text:
\section*{This is an unnumbered first-level section head}
.

\section{Introduction}
Let $d \geq 1$ be the spatial dimension.
Fix a wavenumber $k > 0$.
Let $[\underline \theta, \overline \theta]$ be an interval of angles.
Let $u_0 = u_0({\bf x}, \theta)$, $({\bf x}, \theta) \in \mathbb{R}^d \times [\underline \theta, \overline \theta]$, be the incident  plane wave that illuminates the medium at an angle $\theta$.
This choice of $u_0$ as the incident wave arises from the well-known  inverse scattering problem, stated later.
Let $u({\bf x}, \theta)$, $({\bf x}, \theta) \in \mathbb{R}^d \times [\underline \theta, \overline \theta]$, be the wave function that  is governed by the Helmholtz equation and the Sommerfeld radiation condition
\begin{equation}
	\left\{
		\begin{array}{ll}
			\Delta u({\bf x}, \theta) + k^2 c_0({\bf x})u({\bf x}, \theta) = -k^2 p({\bf x}) u_0({\bf x}, \theta) &{\bf x} \in \mathbb{R}^d,\\
			\partial_{|{\bf x}|} u({\bf x}, \theta) - \ii k u({\bf x}, \theta) = o(|{\bf x}|^{\frac{1-d}{2}})
			&|{\bf x}| \to \infty.
		\end{array}
	\right.
	\label{Heqn}
\end{equation}
Here,  $c_0: \mathbb{R}^d \to [1, \infty)$ and $u_0: \mathbb{R}^d \times [\underline \theta, \overline \theta] \to \mathbb{C}$ are given functions.
The function $c_0$ represents the background dielectric constant of the medium.

Let $\Omega$ be an open and bounded domain of $\mathbb{R}^d$ with a smooth boundary.	Assume that the source function $p$ is compactly contained in $\Omega$.
We are interested in the following problem.
\begin{problem}[Inverse source problem]
	Determine the source function $p({\bf x})$ for all ${\bf x} \in \Omega$ from the following boundary measurements
	\begin{equation}
		f({\bf x}, \theta) = u({\bf x}, \theta)
		\quad
		\mbox{and}
		\quad
		g({\bf x}, \theta) = \partial_{\nu} u({\bf x}, \theta)
		\label{1.2}
	\end{equation}
	for all $({\bf x}, \theta) \in \partial \Omega \times [\underline \theta, \overline \theta].$
	Here, $\nu({\bf x})$ is the unit normal vector to $\partial \Omega$ at ${\bf x}$.
\label{isp}
\end{problem}

This inverse source problem is the linearization of the nonlinear inverse scattering problem. Since the inverse scattering problem has many real-world applications; mostly in exploring some inaccessible regions from external measurement;
for e.g, bio-medical imaging, nano-sciences, security, seismic exploration.
Therefore, Problem \ref{isp} is significant in those fields.
The solver for Problem \ref{isp} proposed in this paper consists of three steps.
\begin{enumerate}
    \item \label{step1} We derive a system of linear elliptic equations. Solution of this system is a vector involving the first $N$ Fourier coefficients of the function $w = -u/(k^2 u_0)$, $ N \in \mathbb{N}$, with respect to a special basis of $L^2$ introduced in \cite{Klibanov:jiip2017}.
    \item \label{step2}  We apply the quasi-reversibility method developed in \cite{NguyenLiKlibanov:2019} to solve the system obtained in Step \ref{step1}.

    \item Directly compute the desired source from the solution obtained in Step \ref{step2}.
\end{enumerate}

The inverse source problem and some of its versions were studied intensively. We cite to \cite{BaoLinTriki:jde2010, BaoLinTriki:CRM2011, BaoLinTriki:cm2011, EntekhabiIsakov:ip2018, IsakovLu:SIAM2018, IsakovLu:ipi2018} for the uniqueness, stability and numerical methods to solve inverse source problems for the case when the medium is homogeneous; i.e. $c_0 = \mbox{constant}$.
The numerical reconstruction methods in those publications are based on the least squares optimization method.
Good quality  reconstructions due to those approaches are achieved only when the wave number $k$ is large.
We observe that when $k$ is large, the data is  very sensitive with noise. This is because of the high oscillation of the data.
Unlike this, the reconstructive method  in this paper and the method in \cite{NguyenLiKlibanov:2019} do not require the data at high frequency. Therefore, we can reconstruct the source with reasonable value of $k$. The difficulty about the noise is overcome.

