future, too. Jim Scholfield, Senior VP for Rand Thompson and a New York City career c
Demonstrate leadership
Great leaders are always indispensable. To demonstrate your leadership skills, volunteer to head a project, no matter how small. When you're able to deliver excellent results on time—and on budget—your colleagues are bound to take no
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redpajama_set_name: RedPajamaCommonCrawl


This timeline is limited to events associated with the steady erosion of personal and intellectual freedoms. For a more complete listing of events, see The Holocaust Project's Timebase.
9/14: 107 National Socialist deputies are elected to the Reichstag (20% of the vote), making the Nazis Germany's second largest party. Social Democrats remain the largest party in the Reichstag, Germany's Parliament.
1931 9/12: Nazi gangs in Berlin attack Jews returning from synagogue.
1/30: Adolf Hitler appointed Chancellor by President Paul von Hindenburg. The Sturm Abteilung (Nazi party private police, also known as "Brown Shirts") celebrate Hitler's accession to power with a torchlight parade through Berlin. Violence breaks out all over Germany between the Sturm Abteilung and communists.
2/1: Under pressure from Hitler, Hindenburg orders the dissolution of the Reichstag. New elections are called for March 5, 1933.
2/4: Law for the Protection of the German People: this law restricted demonstrations, freedom of speech, freedom of press, and ordered the confiscation of literature considered to be dangerous to the state.
2/5: All Communist Party buildings and printing presses are confiscated.
2/24: Nazi police raid the Communist Party headquarters in Berlin and claim to have discovered plans for a Communist uprising. Formerly private armies of the Nazi Party, the Stahlhelm (Steel Helmet), the Sturm Abteilung (SA) and SS are officially granted auxiliary police status.
2/27: Reichstag, the German Parliament building, is burned. Marinus van der Lubbe, a Dutch Communist, is arrested.
2/28: The Nazi party newspaper, the V�lkischer Beobachter accuses communists of a plot to seize power. Law for the Protection of People and the State abolishes the following rights: free speech, free press, sanctity of the home, security of mail and telephone, freedom to assemble or form organizations and the inviolability of private property. This law also cleared the way for the Nazis to put their political opponents in prison and establish concentration camps.
Sometime in March: Librarians Wolfgang Herrmann and Wilhelm Schuster publish the first attack on "un-German" literature in the professional library journals, entitled "Erkl�rung und Aufruf" ("Clarification and Entreaty").
3/3: Hitler (speaking in Frankfurt): "I don't have to worry about justice; my mission is only to destroy and exterminate, nothing more."
3/5: Election creates the Third Reich. The Nazi party has a majority of Parliament.
3/6: The emergency decree For the Protection of the German People restricts the opposition press and information services.
3/9: The SA sponsors anti-Jewish riots throughout Germany.
3/11: "Reich Ministry of People's Enlightenment and Propaganda" is created by law; it is to be headed by Josef Goebbels.
3/20: The first concentration camp is established at Dachau near Munich.
3/22: Berlin: The Gestapo searches Albert Einstein's apartment.
3/23: The Enabling Act ended parliamentary practice in Germany, giving Hitler's government power to enact laws without parliamentary sanction.
4/1: Hitler imposes a nationwide, one-day boycott of Jewish businesses, physicians and lawyers. Armed SA men are deployed to block the entrances to Jewish-owned shops and stores. Signs are posted in English implying that Jewish claims of persecution are false.
4/7: Two new laws are passed: The Law for the Restoration of the Professional Civil Service is which gives the state authority to dismiss all politically unwanted persons from Civil Service jobs; university and library personnel were especially hard hit. The Law concerning State Governors deprived the German states their traditional jurisdiction over cultural and educational affairs.
4/13: The Deutsche Studentenschaft (German Students' Association) begins their cultural war by posting their "Wider den undeutschen Geist" ("Against the un-German Spirit") posters all over Germany. The virulently anti-Semitic poster lists the ways they intend to "cleanse" German language and literature.
4/25: The Law for Preventing Overcrowding in German Schools and Colleges is declared, limiting "non-Aryan" admittance to institutions of higher education to 1.5 %.
4/26: The Gestapo begins it's state-sanctioned reign of terror.
5/2: All independent and Socialist trade unions in Germany are closed down and dissolved on Hitler's orders.
5/5: Cologne: University students burn books on Judaism or those written by Jewish authors.
5/6: Berlin: 80 members of Nazi student organizations and Sturm Abteiling (SA) raid the Institute for Sexual Research. In less than an hour, they gathered nearly half a ton of books, pamphlets, and teaching materials to be burned at the May 10th book-burning.
5/10: Berlin: Goebbels organizes Nazi student organizations and SA troops to ransack public libraries and the library of the Humboldt University, and burn the books at the Opernplatz. Goebbels speaks before the crowd about the harm that "un-German" literature does to society.
5/16: Librarian Wolfgang Herrmann publishes his "Principles for Sanitizing Public and Lending Libraries" in a professional librarianship journal. It contains one of the earliest lists of authors and titles targeted for removal or destruction.
5/18: Heidelberg: "Students" conduct a book-burning on the university campus, preceded by a torchlight procession through the town. By this date, burnings had also been conducted in Frankfurt, Göttingen, Cologne, Hamburg, Dortmund, Halle, Nuremberg, Würzburg, Hannover, Munich, Münster, Königsberg, Koblenz, and Salzburg.
5/22: Berlin: Nazi secret police (Gestapo) raid public and private libraries and confiscate 500 tons of Marxist materials, including works by Karl Marx, Karl Liebknecht, Rosa Luxembourg, as well as that of Bolshevik leaders. The confiscated books and papers were pulped and auctioned off to paper mills.
6/22: Hermann Goering issues a decree instructing all government employees to spy on each other.
7/5: The Catholic Center Party dissolves; the Nazis become the only active political party in the Reichstag.
7/7: Several German universities announce that Jewish students will not receive their degrees.
7/14: The creation of new political parties is prohibited and The Law on Plebiscites is passed. All political opposition to Nazism is now outlawed and it becomes the one and only political party in Germany. Under the Law on the Revocation of Naturalization and Deprivation of German Citizenship of Jews, German citizenship may be taken away from people designated as "undesirables."
7/20: According to The Holocaust Project, "a number of contemporary historians consider this to be the day Hitler's dictatorship of Germany actually began." Germans are required to use the Hitler salute for general greeting.
7/21: Nuremberg: Hundreds of Jewish store owners are arrested by the SA and paraded through the streets for hours.
7/25: Passage of the Law for the Prevention of Genetically Diseased Offspring.
7/29: Professor Fischer, the new Rector of the University of Berlin says in his inaugural address: "The new leadership, having only just taken over the reins of power, is deliberately and forcefully intervening in the course of history and in the life of the nation, precisely where this intervention is most urgently, most decisively, and most immediately needed. To be sure, this need can only be perceived by those who are able to see and to think within a biological framework, but it is understood by these people to be a matter of the gravest and most weighty concern. This intervention can be characterized as a biological population policy, biological in this context signifying the safeguarding by the state of our hereditary endowment and our race, as opposed to the unharnessed processes of heredity, selection, and elimination."
9/22: Reich Chamber of Culture Law is created to control all literature, press, radio, theater, music and art. "Non-Aryans" are restricted from contributing in these areas. Josef Goebbels' Ministry of Propaganda will direct these efforts.
10/4: Law regulating the function of editors of newspapers and periodicals.
10/23: Martin Buber and 51 other Jewish educators are fired from their positions at German universities.
12/18: Another Nazi decree bars Jews from the field of journalism and its associated professions.
6/5: Book reviewing by persons and organizations unaligned with the Nazi party and government is restricted. Goebbels' Chamber of Culture is to "coordinate" official literary criticism.
9/15: Law for the Protection of German Blood and German Honor. This law, one of the notorious Nürnberg laws, excluded all Jewish and Jewish-related authors, publishers, editors, etc. from the cultural life of Germany.


with Ash wood, 6 silver anodized aluminum tubes.
With its mid-range tuning, this chime plays the inspiring tones heard in the opening measures of Amazing Grace, one of America's best loved and most widely known hymns. Made of: Cherry finish ash wood, 6 silver anodized aluminum tubes.


text:
\subsection{Some implementation details}\label{sec:impl}

The tests were run on a computer with an Intel Core i7-920 CPU running at 2.67 GHz,
with 18 GB of RAM clocked
at 1066 MHz. We used the single-core version of MUMPS 4.10, and
the Intel MKL BLAS shipped with MATLAB R2017b.

In all experiments the tolerance $\tau$ for the accuracy of the
Lanczos method was set to $10^{-5}$ while the trust-region procedure was
 stopped when
the norm of the projected gradient norm fell below  $10^{-4}$
(i.e. $\epsilon = 10^{-4}$ in Algorithm \ref{alg:tr}). All the other parameters
in Algorithm \ref{alg:tr} were set to standard values as suggested in
\cite[Chapter 17]{trBible}. The trust-region radius update follows
\cite[Chapter 17, page 782]{trBible} as well. Also, in Algorithm \ref{alg:tr}
the trust-region step $\bf s^{(k)}$ is computed ``exactly'', that is, we
 minimize the reduced model (\ref{eq:modtr}) in ${\cal B}_k \cap \Omega$
 to high accuracy. This is a reasonable choice due to problem low dimension of the
 problem.
To this end, we used
the function $\texttt{fmincon}$ included in the \texttt{MATLAB}'s Optimization
toolbox, setting the built-in \texttt{sqp} solver and using the
default parameter settings.

Furthermore, the parameter space is always preliminary
scaled, so that the initial point $\mathbf x^{(0)}$ is the vector of all ones. This ensures
that checking norm-wise conditions on the eigenvalues and on the gradient
yields relative accuracy on the parameters, independently of their scaling.

The weight vector $\mathbf w$ is always chosen as $w_i = f_i^{-1}$, which ensures
relative accuracy on the recovered frequency, except where otherwise stated.
In particular, in the clock tower example, we emphasize the consequences
of a different choice of $\mathbf w$.

\subsection{Arch on piers}

In this section we consider a simple example for demonstration purposes. We built
a 2D discretization of the arch on piers, shown in Figure~\ref{fig:portal}.
The arch spans $4$ meters, and rests on two $4$-meters-high lateral piers. The structure is modeled by means of $336$ finite elements, and clamped
at the base of the two piers. This corresponds to $851$ total degrees of freedom
in the structure. We have used plane strain $4$-node quadrilateral elements.

The
three materials composing the structure are depicted with different colors
in Figure~\ref{fig:portal}.
Material $1$ is used for the arch (green in the figure),
and materials $2$ and $3$ for the piers (red and blue, respectively).
Poisson's ratio $\nu$ is set to $0.2$ for all the materials, and assumed
to be known a priori.
\begin{figure}
    \centering
    \includegraphics[height=.35\textheight]{portale}
    \caption{The arch on piers modeled in
        the experiments. Each differently colored region
        corresponds to a different material. }
    \label{fig:portal}
\end{figure}

\begin{figure} \centering
    \begin{tikzpicture}
    \begin{semilogyaxis}[width=.45\linewidth, height = .35\textheight, ymax = 5,
    ylabel = Obj. function / Proj. gradient, xlabel = Iterations, legend pos = south west,
    ymin = 1e-13]
    \addplot table {dat/convergence_std.dat};
    \addplot table[x index = 0, y index = 7] {dat/convergence_std.dat};
    \addplot[dashed, domain = 0:4] {1e-4};
    \legend{Obj. function, Proj. gradient, Tolerance ($10^{-4}$)}
    \end{semilogyaxis}
    \end{tikzpicture}~~~~~~~~
    \begin{tikzpicture}
    \begin{axis}[width=.45\linewidth, height = .35\textheight,
    ylabel = Frequencies (Hz), xlabel = Iterations, legend pos = north west]
    \addplot table[x index = 0, y index = 2] {dat/convergence_std.dat};
    \addplot[domain = 0 : 4, dashed]{ 9.575048987168955e+00 };
    \addplot table[x index = 0, y index = 3] {dat/convergence_std.dat};
    \addplot[domain = 0 : 4, dashed]{ 1.487136204507410e+01 };
    \addplot table[x index = 0, y index = 4] {dat/convergence_std.dat};
    \addplot[domain = 0 : 4, dashed]{ 2.317329261256074e+01 };
    \addplot table[x index = 0, y index = 5] {dat/convergence_std.dat};
    \addplot[domain = 0 : 4, dashed]{ 3.917029465848877e+01 };
    \addplot table[x index = 0, y index = 6] {dat/convergence_std.dat};
    \addplot[domain = 0 : 4, dashed]{ 6.283760720867433e+01 };
    \end{axis}
    \end{tikzpicture}
    \caption{The left graph shows the convergence of the objective function
        to the minimum for each new reduced model generated in the
        optimization procedure. The dashed line is the tolerance
        set for the projected gradient in the optimization scheme.
        The right graph plots the convergence of the
        frequencies during the process. }
    \label{fig:portal-convergence-near}
\end{figure}

\begin{figure} \centering
    \begin{tikzpicture}
        \begin{semilogyaxis}[width=.45\linewidth, height = .3\textheight,
        ylabel = Obj. function / Proj. gradient, xlabel = Iterations,
        legend pos = south west, ymin = 1e-9]
             \addplot table {dat/convergence_far.dat};
             \addplot table[x index = 0, y index = 7] {dat/convergence_far.dat};
             \addplot[dashed, domain = 0 : 11] {1e-4};
             \legend{Obj. function, Proj. gradient, Tolerance ($10^{-4}$)}
        \end{semilogyaxis}
    \end{tikzpicture}~~~~~~~~
    \begin{tikzpicture}
        \begin{axis}[width=.45\linewidth, height = .3\textheight,
          ylabel = Frequencies (Hz), xlabel = Iterations]
            \addplot table[x index = 0, y index = 2] {dat/convergence_far.dat};
            \addplot[domain = 0 : 11, dashed]{ 9.575048987168955e+00 };
            \addplot table[x index = 0, y index = 3] {dat/convergence_far.dat};
            \addplot[domain = 0 : 11, dashed]{ 1.487136204507410e+01 };
            \addplot table[x index = 0, y index = 4] {dat/convergence_far.dat};
            \addplot[domain = 0 : 11, dashed]{ 2.317329261256074e+01 };
            \addplot table[x index = 0, y index = 5] {dat/convergence_far.dat};
            \addplot[domain = 0 : 11, dashed]{ 3.917029465848877e+01 };
            \addplot table[x index = 0, y index = 6] {dat/convergence_far.dat};
            \addplot[domain = 0 : 11, dashed]{ 6.283760720867433e+01 };
        \end{axis}
    \end{tikzpicture}
    \caption{The left graph shows the convergence of the objective function
        to the minimum for each new reduced model generated in the
        optimization procedure. The dashed line is the tolerance
        set for the projected gradient in the optimization scheme.
        The right graph plots the convergence of the
        frequencies during the process. In this case, the initial points
        were chosen quite far from the original ones, selecting
        $E_2 = 2000\ \mathrm{MPa}, \rho_2 = 1100\ \mathrm{kg\cdot m^{-3}},
          E_3 = 1100\ \mathrm{MPa}$.}
        \label{fig:portal-convergence-far}
\end{figure}

For the three materials we assume
the following values:
\begin{align*}
  E_1 &= 3250\ \mathrm{MPa} & E_2 &= 5000\ \mathrm{MPa} & E_3 &= 4800\ \textrm{MPa} \\
  \rho_1 &= 1800\ \mathrm{\frac{kg}{m^{3}}} & \rho_2 &= 2200\ \mathrm{\frac{kg}{m^{3}}} &
  \rho_3 &= 2100\ \mathrm{\frac{kg}{m^{3}}},
\end{align*}
where the index $j \in \{ 1, 2, 3 \}$ indicates the material under consideration.
We computed the $5$ leading frequencies by running the NOSA-ITACA code  (with
these fixed parameters), to obtain:
\[
  \mathbf f \approx \begin{bmatrix}
    9.575 & 14.87 & 23.17 & 39.17 & 62.84
  \end{bmatrix}.
\]
Our aim is to validate the optimization method by recovering
the original parameters matching
these frequencies, assuming $E_2, \rho_2$, and $E_3$ are unknown.
We consider the following bounds of
realistic parameters:
\begin{align*}
  1000\ \mathrm{MPa} \leq &E_2 \leq 9000\ \mathrm{MPa}, &
  1000\ \mathrm{\frac{kg}{m^{3}}} \leq &\rho_2 \leq 3000\ \mathrm{\frac{kg}{m^{3}}}, &
  1000\ \mathrm{MPa} &\leq E_3 \leq 9000\ \mathrm{MPa},
\end{align*}
which define the corresponding box $\Omega$ in (\ref{eq:Omega})

In the default implementation, the starting points have been chosen the
midpoint of the intervals, which are quite good estimates of the true
values. Convergence in this case is displayed in Figure~\ref{fig:portal-convergence-near}. The initial
points are sufficiently close to the correct ones for the frequencies
to almost match already, and the distance to the optimum during
convergence is not easily distinguishable.  However, the left plot in Figure~\ref{fig:portal-convergence-near}, which shows the convergence
of the objective function in log-scale, clearly demonstrates that we are recovering
the right solution.

To test the robustness of the approach, we modified the starting
points to be relatively far from the correct ones by making the following
choices:
\begin{align*}
     E_2^{(0)} &= 2000\ \mathrm{MPa}, & \rho_2^{(0)} &= 1100\ \mathrm{\frac{kg}{m^{3}}}, &
     E_3^{(0)} &= 1100\ \mathrm{MPa}.
\end{align*}
Convergence of the method is displayed in the graphs in Figure~\ref{fig:portal-convergence-far}, which clearly shows that the
initial frequencies are quite far from the correct ones; yet the method
reaches convergence using only $12$ reduced models. Moreover, a rough estimate (though probably
sufficient for most engineering purposes) is already obtained
after about 6 steps. Moreover, the Young's modulus and the mass density of the
initial model are recovered exactly up to $5$ digits in this example.

Another important feature for a method that must deal with experimentally
measured frequencies is its robustness when the input is subject to noise.
In fact, we would expect the frequencies to be accurate only up to a
certain relative threshold.

To simulate the behavior of the method in a predictable environment, we
perturbed the frequency vector $\hat {\mathbf f} = \mathbf f + \delta \mathbf f$, by imposing
 $|\delta f_i| \leq |f_i| \cdot \delta$, with $\delta$ the prescribed
noise level. We have run tests for $\delta$ ranging from $0.01 \%$ to
$100 \%$: the corresponding error in the retrieved frequencies is
plotted in Figure~\ref{fig:noise}.

The error is measured in relative norm, that is we have plotted the infinity
norm of the vector with components $(x_i - \hat x_i) / x_i$, where $x_i$
is the vector with the actual model parameters, and $\hat x_i$
those estimated by the optimization process.

\begin{figure}
    \centering

    \begin{tikzpicture}
    \begin{loglogaxis}[width = \linewidth, legend pos = north west,
    height = .35\textheight, xlabel = $\norm{\delta \mathbf f}_\infty$,
    ylabel = Relative error]
    \addplot[mark=*, blue, only marks] table {dat/arco_noise.dat};
    \addplot[domain = 1e-4 : 1e0, dashed, black] { 1e-1 * x };
    \addplot[domain = 1e-4 : 1e0, dotted, red] { 10 * x };
    \legend{Relative error, $10^{-1} \cdot \norm{\delta \mathbf f}_\infty$, $10 \cdot \norm{\delta \mathbf f}_\infty$};
    \end{loglogaxis}
    \end{tikzpicture}

\caption{This plot shows the
     relation between the noise affecting the frequency measurements
    and the component-wise relative error on the retrieved optimal
    parameters. The graph shows the linear dependency of the error
    on the estimated parameters and the noise level of the
    input data.}
\label{fig:noise}
\end{figure}

The linear behavior of the relative error with respect to the noise level is
clearly visible. This behavior suggests that,
even if the frequency measurements are contaminated with noise, it
is still possible to retrieve meaningful parameters if enough information
is provided. Note that, in general, correct (or accurate) recovery
of the parameters might not be possible, and a more detailed study of
the condition number of this inverse problem will be object of further
in-depth study in
in future research.

\subsection{A dome}

Here we provide a more complex example, represented by a dome supported by four
14-meters-high pillars. The dome consists of an octagonal shaped cloister
vault 5 meters high, resting on a drum inscribed on a $10 \times 11$ meters rectangle. We assume the structure to be composed of 4 different materials:
material 1 for the vault (red in Figure~\ref{fig:dome-model}), material 2 for the top of the drum
(blue), material 3 for the lower part of the
drum (green) and material 4 for the pillars (gray).

The finite element model consists of $\num{31052}$ elements and $\num{41245}$ nodes.
It is modeled
using 3D $8$-node hexahedron brick elements, which gives rise to a (projected) model with
$\num{122,853}$ degrees of freedom.

\begin{figure}
    \centering
    \includegraphics[height=.45\textheight]{tempietto}
    \caption{Domed temple}
    \label{fig:dome-model}
\end{figure}

We performed a test similar to that carried out for the arch on piers:
we computed the frequencies for
a certain choice of parameters, some of which were assumed
to be unknown, and ran the
optimization scheme trying to recover the original parameters.