As mentioned in step \ref{step2} above, we only compute the first $N$ Fourier coefficients of the function $w = -u/(k^2 u_0)$.
That means we only solve Problem \ref{isp} in an ``Galerkin" approximation context.
Rigorously verifying the convergence of this approximation as $N \to \infty$ is extremely challenging. We assume that this approximation is valid.
In contrast, this numerical approach is very effective  for many kinds of inverse problems; see e.g. \cite{Nguyen:CAMWA2020, KlibanovNguyen:ip2019, NguyenKlibanov:ip2022, Nguyens:jiip2020, LeNguyenNguyenPowell:JOSC2021}. This is the reason we employ this technique again in this paper.

The paper is organized as follows. In Section \ref{motivation}, we provide in details the formulation of Problem \ref{isp}.
In Section \ref{sec_Ana}, we establish the approximation context and derive a system of PDEs that plays an important role in our algorithm.
In Section \ref{s5}, we recall the quasi-reversibility method to solve the system obtained in Section \ref{sec_Ana}. We present numerical study in Section \ref{sec_num}. Section \ref{sec_concl} is for concluding remarks.

\section{The significance of Problem \ref{isp}} \label{motivation}

Problem \ref{isp} is the linearization  of the nonlinear inverse scattering problem, which has many real-world applications. We will list some of the important applications later.
Let $c: \mathbb{R}^d \to [1, \infty)$ be the spatially distributed dielectric constant of the medium.
For each angle $\theta \in [\underline \theta, \overline \theta],$   define
\[
	\xi(\theta) = (\cos \theta, \sin \theta, 0, \dots, 0) \in \mathbb{S}^{d-1}
\] that represents the direction of the angle $\theta$ in the $x_1 x_2$-plane.
Here, $\mathbb{S}^{d-1} = \{{\bf x} \in \mathbb{R}^d, |{\bf x}| = 1\}$ is the unit sphere in $\mathbb{R}^d.$
 We use the plane wave, so-called the incident plane wave, of the form
\begin{equation}
	v_0({\bf x}, \theta) = e^{\ii k {\bf x} \cdot \xi(\theta)}
	\quad
	\mbox{for all } ({\bf x}, \theta) \in \mathbb{R}^d \times [\underline \theta, \overline \theta], k > 0,
	\label{incwave}
\end{equation}
to illuminate the medium.
The incident wave propagates in space and scatters.
 The resulting total wave $v({\bf x}, \theta)$ is governed by the Helmholtz equation
\begin{equation}
	\Delta v({\bf x}, \theta) + k^2 c({\bf x}) v({\bf x}, \theta) = 0
	\quad \mbox{for all } {\bf x} \in \mathbb{R}^d, \theta \in [\underline \theta, \overline \theta]
	\label{2.1}
\end{equation}
and the Sommerfeld radiation condition
\[
	\partial_{|{\bf x}|} v_{\rm sc}({\bf x}, \theta) - \ii k v_{\rm sc}({\bf x}, \theta) =  o(|{\bf x}|^{\frac{1 - d}{2}})
\]
where
\begin{equation}
	v_{\rm sc}({\bf x}, \theta) = v({\bf x}, \theta) - v_0({\bf x}, \theta)
	\quad
	\mbox{for all } ({\bf x}, \theta) \in \mathbb{R}^d \times [\underline \theta, \overline \theta]
\end{equation}
is the scattering wave.