In order to obtain the reference frequencies the characteristics
of the 4 materials (Young's modulus and mass density)
were set as follows:
\begin{align*}
    E_1 &= 3000\ \mathrm{MPa} & \rho_1 &= 1800\ \mathrm{\frac{kg}{m^3}} &
    E_2 &= 4000\ \mathrm{MPa} & \rho_2 &= 2000\ \mathrm{\frac{kg}{m^3}} \\
    E_3 &= 3500\ \mathrm{MPa} & \rho_3 &= 1900\ \mathrm{\frac{kg}{m^3}} &
    E_4 &= 5000\ \mathrm{MPa} & \rho_2 &= 2200\ \mathrm{\frac{kg}{m^3}} \\
    \nu_j &= 0.25, \qquad  j = 1, \ldots, 4.
\end{align*}
The leading $10$ frequencies have been computed using the NOSA-ITACA code. The
result, expressed in Hertz, are the frequencies in the vector:
\[
  \mathbf f \approx \begin{bmatrix}
    2.19 & 2.23 & 3.76 & 3.83 & 4.32 & 4.60 &
    4.72 & 8.26 & 8.30 & 9.21
  \end{bmatrix}.
\]
The optimization code was run setting the bounds as
\[
  2000\ \mathrm{MPa} \leq E_j \leq 6000\ \mathrm{MPa} \qquad
  1600\ \mathrm{\frac{kg}{m^3}} \leq \rho_j \leq 2400\ \mathrm{\frac{kg}{m^3}} \qquad
  j = 1,\ldots, 4,
\]
with the sole exception of $\rho_3$ which was set to the correct value.
This leaves $7$ parameters to be optimized. No
particular starting conditions were specified. In this case, the algorithm
chooses the midpoint of the intervals. The convergence history is reported in Figure~\ref{fig:dome}, which reveals that high accuracy is attained with only
$4$ outer iterations --- with the residual of the objective function
being very small even after two steps.
The weights have been chosen in order to reach relative accuracy\footnote{A detailed discussion of the influence of the weights on the accuracy of the recovered frequencies is presented
    in the next section, where its relevance is discussed for a real example.} (i.e., $w_i = f_i^{-1}$)
on the target frequencies, with the usual tolerance $\epsilon = 10^{-4}$.
\begin{figure}
\centering
\begin{tikzpicture}
\begin{semilogyaxis}[
xtick={0,1,2,3,4},
width=.45\linewidth, height = .35\textheight,
ylabel = Obj. function / Proj. gradient, xlabel = Iterations,
legend pos = north east, ymax = 1e3]
\addplot table[x index = 0, y index = 1] {dat/convergence_massa_relative.dat};
\addplot table[x index = 0, y index = 12] {dat/convergence_massa_relative.dat};
\addplot[dashed, domain = 0:4] {1e-4};
\legend{Obj. function, Proj. gradient, Tolerance ($10^{-4}$)}
\end{semilogyaxis}
\end{tikzpicture}~~~~~~~~
\begin{tikzpicture}
\begin{axis}[width=.45\linewidth, height = .35\textheight,
ylabel = Frequencies (Hz), xlabel = Iterations]
\addplot table[x index = 0, y index = 2] {dat/convergence_massa_relative.dat};
\addplot[domain = 0 : 4, dashed]{ 2.19 };
\addplot table[x index = 0, y index = 3] {dat/convergence_massa_relative.dat};
\addplot[domain = 0 : 4, dashed]{ 2.23 };
\addplot table[x index = 0, y index = 4] {dat/convergence_massa_relative.dat};
\addplot[domain = 0 : 4, dashed]{ 3.76 };
\addplot table[x index = 0, y index = 5] {dat/convergence_massa_relative.dat};
\addplot[domain = 0 : 4, dashed]{ 3.83 };
\addplot table[x index = 0, y index = 6] {dat/convergence_massa_relative.dat};
\addplot[domain = 0 : 4, dashed]{ 4.32 };
\addplot table[x index = 0, y index = 7] {dat/convergence_massa_relative.dat};
\addplot[domain = 0 : 4, dashed]{ 4.6 };
\addplot table[x index = 0, y index = 8] {dat/convergence_massa_relative.dat};
\addplot[domain = 0 : 3, dashed]{ 4.72 };
\addplot table[x index = 0, y index = 9] {dat/convergence_massa_relative.dat};
\addplot[domain = 0 : 3, dashed]{ 8.26 };
\addplot table[x index = 0, y index = 10] {dat/convergence_massa_relative.dat};
\addplot[domain = 0 : 3, dashed]{ 8.30 };
\addplot table[x index = 0, y index = 11] {dat/convergence_massa_relative.dat};
\addplot[domain = 0 : 3, dashed]{ 9.21 };
\end{axis}
\end{tikzpicture}

    \caption{Convergence history for the dome model, matching
    	$10$ frequencies with $7$ free parameters. The left plot shows the convergence of the objective function
        and the norm of the projected gradient, and in the right plot the convergence
        of frequencies to the exact ones is reported. }
    \label{fig:dome}
\end{figure}

The estimated mechanical parameters obtained through
the optimization are as follows:
\begin{align*}
E_1 &= 2953\ \mathrm{MPa} & \rho_1 &= 1769\ \mathrm{\frac{kg}{m^3}} &
E_2 &= 3932\ \mathrm{MPa} & \rho_2 &= 1960\ \mathrm{\frac{kg}{m^3}} \\
E_3 &= 3426\ \mathrm{MPa} & \rho_3 &= 1900\ \mathrm{\frac{kg}{m^3}} &
E_4 &= 4951\ \mathrm{MPa} & \rho_2 &= 2174\ \mathrm{\frac{kg}{m^3}}.
\end{align*}
This corresponds to a relative error between $0.97\%$ and $2.1\%$, with
an average of about $1.6\%$. Although such accuracy is a quite good accuracy for practical purposes,  even more
precise estimates can be obtained by reducing the tolerance to below
$10^{-4}$ that we have used in these tests.

\subsection{The Clock tower}

As a real example, we consider the case study of the Clock tower
(``Torre delle Ore'') in Lucca, Italy. The 48.4 m high masonry
structure, dating back to the 13th century, has a rectangular cross
section of about 5.1$\times$7.1 meters and walls of thickness varying
from 1.77 m at the base to 0.85 m at the top. Two barrel vaults are
set at heights of  about 12.5 m and 42.3 m. The bell
chamber is covered by a pavillon roof made up of wooden trusses and
rafters. At a height of about 33 m the walls have been instrumented with 4
steel tie rods of 30$\times$30 mm rectangular section. The adjacent
buildings abut the tower on two sides for a height of about 13 m and
constitute asymmetric boundary conditions. The modal behavior of the
tower has already been analyzed using the NOSA-ITACA code and
through experimental measurements conducted in November 2016, when
the tower was fitted with 4 triaxial seismometric stations. The two
procedures yielded similar, albeit not perfectly matching results
\cite{pellegrini2017anew}. Figure~\ref{fig:tore} shows a picture of the tower,
together with the mesh
used for its discretization.

\begin{figure}
	\centering
	\begin{minipage}{.45\linewidth}
    	\centering
       \includegraphics[width = .35\textheight, angle = 90]{tore_22}
	\end{minipage}~~\begin{minipage}{.5\linewidth}
	   \centering
       \includegraphics[height = .35\textheight]{tore_mats}
     \end{minipage}
     \caption{On the left, a picture of the clock tower in Lucca. On the right,
     	the mesh used for the finite element discretization, where the
     	different materials for the bell chamber and the lower part of the
     	tower are reported in different colors (respectively red and light blue).}
     \label{fig:tore}
\end{figure}

The frequencies obtained by analyzing data from the fitted instruments
via Operational Modal Analysis techniques, are as follows:
\[
  \mathbf f \approx \begin{bmatrix}
    1.05 & 1.3 & 4.19 & 4.50
  \end{bmatrix}.
\]
The frequencies obtained with the
finite element model in \cite{pellegrini2017anew}
are, instead, $[ 0.98 \ 1.24 \ 4.32 \ 4.38 ]$.
The materials constituting
the tower are wood, steel, and masonry, with the latter being different
in the bell chamber and the lower part of the tower.
The mechanical properties used in the finite
element model are
\begin{align*}
   E_{w} &= 10000\ \mathrm{MPa} & \rho_{w} &= 800\ \mathrm{\frac{kg}{m^{3}}} & \nu_{w} &= 0.35 \\
   \rho_{m_l} &= 2100\ \mathrm{\frac{kg}{m^3}} & \rho_{m_c} &= 1700\ \mathrm{\frac{kg}{m^3}} & \nu_{m_l} &= \nu_{m_c} = 0.2 \\
   E_{s} &= \num{210000}\ \mathrm{MPa} & \rho_s &= 7850\ \mathrm{\frac{kg}{m^3}} & \nu_s &= 0.3,
\end{align*}
and we leave the Young's moduli $E_{m_c}$ and $E_{m_l}$ as free
parameters, since they are not known explicitly
due to the lack of experimental information.
Here, the subscript $m_l$ identifies the lower part
of the tower, and $m_c$ the bell chamber.

We can reformulate the problem as an optimization one, trying
to match the frequencies of the system to the measured ones,
allowing the parameters to vary in a reasonable range. In our case we
can set
\[
  2500\ \mathrm{MPa} \leq E_{m_l} \leq 5500\ \mathrm{MPa} \qquad
  1000\ \mathrm{MPa} \leq E_{m_c} \leq 5500\ \mathrm{MPa}.
\]
As a first test, we choose the weights as the vector of all ones.
The convergence of the optimization
scheme, obtained with these initial parameters, is reported in Figure~\ref{fig:convergence-tore}. Note that
the method optimizes the third and fourth frequencies to match quite closely,
but at the price of decreasing the quality of the match for the first two (see Table~\ref{tab:tore-relative-vs-absolute}).

\begin{figure} \centering
    \begin{tikzpicture}
    \begin{semilogyaxis}[
    xtick={0,1,2,3,4,5},
    width=.45\linewidth, height = .3\textheight,
    ylabel = Obj. function / Proj. gradient, xlabel = Iterations,
    legend pos = north east, ymax = 1e4]
    \addplot table[x index = 0, y index = 1] {dat/convergence_tore.dat};
    \addplot table[x index = 0, y index = 6] {dat/convergence_tore.dat};
    \addplot[dashed, domain = 0:5] {1e-4};
    \legend{Obj. function, Proj. gradient, Tolerance ($10^{-4}$)}
    \end{semilogyaxis}
    \end{tikzpicture}~~~~~~~~
    \begin{tikzpicture}
    \begin{axis}[width=.45\linewidth, height = .3\textheight,
    ylabel = Frequencies (Hz), xlabel = Iterations, xtick = {0,1,2,3,4,5}]
    \addplot table[x index = 0, y index = 2] {dat/convergence_tore.dat};
    \addplot[domain = 0 : 5, dashed]{ 1.05 };
    \addplot table[x index = 0, y index = 3] {dat/convergence_tore.dat};
    \addplot[domain = 0 : 5, dashed]{ 1.30 };
    \addplot table[x index = 0, y index = 4] {dat/convergence_tore.dat};
    \addplot[domain = 0 : 5, dashed]{ 4.19 };
    \addplot table[x index = 0, y index = 5] {dat/convergence_tore.dat};
    \addplot[domain = 0 : 5, dashed]{ 4.50 };
    \end{axis}
    \end{tikzpicture}
    \caption{The left graph reports the convergence history
        of the objective function
        to the minimum for each new reduced model generated in the
        optimization procedure. The norm of the projected gradient
        is reported as well, and the dashed line is the tolerance of the
        optimization scheme. The right plot represents the convergence of the
        frequencies during the process (the dashed lines are the experimental
        frequencies). Iteration $0$ is the evaluation
        function at the starting point. In this example, the
        free parameters are $E_{m_l}$ and $E_{m_c}$.}
    \label{fig:convergence-tore}
\end{figure}

\begin{figure} \centering
	\begin{tikzpicture}
	\begin{semilogyaxis}[width=.45\linewidth, height = .3\textheight,
	ylabel = Obj. function / Proj. gradient, xlabel = Iterations, legend pos = north east,
	ymax = 300]
	\addplot table {dat/convergence_tore_relative.dat};
	\addplot table[x index = 0, y index = 6] {dat/convergence_tore_relative.dat};
	\addplot[dashed, domain = 0:4] {1e-4};
	\legend{Obj. function, Proj. gradient, Tolerance ($10^{-4}$)}
	\end{semilogyaxis}
	\end{tikzpicture}~~~~~~~~
	\begin{tikzpicture}
	\begin{axis}[width=.45\linewidth, height = .3\textheight,
	ylabel = Frequencies (Hz), xlabel = Iterations]
	\addplot table[x index = 0, y index = 2] {dat/convergence_tore_relative.dat};
	\addplot[domain = 0 : 4, dashed]{ 1.05 };
	\addplot table[x index = 0, y index = 3] {dat/convergence_tore_relative.dat};
	\addplot[domain = 0 : 4, dashed]{ 1.30 };
	\addplot table[x index = 0, y index = 4] {dat/convergence_tore_relative.dat};
	\addplot[domain = 0 : 4, dashed]{ 4.19 };
	\addplot table[x index = 0, y index = 5] {dat/convergence_tore_relative.dat};
	\addplot[domain = 0 : 4, dashed]{ 4.50 };
	\end{axis}
	\end{tikzpicture}
	\caption{The left graph reports the convergence history
		of the objective function
		to the minimum for each new reduced model generated in the
		optimization procedure. The norm of the projected gradient
		is reported as well, and the dashed line is the tolerance
		set for the projected gradient in the optimization scheme.
		The right plot represents the convergence of the
		frequencies during the process (the dashed line are the experimental
		frequencies). Iteration $0$ is the evaluation
		function at the starting point. In this example, the
		free parameters are  $E_{m_l}$ and $E_{m_c}$, and the objective
		function is defined using the weights $w_i = f_i^{-1}$. }
	\label{fig:convergence-tore-relative}
\end{figure}

This is connected to the choice of $\mathbf w$, which ensures
norm-wise accuracy, but not a relative one. Often, we prefer to have a low
\emph{relative} error on the frequencies, instead of an absolute one.
Therefore, it is natural to choose $w_i = f_i^{-1}$ (in fact, this
is the default choice in the implementation). This yields
different results, and different approximations for both the parameters
and the frequencies. More specifically, we expect a better match
on the lowest end of the spectrum, and a worse one on the highest
frequencies.

The convergence history with this choice of $\mathbf w$ is reported in Figure~\ref{fig:convergence-tore-relative}, and the results
are shown in Table~\ref{tab:tore-relative-vs-absolute}. It appears
immediately evident that the relative errors are balanced with $w_i = f_i^{-1}$,
whereas they tend to increase on the small frequencies when $w_i = 1$.

The parameters identified by the model updating phase give
$E_{m_l} \approx \num{3182}\ \mathrm{MPa}$ and $E_{m_c} \approx \num{1873}\ \mathrm{MPa}$ when using
weights equal to one, and $E_{m_l} \approx \num{3076}\ \mathrm{MPa}$ and $E_{m_c} \approx \num{1950}\ \mathrm{MPa}$ when aiming for relative accuracy.

We conclude this section by comparing the overall CPU time of
our implementation with those of alternative
solvers, which in various ways fail to exploit the problem structure.
Table~\ref{tab:timings} compares 4 different implementations:
\begin{description}
   \item[RM] This implementation corresponds to the current proposal, that is,
     a solver based on the trust-region and
     projection approach described in the previous section and used in the
     tests presented herein. This strategy is labeled as ``RM'' in the table --- highlighted with the bold font.
   \item[BB] Here we interpret the finite element code as a black-box function and solve
   the optimization problem using a general purpose optimizer.
     This is done to reflect a scenario in which the user has no access to the
     internals of the finite element code --- and requires an assembly phase at
     every function evaluation. Derivatives are computed here
     by finite differences. This is column ``BB'' in the table.
   \item[A] In this approach, we separately assemble the parametric generalized eigenvalue
     problem $K(\mathbf x) - \lambda M(\mathbf x)$ and then pass to a general
     optimizer a function that evaluates the frequencies for a given value
     of $\mathbf x$. This function corresponds to the Lanczos projection
     routine used internally in RM.  Derivatives are computed here
     by finite differences. This is approach ``A'' in the table.
   \item[AD] The same as A, but derivatives are evaluated analytically
     as described to Section~\ref{sec:derivatives}. This is ``AD'' in the table.

\end{description}
The general optimizer chosen for the BB, A and AD implementations is again
{\tt fmincon} with the \texttt{sqp} option available in the Optimization toolbox of MATLAB.
In our experience, this choice turns out to be the most efficient
among those available in the toolbox. This same function
has been used for
computation of the step in the trust-region scheme of Algorithm \ref{alg:tr}.

In Table~\ref{tab:timings}, the time for assembly (which needs to be added to the costs in columns
A, AD, and RM) is reported separately, as well as the problem's degrees of freedom.

It is immediately apparent that the proposed RM method is the fastest in all cases
except the arch on piers, where given the small dimension incurs in some overhead.
Moreover, it is worth stressing that our approach exhibits a complexity that
seems to grow nicely with the dimension. The general optimization approach BB, instead,
becomes costly when the number of degrees of freedom increases.

For instance, in the dome example, the timings range from about 7 hours for BB
to slightly more than 3 minutes with our approach. The
 case where we supply derivatives to the optimizer (AD)
still takes more than 30 minutes.

\begin{table}
    \begin{tabular}{c|cc|cc}
      Exp. freq. & Freq. ($w_i = 1$) & Rel. error ($w_i = 1)$ & Freq. ($w_i = f_i^{-1}$) & Rel. error ($w_i = f_i^{-1}$) \\ \hline
      $1.05$ Hz & $1.0621$ Hz & $1.15$\% & $1.0449$ Hz & $0.49$\% \\
      $1.30$ Hz & $1.3366$ Hz & $2.82$\% & $1.315$ Hz & $1.15$\% \\
      $4.19$ Hz & $4.2041$ Hz & $0.33$\% & $4.2154$ Hz & $0.61$\% \\
      $4.50$ Hz & $4.4729$ Hz & $0.60$\% & $4.4409$ Hz  & $1.31$\% \\
    \end{tabular}
    \caption{Frequencies and corresponding relative errors computed
        using the weights $w_i = 1$ and $w_i = f_i^{-1}$ for
        the problem of the clock tower in Lucca, Italy.}
    \label{tab:tore-relative-vs-absolute}
    %
    %
    %
\end{table}

\begin{table}
    \centering
    \begin{tabular}{c|cccccc}
               & \# dof& BB & A & AD & \textbf{RM} & Assembly \\ \hline
        Arch on piers   & 851 & 16.40s    & 1.17s     & 0.62s  & \textbf{0.68s}   & 0.42s \\
        Dome        & 122,853 & 25,032s & 7,662.4s  & 1,990.5s & \textbf{226.02s} & 134.32s \\
        Clock tower & 45,511 & 476.9s   & 305.39s   & 202.87s & \textbf{44.10s}  & 32.22s \\
    \end{tabular}
    \caption{Timings for different optimization approaches. The timings
        reported for the approaches ``Assembled'' (A), ``Assembled $+$ derivatives'' (AD), and
        ``Reduced model'' (RM) do not include the time needed to assembly the
        parametric finite element model, which is reported in the
        last column. The Portal model has been run with the default
        starting points. }
    \label{tab:timings}
\end{table}

The timings suggest the unsurprising conclusion that better knowledge of the underlying
optimization problem leads to more efficient and faster optimization routines.
It should be noted that
the stopping criterion for all the approaches is set to ensure that the first order
optimality condition (given by the norm of the projected gradient) is smaller than
$\epsilon = 10^{-4}$ used in the experiments. All the approaches provide solutions
with comparable accuracy.

\subsection{Formulation of the optimization problem} \label{sec:formulation-optimization}

The model updating problem can be reformulated as an optimization problem
by assuming that the (projected) stiffness and mass matrices $K$ and $M$
are functions of the parameter vectors $\mathbf x$. We use the
notation
\[
K := K(\mathbf x), \qquad
M := M(\mathbf x), \qquad
\mathbf x \in \mathbb R^\ell,
\]
to denote this dependency. The set of valid choices for the parameters is
denoted by $\Omega$, which we assume to be an $\ell$-dimensional
box, that is
\begin{equation} \label{eq:Omega}
\Omega = [ a_1 , b_1 ] \times \ldots \times [ a_\ell, b_\ell ],
\end{equation}
for certain values $a_i < b_i$, $i = 1, \ldots, \ell$. We also assume
that initial estimates for the parameter values are available, and
we denote these by $\mathbf x^{(0)}$. When this is not the case, we choose
$\mathbf x^{(0)}$ as the center of the box $\Omega$ (i.e., the vector with the midpoints
of the intervals $[a_i, b_i]$ as coordinates).

Our ultimate aim is to determine the optimal value of
$\mathbf x$ that minimizes a certain cost functional $\phi(\mathbf x)$
within the box $\Omega$.
This is an instance of the optimization problem
\begin{equation}
\label{eq:optpb}
\min_{\mathbf x \in \Omega} \phi(\mathbf x).
\end{equation}

The choice of the objective function $\phi(\mathbf x)$ is related
to the frequencies that we want to match. If we need to match
$s$ frequencies
of the model, we choose a suitable weight vector
$\mathbf w := [ w_1 , \ldots, w_s ]$, with $w_i \geq 0$,
and define the functional
$\phi(\mathbf x)$ as follows:
\begin{equation}
\label{eq:obj}
\phi(\mathbf x) :=
\norm*{
	\frac{\sqrt{\Lambda_s(K, M)}}{2\pi} - \mathbf f }_{\mathbf w,2}^2, \qquad
\norm{\mathbf y}_{\mathbf w, 2} := \sqrt{\mathbf y^T D_{\mathbf w}^2 \mathbf y},
\end{equation}
where $\mathbf f$ is the vector of the measured frequencies,
$D_{\mathbf w} = \mathrm{diag}(w_1, \ldots, w_s)$
and $\Lambda_s(K, M)$ is the vector containing the smallest $s$
eigenvalues of the pencil $K - \lambda M$,
ordered according to their magnitude.