	By \eqref{incwave}, the angle between the $x_1$ axis and the propagation of incident wave in the $x_1 x_2$ plane is $\theta$. Roughly speaking, $\theta$ can be considered as the angle of view. Since we consider the case of complete angles of view, we  choose $[\underline \theta, \overline \theta] = [0, 2\pi].$ The study of Problem \ref{isp} in the case of partial angle of views will be studied  later.

The inverse scattering problem is formulated as follows.
\begin{problem}[The inverse scattering problem]
Let $\Omega$ be a bounded and open domain of $\mathbb{R}^d$ with smooth boundary.
Given the boundary measurement $v_{\rm sc}({\bf x}, \theta)$ and $\partial_{\nu} v_{\rm sc}({\bf x}, \theta)$ for all $({\bf x}, \theta) \in \partial \Omega \times [\underline \theta, \overline \theta],$
determine the spatially distributed dielectric constant $c({\bf x})$ for all ${\bf x} \in \Omega$,
\label{cip}
\end{problem}

{\it Why is the inverse scattering problem interesting and significant?}
According to  the formulation of Problem \ref{cip},  we want to compute the information of the spatially distributed dielectric constant of the medium from the external measurement of the wave field.
 The knowledge of  the reconstructed spatially distributed dielectric constant of a medium directly provides significant information about unknown objects inside that medium; for e.g, position, shape, size and physical properties. Typical examples of those objects are anti-personnel explosive devices buried under the ground, cancerous tumors inside living tissues, and nano structures.
Therefore, solving the inverse scattering problem has important applications in  bio-medical imaging, nondestructive testing, radar, security, optical physics, seismic exploration, nano science.
Hence, the inverse scattering problem and the related ones have been intensively studied.
We list here several approaches: the imaging techniques based on sampling and the factorization methods \cite{ AmmariChowZou:sjap2016, AmmariKang:lnim2004,  Tan1:ip2012, Burge2005,   Colto1996, HarrisLiem:SIAM2020, Kirsc1998, LiLiuZou:smms2014, LiLiuWang:jcp2014, Liem:jiiptoappear, Soumekh:SAR}, the methods based on optimization \cite{Bakushinsii:springer2004, Tan2:ip2012, Chavent:springer2009, Engl:Kluwer1996, Gonch2013, Tihkonov:kapg1995}, the methods based on Born series \cite{Bleistein:ap1984, Chew:vnr1990, Devaney:cup2012, Kirsch:aa2017, Langenberg:1987, Moskow:Ip2008}, the method based on linearization \cite{BaoLi:ip2005, BaoLi:SIAM2005, Bao:ip2015, Chen:ip1997}, and the convexification method \cite{VoKlibanovNguyen:IP2020, Khoaelal:IPSE2021, KhoaKlibanovLoc:SIAMImaging2020}.
See \cite{ColtonKress:2013} for a more complete list.

The current paper contributes to the field by solving the linearization of the inverse scattering problem.
Assume that the dielectric constant $c$ is a perturbation of a background function $c_0$.
For simplicity, in this section, we choose $c_0$ to be identically 1.
That means, the function $c$ is of the form
\begin{equation}
	c({\bf x}) = 1 + \eta p({\bf x})
	\quad
	\mbox{for all } {\bf x} \in \mathbb{R}^d
	\label{2.3}
\end{equation}
where $0 < \eta \ll 1$  and $p$ is a function that indicate the unknown inclusion.
Therefore, the wave function $v$ satisfies
\begin{equation}
	\Delta v({\bf x}, \theta) + k^2 (1 + \eta p({\bf x}))v({\bf x}, \theta) = 0
	\quad
	\mbox{for all }
	({\bf x}, \theta) \in \mathbb{R}^d \times [\underline \theta, \overline \theta].
	\label{2.4}
\end{equation}
It is not hard to verify that
\begin{equation}
	\Delta v_0({\bf x}, \theta) + k^2 v_0({\bf x}, \theta) = 0
	\quad
	\mbox{for all }
	({\bf x}, \theta) \in \mathbb{R}^d \times [\underline \theta, \overline \theta].
	\label{2.5}
\end{equation}
Let $u = v_{\rm sc}/\eta$. It follows from \eqref{2.4} and \eqref{2.5} that
\begin{equation}
	\Delta u({\bf x}, \theta) + k^2 u({\bf x}, \theta) = -k^2  p({\bf x}) v({\bf x}, \theta)
	\quad
	\mbox{for all }
	({\bf x}, \theta) \in \mathbb{R}^d \times [\underline \theta, \overline \theta].
	\label{2.6}
\end{equation}
It is obvious that $\lim_{\eta \to 0} v({\bf x}, \theta) = v_0({\bf x}, \theta)$. Thus,
since $\eta$ is a small number, equation \eqref{2.6} can be approximated as
\begin{equation}
	\Delta u({\bf x}, \theta) + k^2 u({\bf x}, \theta) = -k^2 p({\bf x}) v_0({\bf x}, \theta)
	\quad
	\mbox{for all }
	({\bf x}, \theta) \in \mathbb{R}^d \times [\underline \theta, \overline \theta].