The vector $\mathbf w$ encodes the weights that should be given
to each frequency in the optimization scheme. If the aim is to
minimize the distance between the vector of measured and computed
frequencies in the usual Euclidean norm, then $\mathbf w = \mathbbm{1}_s$,
the vector of all ones, should be chosen.
If, instead, a relative accuracy on the frequencies is desired,
$w_i = f_i^{-1}$ is the natural choice. To avoid scaling issues in the
objective function, we always normalize $\mathbf w$ to have
Euclidean norm $1$.

\section{Introduction}
\label{sec:introduction}
\input{introduction}

\section{The model: modal analysis of masonry buildings and frequency matching}
  \label{sec:model}
\input{model}

\section{The optimization framework}
\label{sec:trustregion}
\input{trustregion}

\section{Building the reduced model by Lanczos projection}
\label{sec:reducedmodel}
\input{reducedmodel}

\section{Numerical experiments and real examples}
\label{sec:examples}
\input{examples}

\section{Concluding remarks}
\label{sec:concluding-remarks}
\input{conclusions}

\section*{References}

\bibliographystyle{elsarticle-harv}

\subsection{The Lanczos method}
\label{sec:lanczos}

The Lanczos iteration, for a symmetric
$n \times n$
matrix $A$, can be summarized as follows.
We construct an orthogonal basis $W_m$
for the subspace $\mathcal K_{m}(A, v) \subseteq \mathbb R^{n}$
defined as:
\[
  \mathcal K_{m}(A, v) := \mathrm{span}(v, Av, \ldots, A^{m-1} v ).
\]
Unless a breakdown occurs, this space has dimension $m$,
and the basis $W_m$ is represented as a $n \times m$ orthogonal matrix
(that is, $W_m^T W_m = I_m$). Given the basis, one then computes the projection
$T_m = W_m^T A W_m$. The matrix $T_m$ is
tridiagonal, and its eigenvalues (known as Ritz values) are
good approximations of the extremal eigenvalues of $A$ even when
$m \ll n$. This procedure is appealing for several reasons.
\begin{enumerate}[(i)]
\item Computation of the space $W_m$ only requires matrix-vector
  multiplications, and therefore it is easy to exploit the sparsity or,
  more generally, the structure of $A$.
\item The orthogonalization of the basis requires only scalar products,
  which can be computed very efficiently if $m$ does not increase
  too much.
\item The algorithm can be carried out iteratively, i.e.,
  the optimal value of $m$ does not need to be known a priori,
  and $T_m$ can be built incrementally. In our framework,
  we stop the iteration when the error on the computed
  eigenvalues is guaranteed to be smaller than a certain
  relative threshold $\tau$ (see \cite{demmel1997applied} for a classical
  reference on the stopping criterion of Lanczos for computing eigenvalues).
  The actual value of $\tau$ is reported in Section~\ref{sec:examples}.
\end{enumerate}

When the matrix $A$ arises from a FE
discretization of a PDE involving only local operators
and elements with compact support and few
overlappings, the number of nonzero elements is only $\mathcal O(n)$;
hence, the matrix vector
multiplication can be carried out with linear complexity. Moreover,
the matrix usual has a band or multiband structure, and using good
sparse solvers or appropriate preconditioners we can solve linear
systems in $\mathcal O(n)$ time.
Therefore, the smallest eigenvalue of $A$ can be extracted applying
Lanczos to $A^{-1}$.
Since we are interested in modal analysis of structures discretized by FE,
we need
to approximate the smallest eigenvalues of a pencil
$K - \lambda M$, with $K$ and $M$ symmetric
positive definite;
the same technique to extract the smallest eigenvalues can be applied
(implicitly) to the symmetric matrix $M^{\frac 1 2} K^{-1} M^{\frac 1 2}$,
obtaining:
\[
  T_m = W_m^T M^{\frac 1 2} K^{-1} M^{\frac 1 2} W_m = U_m^T M K^{-1} M U_m, \qquad
  U_m = M^{-\frac 1 2} W_m.
\]
It is immediately verifiable that $U_m$ contains a basis
of the same subspace spanned by $W_m$, but this matrix is
$M$-orthogonal (that is, orthogonal with respect to the inner
product defined by $M$), so $U_m^T M U_m = I_m$ \cite{ericsson1980spectral}.

\begin{remark}
  In the above framework, there is no need to explicitly compute the square
  root of the positive definite matrix $M$, which would be a very
  expensive operation. The matrix $T_m$ is implicitly obtained
  using the matrix-vector product with $M K^{-1} M$ and
  re-orthogonalization by the scalar-product induced by $M$,
  computing the basis $U_{m}$ directly.
\end{remark}

However, using this procedure as a black-box to evaluate the
objective function in an optimization method can easily become
too onerous: even if the complexity is only linear, the size of the problem can make
this operation unpractical (we often deal with $n \approx 10^6$ or
even more in finite element models).

Therefore, we wish to investigate the use of the Lanczos process
to build a local parametric reduced model for the case when $K$ and
$M$
are not constant matrices, but depend on $\ell$ parameters
$x_1, \ldots, x_\ell$.

\subsection{Parametric finite element models}

Let us assume that a finite element model
depending on a variable $\mathbf x \in \mathbb R^{\ell}$ is given. These parameters
$\mathbf x$
may encode different properties of the system (our method does not
make any assumption about what they mean --- but only on
their algebraic properties). Most of the time, these will
be some materials' physical characteristics, such as
Young's modulus, or mass density. In almost all the cases of
interest, the dependency is \emph{linear} in $\mathbf x$. However,
for our discussion we just assume it to be smooth, i.e., at least
$\mathcal C^2$.

\subsection{A first-order approximation to the updated Lanczos projection}

In this section we develop a strategy that, given a certain
initial value $\mathbf x^{(0)}$
for the model parameters, builds a local model to be
used in the trust-region scheme presented in Section~\ref{sec:trustregion}.

Let us denote with $F(\mathbf x)$ the matrix valued function that,
for a given choice of the parameters $\mathbf x$, returns the
tridiagonal matrix obtained running the Lanczos process
on $M^{\frac 1 2}(\mathbf x) K(\mathbf x)^{-1} M^{\frac 1 2}(\mathbf x)$.

We shall approximate $F(\mathbf x)$ in a neighborhood of $\mathbf x^{(0)}$.
This approximation calls for two steps: first, we fix
the subspace used for the projection, as described in
Lemma~\ref{lem:f0}, obtaining an approximation $F_0(\mathbf x)$ of $F(\mathbf x)$.
Then, we show how to further approximate this function in order to make
it cheap to evaluate. Lemma~\ref{lem:objectivef} describes how to
combine these approximations to obtain a
 first-order correct
local model  for the trust-region scheme.

\begin{remark}
    The outcome of the Lanczos process depends on the initial vector
    chosen for the scheme. Here and in the following, we assume that this
    vector is fixed whenever we keep $\mathbf x^{(0)}$ unchanged. This makes
    the subspace generated by the Lanczos projection unique.
\end{remark}

\begin{lemma} \label{lem:f0}
	Let $F(\mathbf x) = U_m(\mathbf x) M(\mathbf x) K^{-1}(\mathbf x) M(\mathbf x) U_m(\mathbf x)$ be the matrix-valued function that
	associates the point $\mathbf x$ with the
	Lanczos projection of the pencil $K(\mathbf x) - \lambda M(\mathbf x)$ previously defined.
	Then, there exists a matrix-valued function $U_{m,0}(\mathbf x)$ such
	that, in a neighborhood $\mathcal N$ of $\mathbf x^{(0)}$,
	\begin{enumerate}
        \item For any $\mathbf x \in \mathcal N$ the
          matrix-valued function $U_{m,0}(\mathbf x)$ is an
          $M(\mathbf x)$-orthogonal
          basis for the column-span of $U_m(\mathbf x^{(0)})$.
        \item The function $F_0(\mathbf x)$ defined as
        \[
          F_0(\mathbf x) := U_{m,0}(\mathbf x)^T M(\mathbf x) K(\mathbf x)^{-1} M(\mathbf x) U_{m,0}(\mathbf x)
        \]
        is  $\mathcal C^2(\mathcal N)$
        and $F_0(\mathbf x^{(0)}) = F(\mathbf x^{(0)})$, that
        is it is a zeroth-order approximation of $F(\mathbf x)$.
	\end{enumerate}
\end{lemma}

\begin{proof}
    By construction, the basis generated by the Lanczos method
    at $\mathbf x^{(0)}$ returns an $M(\mathbf x^{(0)})$-orthogonal
    basis $U_{m}(\mathbf x^{(0)})$.
    We have
    \[
      U_m(\mathbf x^{(0)})^T M(\mathbf x) U_m(\mathbf x^{(0)}) =
        Z(\mathbf x), \qquad
        Z(\mathbf x^{(0)})  = I,
    \]
    and
    since $M(\mathbf x)$ is of class $\mathcal C^2$, the same can be said of
    the symmetric matrix $Z(\mathbf x)$. Therefore, we can define
    $U_{m,0}(\mathbf x) := U_m(\mathbf x^{(0)}) Z(\mathbf x)^{-\frac 12}$.
    For $\mathbf x$ close enough to $\mathbf x^{(0)}$, the value
    of $Z(\mathbf x)$ is bounded away from singular matrices (recall that
    $Z(\mathbf x^{(0)}) = I$), so the inverse square-root is locally analytic
    and therefore $U_{m,0}(\mathbf x)$ is at least $\mathcal C^2(\mathcal N)$
    in a neighborhood $\mathcal N$. Direct substitution yields $U_{m,0}^T(\mathbf x) M(\mathbf x) U_{m,0}(\mathbf x) = I$
    and $F_0(\mathbf x^{(0)}) = F(\mathbf x^{(0)})$,
    as requested.
\end{proof}

The idea behind the approximation of Lemma~\ref{lem:f0} is
to obtain the eigenvalues of
$F_{0}(\mathbf x)$ by a subspace projection
method, where the subspace is approximated by choosing the one
obtained by running Lanczos
for the parameters at $\mathbf x^{(0)}$ instead of the one
at $\mathbf x$. If the two values are close, this subspace
is still ``good enough'' to provide accurate approximations of the
spectrum. In fact a similar strategy underlies
several methods known as \emph{Krylov subspace recycling}
\cite{parks2006recycling,soodhalter2014krylov}. These techniques find
applications in solving sequences of shifted linear systems, which is
a step required, for instance, in model reduction algorithms.

The definition of $F_0(\mathbf x)$, as is, does not make it
particularly easier to evaluate with respect to $F(\mathbf x)$.
In Section~\ref{sec:firstorder}, we will show how to approximate
$F_0(\mathbf x)$ at the first-order in a way that makes its
computation very efficient. The next result justifies this approach.

\begin{lemma} \label{lem:firstorderf}
    Let $\hat F_0(\mathbf x)$ be any locally $\mathcal C^2$
    first-order symmetric positive definite
    approximation of $F_0(\mathbf x)$, and let $\hat\Lambda_s(\cdot)$ be
    the function that associates a symmetric
    matrix with the inverse of its largest $s$ eigenvalues. If the eigenvalues
    at $\mathbf x^{(0)}$ of $F(\mathbf x^{(0)})$ are all
    distinct then
    \[
      \hat\Lambda_s(F(\mathbf x)) = \hat\Lambda_s(\hat F_0(\mathbf x)) +
      O(\norm{\mathbf x - \mathbf x^{(0)}}^2), \qquad
      \mathbf x \in \mathcal N,
    \]
    where $\mathcal N$ is an appropriate neighborhood of $\mathbf x^{(0)}$.
\end{lemma}

\begin{proof}
    When the eigenvalues of a symmetric matrix are all distinct, they
    locally depend analytically on the entries of the matrix (see, for instance,
    \cite{kato2013perturbation,sun1990multiple}). Since the
    matrices involved are all positive definite, the same holds for
    their inverses $\lambda_{i}^{-1}$.
     Therefore, the result follows by composing
    $\hat\Lambda_s(\cdot)$ with the functions $F(\mathbf x)$ and
    $\hat F_0(\mathbf x)$.
\end{proof}

\begin{remark}
    In the event two or more eigenvalues match the situation is slightly
    more problematic,
    since the dependency of the eigenvalues is only continuous in general,
    and although partial derivatives exist,
    a higher-order smoothness in the eigenvalue functions cannot be claimed \cite{sun1990multiple}.
    For simplicity, we restrict our attention to the case where the eigenmodes are
    separated enough not to collide while optimizing the
    parameters. This is the
    case in all practical applications analyzed in Section~\ref{sec:examples}.
    However, a greater generality could be treated by replacing
    eigenvectors with deflating subspaces, at the price of a
    more involved treatment.
\end{remark}

\subsubsection{First-order expansion of the projection}
\label{sec:firstorder}

In view of Lemma~\ref{lem:firstorderf}, in this Section we aim
at constructing an approximation to $F_0(\mathbf x)$ that matches
at the first-order, and that is numerically undemanding to evaluate. More
precisely, we aim at complexity $\mathcal O(m^3)$,
where $m$ is the dimension of the Krylov subspace. Let us write
the value of $K(\mathbf x)$ and $M(\mathbf x)$ as small perturbations
of their values at $\mathbf x^{(0)}$:
\[
  K(\mathbf x) = K(\mathbf x^{(0)}) + \delta K(\mathbf x), \qquad
  M(\mathbf x) = M(\mathbf x^{(0)}) + \delta M(\mathbf x).
\]
We can expand the above expressions to the
first-order, which yields
\begin{equation}
  \label{eq:secondorderK}
  K(\mathbf x) = K(\mathbf x^{(0)}) +
  \sum_{j = 1}^{\ell} (x_j - x^{(0)}_j) \frac{\partial K(\mathbf x^{(0)})}{\partial x_j}
  + O(\norm{\mathbf x - \mathbf x^{(0)}}^2),
\end{equation}
and the analogous formula for $M(\mathbf x)$. Let us assume that we are
given an $M(\mathbf x)$-orthogonal basis $U_{m,0}(\mathbf x)$
spanning the same subspace of $U_m(\mathbf x^{(0)})$ that we have computed
with the Krylov projection method at $\mathbf x^{(0)}$. In this case,
 we can compute the new
projected counterpart of $M^{\frac 12}(\mathbf x) K(\mathbf x)^{-1} M^{\frac 12}(\mathbf x)$ as follows:
\begin{equation}
  \label{eq:projection}
  F_{0}(\mathbf x) = U_{m,0}^T(\mathbf x) M(\mathbf x) K(\mathbf x)^{-1} M(\mathbf x) U_{m,0}(\mathbf x).
\end{equation}
In order to derive a first-order expansion for the above formula we need
first-order expansions for all the terms involved. However, we still do
not have such an expression for the inverse of $K(\mathbf x)$.
For $\mathbf x$ sufficiently close to $\mathbf x^{(0)}$
we can write the Neumann expansion:
\[
  K(\mathbf x)^{-1} = (I + K(\mathbf x^{(0)})^{-1} \delta K(\mathbf x))^{-1} K(\mathbf x^{(0)})^{-1} =
  \sum_{j = 0}^{\infty} (-1)^j \left[ K(\mathbf x^{(0)})^{-1} \delta K(\mathbf x) \right]^j K(\mathbf x^{(0)})^{-1}.
\]
Relying on Equation~(\ref{eq:secondorderK}), and dropping the second-order
terms yields
\begin{align*}
  K(\mathbf x)^{-1} = K(\mathbf x^{(0)})^{-1} &-
  \sum_{j = 1}^\ell (x_j - x_j^{(0)}) K(\mathbf x^{(0)})^{-1} \frac{\partial K(\mathbf x^{(0)})}{\partial x_j} K(\mathbf x^{(0)})^{-1}
+ O(\norm{\mathbf x - \mathbf x^{(0)}}^2),
\end{align*}
which in turn can be rewritten in the following form:
\begin{equation}
  \label{eq:kexpansion}
K(\mathbf x)^{-1} = K(\mathbf x^{(0)})^{-1} \left( I - \sum_{j = 1}^\ell (x_j - x_j^{(0)}) \frac{\partial K(\mathbf x^{(0)})}{\partial x_j} K(\mathbf x^{(0)})^{-1} \right) + O(\norm{\mathbf x - \mathbf x^{(0)}}^2).
\end{equation}
Equation~(\ref{eq:kexpansion}) has the desirable property that all the
inverses of $K(\mathbf x)$ appear evaluated at $\mathbf x^{(0)}$,
and the dependency on $\mathbf x$ is linear.

We now have all the ingredients to derive a cheap formula for computing $F_{0}(\mathbf x)$,
by substituting
the expansions in Equation~(\ref{eq:projection}). This leads to:
\begin{equation} \label{eq:projected-expanded}
  \begin{split}
  F_{0}(\mathbf x) &= U_{m,0}^T(\mathbf x) M(\mathbf x^{(0)}) K(\mathbf x^{(0)})^{-1} M(\mathbf x^{(0)}) U_{m,0}(\mathbf x) \\
                   &+ \sum_{j = 1}^\ell (x_j - x_j^{(0)}) U_{m,0}(\mathbf x)^T G_j(\mathbf x^{(0)}) U_{m,0}(\mathbf x) + O(\norm{\mathbf x - \mathbf x^{(0)}}^2),
\end{split}
\end{equation}
where $G_j(\mathbf x^{(0)})$ is obtained by grouping together all the first-order contributions, that is,
\begin{align*}
  G_j(\mathbf x^{(0)}) &= \frac{\partial M(\mathbf x^{(0)})}{\partial x_j} K(\mathbf x^{(0)})^{-1} M(\mathbf x^{(0)})
  - M(\mathbf x^{(0)}) K(\mathbf x^{(0)})^{-1}\frac{\partial K(\mathbf x^{(0)})}{\partial x_j} K(\mathbf x^{(0)})^{-1} M(\mathbf x^{(0)}) \\
  &+ M(\mathbf x^{(0)}) K(\mathbf x^{(0)})^{-1} \frac{\partial M(\mathbf x^{(0)})}{\partial x_j}.
\end{align*}
The terms $G_j(\mathbf x^{(0)})$ are
large matrices, and computing them explicitly would be
unfeasible. However,
by observing Equation~(\ref{eq:projected-expanded}) closely,
we immediately notice that
we do not need the complete matrices, but just their projected counterparts
$\hat G_j(\mathbf x) := U_{m,0}(\mathbf x)^T G_j(\mathbf x^{(0)}) U_{m,0}(\mathbf x)$,
which are much smaller.
Moreover, in the definition of $G_j(\mathbf x^{(0)})$
the only matrix
that appears as an inverse is $K(\mathbf x^{(0)})$. In our Lanczos implementation
we use a direct sparse solver so, after the projection, we already have
a sparse Cholesky factorization of $K(\mathbf x^{(0)})$, and we compute
$\hat G_j(\mathbf x^{(0)})$
at the cost of some extra back-substitutions and
sparse matvec products (which have a comparable cost).
Similar savings could be obtained recycling the preconditioner
when using an iterative method.

However, the same computational cost would be needed to calculate $\hat G_j$
at $\mathbf x \neq \mathbf x^{(0)}$. We would like to avoid this extra
computation. To this end, observe that according to the proof
of Lemma~\ref{lem:f0}, we have $U_{m,0}(\mathbf x) = U_m(\mathbf x^{(0)}) Z(\mathbf x)^{- \frac 12}$.
Therefore, we can write
the formulas to compute $\hat G_j(\mathbf x)$
as follows:
\[
  \hat G_j(\mathbf x) = Z(\mathbf x)^{-\frac 12} U_m(\mathbf x^{(0)})^T G_j(\mathbf x^{(0)}) U_m(\mathbf x^{(0)}) Z(\mathbf x)^{-\frac 12}, \qquad
\]
which shifts the problem to that of computing
$Z(\mathbf x) = U_m(\mathbf x^{(0)})^T M(\mathbf x) U_m(\mathbf x^{(0)})$. As in the previous equations, we can obtain
a first-order approximation of $Z(\mathbf x)$ at any $\mathbf x$
with a first-order
truncation by precomputing some small projected matrices at $\mathbf x^{(0)}$:
\begin{align*}
  \hat Z(\mathbf x) &:= U_m(\mathbf x^{(0)})^T \left(
    M(\mathbf x^{(0)}) + \sum_{j = 1}^\ell (x_j - x_j^{(0)}) \frac{\partial M(\mathbf x^{(0)})}{\partial x_j} \right)
    U_m(\mathbf x^{(0)}) \\
    &= I + \sum_{j}^\ell (x_j - x_j^{(0)}) U_m(\mathbf x^{(0)}) ^T\frac{\partial M(\mathbf x^{(0)})}{\partial x_j} U_m(\mathbf x^{(0)}).
 \end{align*}

 We can now derive a formula for an approximation $F_{m,0}(\mathbf x)$
 of $F_0(\mathbf x)$
by combining all these considerations:
 \begin{equation}\label{eq:Tapp}
  F_{m,0}(\mathbf x) = \hat Z(\mathbf x)^T T_m(\mathbf x_0) \hat Z(\mathbf x) + \sum_{j = 1}^\ell (x_j - x_j^{(0)}) \hat G_j(\mathbf x)
 \end{equation}
 According the the previous remarks, $\hat Z(\mathbf x)$,
 and $\hat G_{j}(\mathbf x)$,
 can be computed cheaply in
 $\mathcal O(m^3)$ flops, so with a computational cost that is
 \emph{independent} of the size of the original problem.
In view of this analysis, we arrive at the following result.

\begin{lemma} \label{lem:stage2}
    The matrix-valued function $F_{m,0}(\mathbf x)$
    defined in \eqref{eq:Tapp} is a first-order approximation
    of $F_0(\mathbf x)$ at $\mathbf x = \mathbf x^{(0)}$. More precisely,
    there exists a neighborhood $\mathcal N$ containing $\mathbf x^{(0)}$
    such that
    \[
      F_{0}(\mathbf x) = F_{m,0}(\mathbf x) + O(\norm{\mathbf x - \mathbf x^{(0)}}^2), \qquad
      \mathbf x \in \mathcal N.
    \]
\end{lemma}

\subsubsection{Restarted Lanczos and other subspace iterations}

At this stage, at no time we have used
the property that $W_{m}(\mathbf x)$ spans a Krylov subspace
generated by $K(\mathbf x)^{-1}$. In fact, this is not a requirement at all.
Any subspace $W_{m}(\mathbf x)$ generated through a subspace iteration
method will work equally well in this framework.

In particular, we could rely on well-known large-scale eigenvalue
solver such as ARPACK \cite{lehoucq1998arpack}, which provides a tried-and-tested
implementation of a restarted Lanczos method to compute the smaller
end of the spectrum. This is what we currently use in the NOSA code.

Other approaches can be considered as well. The only requirement is that
the projected matrix have the form
\[
  W_m(\mathbf x)^T M(\mathbf x)^{\frac 12} K(\mathbf x)^{-1} M(\mathbf x)^{\frac 12} W_m(\mathbf x),
\] which
in turn implies that we are interested in the lowest end of the spectrum.
For this reason, and in order to keep the explanation simple and self-contained,
we decided to consider only the Lanczos method herein. Even if we
never mention restarted variants explicitly, their use does not require any
change in the proposed strategy.

\subsubsection{Computing derivatives} \label{sec:derivatives}

In order to apply Algorithm \ref{alg:tr} and, in particular, to perform the first-order
correction  (\ref{eq:corr}), we need to be able
to evaluate $\nabla\phi(\mathbf x^{(k)})$ and $\nabla\phi_k^{\mathcal R}(\mathbf x^{(k)})$.
In both cases, the problem can be rephrased in terms of the computation of
a derivative of the eigenvalues of a pencil $K - \lambda M$, in view of
\begin{equation} \label{eq:scalar-product-eigs}
  \frac{\partial}{\partial x_j} \norm*{\frac{\sqrt{\Lambda_s(K(\mathbf x), M(\mathbf x))}}{2 \pi} - \mathbf f}_{\mathbf w, 2}^2 =
      \left(
       \frac{\sqrt{\Lambda_s(K(\mathbf x), M(\mathbf x))}}{2 \pi} - \mathbf f
       \right)^T
    D_{\mathbf w}^2
      \frac{
      	  \frac{\partial}{\partial x_j} \Lambda_s(K(\mathbf x), M(\mathbf x))
      	}{
      	  2 \pi \sqrt{\Lambda_s(K(\mathbf x), M(\mathbf x))}
      	},
\end{equation}
where $D_{\mathbf w} = \mathrm{diag}(w_1, \ldots, w_s)$, as in
\eqref{eq:obj}.
Classical perturbation theory for eigenvalues yields the following, for $i = 1, \ldots, s$:
\[
  \left[ \frac{\partial}{\partial x_j} \Lambda_s(K(\mathbf x), M(\mathbf x)) \right]_i =
    \left(v_i^T \left( \frac{\partial K(\mathbf x)}{\partial x_j} - \lambda_i \frac{\partial M(\mathbf x)}{\partial x_j} \right) v_i\right) \cdot \left(
      v_i^T M(\mathbf x) \, v_i
    \right)^{-1},
\]
where $v_i$ is an eigenvector relative to the eigenvalues $\lambda_i$.

\subsection{Constructing a local model for the trust-region scheme}

Now that we have an approximation $F_{m,0}(\mathbf x)$ of $F(\mathbf x)$,
we can obtain a local model for the trust-region scheme by replacing
$\hat\Lambda_s(F(\mathbf x))$ with $\hat\Lambda_s(F_{m,0}(\mathbf x))$.

In view of Lemma~\ref{lem:f0}, the approximation of $F(\mathbf x)$
with $F_{m,0}(\mathbf x)$ is correct only up to the zeroth order, and
the same
holds when composing it with $\hat\Lambda_s(\cdot)$, and so
for $\phi(\mathbf x)$ and $\phi^{\cal R}(\mathbf x)$ as well. However,
using the strategy described in Section~\ref{sec:firstorder},
we can modify $\phi^{\cal R}(\mathbf x)$ a posteriori to match at the first-order.

Since both
$\phi(\mathbf x)$ and $\phi^{\cal R}(\mathbf x)$ are of the form described in equation
\eqref{eq:scalar-product-eigs}, and in both cases the eigenvectors are available
while computing the eigenvalues, the same formula can be used directly.

\begin{lemma} \label{lem:objectivef}
    Let $\phi^{\cal R}(\mathbf x)$ be the function defined as follows
    \begin{equation}\label{eq:modtr}
      \phi^{\cal R}(\mathbf x) = \tilde\phi^{\cal R}(\mathbf x) + (\nabla \phi(\mathbf x^{(0)}) -
      \nabla \tilde \phi^{\cal R}(\mathbf x^{(0)}))^T (\mathbf x - \mathbf x^{(0)}), \quad
      \tilde \phi^{\cal R}(\mathbf x) = \norm*{\frac{\sqrt{\hat\Lambda_s(F_{m,0}(\mathbf x))}}{2\pi} - \mathbf f}_{\mathbf w, 2}^2.
    \end{equation}
    Then, $\phi^{\cal R}(\mathbf x)$ is a first-order approximation of $\phi(\mathbf x)$
    at $\mathbf x = \mathbf x^{(0)}$.
\end{lemma}

\begin{proof}
    In view of Lemma~\ref{lem:f0}, \ref{lem:firstorderf}, \ref{lem:stage2}
     we have $\phi(\mathbf x^{(0)}) = \tilde \phi^{\cal R}(\mathbf x^{(0)})$. Given this condition, the
    first-order matching is ensured by the gradient correction
    done as described with Section~\ref{sec:firstorder}.
\end{proof}

\begin{remark}
    Although the formulas for $\nabla \phi(\mathbf x)$ and $\nabla \phi^{\cal R}(\mathbf x)$ are not reported explicitly in Lemma~\ref{lem:objectivef}, they
    are directly implementable in view of Equation~\eqref{eq:scalar-product-eigs}, and require almost
    no computational effort when the eigenvectors are obtained through
    the Lanczos process.
\end{remark}

\subsection{Trust region methods}
Trust-region methods are well-known iterative methods for solving
bound-constrained optimization problems, see e.g. \cite{conn1988global,trBible,porcelli13}.
Their main distinctive features are robustness and
convergence to first-order critical points regardless of the choice of
the initial estimate.

We now review the basic steps in a trust-region algorithm for solving
the
bound-constrained optimization problem of Equation~\eqref{eq:optpb},
where $\phi:\mathbb R^\ell \rightarrow \mathbb R$ is a smooth function and
$\Omega$ is defined as in \eqref{eq:Omega}.

The key idea of a trust-region method is to define, at iteration $k$,
a model $\phi^{\cal R}_k$ for the objective function $\phi$
around the
current  iterate $\mathbf x^{(k)}$, together with a
region within such model can be trusted to provide an
adequate representation of $\phi$.
The trial step $\mathbf s^{(k)}$ is then computed, either exactly
or approximately, by minimizing this model within the trust-region.
The trust-region is the set of all points
$$
{\mathcal B}_k = \{\mathbf x \in \mathbb R^\ell |\, \|\mathbf x - \mathbf x^{(k)}\|\le \Delta_k\},
$$
where $\Delta_k$ is called the trust-region radius and $\|\cdot\|$ is a  norm
equivalent to the Euclidean norm.

Due to the presence of bound-constraints, all the generated
iterates $\mathbf x^{(k)}$ must be ensured to be feasible, that is, $\mathbf x^{(k)} \in \Omega$ for all $k\ge 0$.
One strategy consists in minimizing the model within the set $\mathcal B_k \cap \Omega$.
When the feasible set $\Omega$ is a box,
the trust-region is generally defined using the $\infty$ norm.

Having minimized the model on $\mathcal B_k \cap \Omega$, it must be decided
whether to accept the trial step or to change the trust-region radius.
Usually, the
trust-region radius and the new point $\mathbf x^{(k)} + \mathbf s^{(k)}$ are tested simultaneously
to assess the quality of the approximation yielded by the local model. This is
measured
using the ratio $\rho^{(k)}$ between the actual and the predicted reduction defined as follows:
\begin{equation}\label{eq:ratio}
 \rho^{(k)} = \frac{\phi(\mathbf x^{(k)})-\phi(\mathbf x^{(k)}+\mathbf s^{(k)})}{\phi^{\cal R}_k(\mathbf x^{(k)}) - \phi^{\cal R}_k(\mathbf x^{(k)}+ \mathbf s^{(k)})}
\end{equation}
If $\rho^{(k)}$ is close to 1, there is good agreement between the
model $\phi^{\cal R}$ and the function $\phi$ over this step, so $\mathbf x^{(k)} + \mathbf s^{(k)}$ is accepted
as the new iterate and it is safe to increase the radius of
the trust-region for the next iteration.
If $\rho^{(k)}$ is positive but not close to 1, then $\mathbf x^{(k+1)} = \mathbf x^{(k)} + \mathbf s^{(k)}$ but the
trust-region radius is not altered. If  $\rho^{(k)}$ is close to zero or
negative, step $\mathbf s^{(k)}$ is rejected and the trust-region radius is shrunk.

Convergence of the iterative process is declared when a suitable
criticality measure is sufficiently small. We consider the
measure
\begin{equation}\label{eq:crit}
  \chi(\mathbf x^{(k)}) = \|P_{\Omega}(\mathbf x^{(k)} - \nabla \phi(\mathbf x^{(k)})) - \mathbf x^{(k)})\|
\end{equation}
where $P_{\Omega}$ is the projection onto the feasible set, and  $\chi(\mathbf x^{(k)})$ is the
norm of the projected gradient onto the box, and reduces to $\norm{\nabla \phi(\mathbf x^{(k)})}$
when $\mathbf x^{(k)}$ is in the interior of $\Omega$.
The approach is summarized in the pseudocode of Algorithm \ref{alg:tr} where we assume that the
model functions are minimized ``exactly'' (as it  will be the case in our applications, see Section
\ref{sec:impl}).

\begin{algorithm}
  \begin{algorithmic}
  \Require The initial point $\mathbf x^{(0)} \in \Omega$,
     an initial trust-region radius $\Delta_0$,
    the constants $\eta_1, \eta_2, \gamma_1, \gamma_2$ such that
  \[
  0 < \eta_1 \le \eta_2 <1 \mbox{ and } 0 < \gamma_1\le \gamma_2 <1,
  \]
  $k_{max}>0$ and $\epsilon>0$.

  \State Compute $\phi(\mathbf x^{(0)})$
  \For{$k = 1, \ldots, k_{max}$}
      \State Define a model $\phi^{\cal R}_k$ in $\mathcal B_k\cap \Omega$.
      \State Compute a step $\mathbf s^{(k)}$ that minimizes the model and
         such that $\mathbf x^{(k)} + \mathbf s^{(k)} \in \mathcal B_k\cap \Omega$.
      \State Compute $\phi(\mathbf x^{(k)}+\mathbf s^{(k)})$ and define the ratio (\ref{eq:ratio}).
      \If{$\rho^{(k)} \ge 1$}
      \State define $\mathbf x^{(k+1)}= \mathbf x^{(k)} + \mathbf s^{(k)}$;
      \Else
      \State define $\mathbf x^{(k+1)}=\mathbf x^{(k)}$.
      \EndIf
      \State Update the trust-region radius setting
             \[
              \Delta_{k+1} \in \left \{ \begin{array}{ll}
                                       \left[\Delta_k, \infty \right) & \mbox{ if } \rho^{(k)} \ge \eta_2 \\
                                       \left [\gamma_2\Delta_k, \Delta_k \right ] & \mbox{ if } \rho^{(k)} \in [\eta_1, \eta_2) \\
                                      \left [\gamma_1\Delta_k, \gamma_2 \Delta_k \right] & \mbox{ if } \rho^{(k)} < \eta_1
                                      \end{array}
                             \right.
             \]

      \State Compute the criticality measure (\ref{eq:crit}).
      \If{ $\chi(\mathbf x^{(k+1)}) \le \epsilon$}
      \State \textbf{return} An approximate first-order critical point $\mathbf x^{(k+1)}$.
      \EndIf
      \EndFor
      \end{algorithmic}
  \caption{Basic trust-region algorithm}
  \label{alg:tr}
\end{algorithm}

In order to obtain a robust and globally convergent trust-region algorithm
involving approximate models, the following assumptions should hold \cite{trBible}
for the objective $\phi$ and the model $\phi^{\cal R}_k$ functions:
\begin{description}
 \item{(AF.1)} $\phi:\mathbb R^\ell \rightarrow \mathbb R$ is twice-continuously differentiable on $\Omega$.
 \item{(AF.2)} The function $\phi$ is bounded below for all $\mathbf x \in \Omega$.
 \item{(AF.3)} The second derivatives of $\phi$ are uniformly bounded for all $\mathbf x \in \Omega$.
\vskip 8pt
 \item{(AM.1)} For all $k$, $\phi^{\cal R}_k$ is twice differentiable on ${\mathcal B}_k$.
 \item{(AM.2)} The values of the objective and the model function coincide at the current iterate, i.e., for
all $k$,
$$\phi (\mathbf x^{(k)} ) = \phi^{\cal R}_k (\mathbf x^{(k)} ).$$
\item{(AM.3)} The gradients of the objective and the model function coincide at the current iterate,
i.e., for all $k$,
$$\nabla \phi (\mathbf x^{(k)} ) = \nabla \phi^{\cal R}_k (\mathbf x^{(k)} ).$$
\item{(AM.4)} The second derivatives of $\phi^{\cal R} (\mathbf x^{(k)} )$ remain bounded
within the trust-region ${\mathcal B}_k$ for all $k$.
\end{description}

\begin{theorem}\cite[Section 12.2.2]{trBible}
Under Assumptions (AF.1)-(AF.3) and (AM.1)-(AM.4), every limit point of the sequence $\{\mathbf x^{(k)}\}$ generated by Algorithm
 \ref{alg:tr} is  first-order critical, i.e.
\[
 \lim_{k\rightarrow \infty} \|P_{\Omega}(\mathbf x^{(k)} - \nabla \phi(\mathbf x^{(k)})) - \mathbf x^{(k)})\| = 0.
\]
\end{theorem}

\subsection{Enforcing first-order matching} \label{sec:first-order}

Here we use the trust-region framework described in the previous section to solve
problem (\ref{eq:optpb}) with objective function (\ref{eq:obj}). Note
that conditions
(AF.1) -- (AF.3) are automatically satisfied by our definition of objective function.
As discussed in Section~\ref{sec:lanczos}, the local model
$\tilde \phi_k^{\mathcal R}(\mathbf x)$
that we
build does not necessarily agree at the first order with the original function
$\phi(\mathbf x)$, which apparently breaks assumption (AM.3) --- and only satisfies
(AM.2). However, if the gradient $\nabla \phi_k^{\cal R}(\mathbf x)$ is known at the
expansion point $\mathbf x^{(k)}$, this can be easily fixed by defining
\begin{equation}\label{eq:corr}
 \phi_k^{\cal R} (\mathbf x) := \tilde \phi_k^{\cal R}(\mathbf x)  + (\phi(\mathbf x^{(k)}) -  \tilde \phi_k^{\cal R}(\mathbf x^{(k)})) +
                             (\nabla \phi(\mathbf x^{(k)}) - \nabla \tilde \phi_k^{\cal R}(\mathbf x^{(k)}))^T(\mathbf x-\mathbf x^{(k)}),
\end{equation}
see, e.g., \cite{Biegler03,Alex95,giunta00}

redpajama_set_name: RedPajamaArXiv


text:
KMC solicits NAMs support for government to pay council's subvention
Gamtel/Gamcel Owes Over One Billion to Broad Band Network Project
Sheikh Omar Jengs says ex-President Jammeh Ordered for Solo Sandeng Burial
Five COVID-19 Patients Currently on Oxygen Therapy, 12 New Cases Recorded
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Gambia's Security Adviser Launches Two Security Strategy Documents
Your Representative January 21, 2021
By Makutu Manneh Officials of the Kanifing Municipal Council on Thursday 21st January 2020, sought the support of National Assembly select...
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By Isatou Kanyi
The Gambia's National Security Adviser, Retired Colonel Momodou Badjie, has officially launched the National Security Strategy (NDS) and Security Sector Reform Strategy (SSRS) in collaboration with the Government of the Gambia and SSR Partners.
Held on Tuesday 24th, 2020, at the Sir Dawda Kairaba Jawara Conference Center, the ceremony was chaired by The Gambia Government's Spokesperson, Mr Ebrima G. Sankareh.
Sankareh presented a certificate of appreciation to persons and institutions for their immense and invaluable contributions towards the realization of the NSS and SSR.
Retired Colonel Badjie speaking at the ceremony also expressed his appreciation to the invaluable support provided by the Government and SSR partners.
"I encourage all of you to leave this launching ceremony with an understanding of the importance of our individual, personal and institutional contributions to make the Gambia peaceful, safe, stable and wonderful country", he remarked.
Badjie further said the Security Sector Reform focuses on human security center approach with the aim of restructuring the sector to achieve an effective, robust, professional, apolitical, accountable, transparent and responsive sector.
He highlighted the importance of the strategy, which he said will also enhance the prevention, protection and response capabilities to security threats in an increasingly complex environment.
Badjie indicated that President Adama Barrow, who is also the Commander-in-Chief of the Armed Forces, took the first step in launching the Security Sector Reform process in 2017 and subsequently the National Security Policy in May 2019.
He said these initiatives are with a view to ensure he laid a solid foundation for his national development ambitions, programmes and projects.
Dawda Jallow, the Minister of Justice in his capacity as Chairperson of the Security Sector Reform Committee, said the launching of the two documents is yet another milestone in the trajectory of the events in security architecture of the country.
"It could be recalled that shortly after President Adama Barrow took over the political leadership of this country in 2017, a technical working group was duly commissioned to conduct a comprehensive Security Sector Reform (SSR) assessment to ascertain the gaps and malfunctions of the Security Sector that his government inherited.
"The assessment was completed in December 2017 and the report came up with recommendations that would enable the sector to be transformed to meet the needs of the state and people's aspirations," Minister Jallow said.
Minister Jallow expressed his gratitude to the EU, AU and all other partners for their contributions and Coordination efforts towards the security sector reform process.
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There are different kinds of herbs available. But they are classified according to their use.
Ayurveda is the ancient science of life, which aims to promote a healthy lifestyle, free of diseases. The Ayurvedic rishis in the ancient times have discovered the medicinal use of many herbs, which are effective in curing a number of diseases that are otherwise difficult to get rid of. The herbs have been in use since the ancient times, when people were not familiar with allopath and other forms of treatment and resorted to the herbs and Ayurvedic medicines that were available. The Ayurvedic herbs can be classified into five types: according to origin, according to habitat, according to various actions, according to action on doshas and according to their use. In this article, we have discusses about the different kinds of Ayurvedic herbs.
Jalaj (Aquatic) - Plants that grow in water, e.g. Lotus, Spyrogyra.
Vruksharuha (Epiphytic, living on the surface of trees) - Plants that grow on other trees but do not depend on that tree for their nutrition, e.g. Rasana.
Vrukshadan (Parasitic) - Plant that dwells on another tree and obtains its nutrition from its host.
Swasthahita - Dravyas which have a tendency to increase or decrease vatadi doshas but do not produce vitiation of these doshas in a healthy person.


Aeria Games has opened up the gates of their new MOBA Chaos Heroes Online today. The open beta has started and the shop is also online. To celebrate, players were also shown a stylized, shiny new trailer complete with heroes pulling off stunning moves and armor that's of course a bit on the skimpy, "not so protective" side.
If you've been playing the game prior to this open beta your data has of course been wiped but there will not be a wipe after open beta so feel free to advance your heroes as much as you want. Founder's Pack purchasers have also had their packs redistributed to them.
When jumping in, make sure to carefully select your sever (as the game features both US and EU servers) and get ready to gank until your little heart is content.
If you missed this title up until now, be sure to check out Spunkify's First Look before you head into battle!
another moba with pathetic cam and controls like in dota 2 or lol. We want to use WASD to move the hero. but even if it had a good cam and controls it is still hosted by aera= the worst company ever.
the game will be pay to win garbage with no support and full with cheaters.
Not a bad game at all. Wish there were more players to play it!
As someone who generally hates the moba genera for being cancerous to videogames as a whole, this is the least cancerous version of it. Being able to buy items on the fly fixes one of dota/lol/hon biggest flaws, which is the first 15 minutes of the game. I'm sorry the first 15 minutes of a pvp game should not be spent sitting there maybe hitting someone once or twice while you farm monsters and run like a panzy anytime someone comes close to you. Antimagic and disspell potions are also add a nice layer of depth to combat.


New!! do you need the internet on site? Just ask, cause ya we have that too! Up to 10 devices can run at 4G speeds.
Winter Weather Warning: Its going to be cold this week!
Pipe and draip, truss, floor mats? yes we have it!!!
and yes we have the trucks to get it there.
Chain Saw !!!!!!!!!!! Truth be told I've been known to run this thing for free!!!!