\end{equation}
By choosing $u_0$ as the incident wave $v_0$, we derive Problem \ref{isp}.
While the argument above only give an example for the motivation to solve Problem \ref{isp}, our method to solve inverse problem is not limited in the context of the inverse scattering problem. That means, the proposed method in this paper can be applied in the general case when $u_0$ is not necessary to be the incident wave $v_0$.

\section{An approximation context} \label{sec_Ana}

Define
\begin{equation}
	w({\bf x}, \theta) = -\frac{u({\bf x}, \theta)}{k^2 u_0({\bf x}, \theta)}
	\quad
	\mbox{or }
	\quad
	u({\bf x}, \theta) = -k^2w({\bf x}, \theta) u_0({\bf x}, \theta)
	\label{w}
\end{equation}
for all $ ({\bf x}, k) \in \Omega \times [\underline \theta, \overline \theta].$
Since $\Delta u({\bf x}, \theta) = -k^2 \Delta [w({\bf x}, \theta) u_0({\bf x}, \theta)]$, we have for all $({\bf x}, \theta) \in \Omega \times [\underline \theta, \overline \theta]$
\begin{equation}
	\Delta u({\bf x}, \theta)
	= -k^2\big[w({\bf x}, \theta) \Delta u_0({\bf x}, \theta) + u_0({\bf x}, \theta) \Delta w({\bf x}, \theta) + 2\nabla w({\bf x}, \theta) \cdot \nabla u_0({\bf x}, \theta)\big].
	\label{3.1}
\end{equation}
On the other hand,
it follows from the Helmholtz equation in \eqref{Heqn} and the second identity in \eqref{w} that
\begin{align*}
	\Delta u({\bf x}, \theta) &= - k^2\big[ c_0({\bf x}) u({\bf x}, \theta) +  p({\bf x})u_0({\bf x}, \theta)\big]\\
	&= - k^2 \big[-k^2 c_0({\bf x}) w({\bf x}, \theta) u_0({\bf x}, \theta) + p({\bf x})u_0({\bf x}, \theta)\big]
\end{align*}
for all $({\bf x}, \theta) \in \Omega \times [\underline \theta, \overline \theta].$
This and \eqref{3.1} imply
\begin{multline*}
w({\bf x}, \theta) \Delta u_0({\bf x}, \theta) + u_0({\bf x}, \theta) \Delta w({\bf x}, \theta) + 2\nabla w({\bf x}, \theta) \cdot \nabla u_0({\bf x}, \theta)
\\
=
	- k^2 c_0({\bf x}) w({\bf x}, \theta) u_0({\bf x}, \theta) + p({\bf x})u_0({\bf x}, \theta)
\end{multline*}
or all $({\bf x}, \theta) \in \Omega \times [\underline \theta, \overline \theta].$
Therefore,
\begin{equation}
	\Delta w({\bf x}, \theta)
	+ \Big[\frac{\Delta u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)} + k^2 c_0({\bf x})  \Big]w({\bf x}, \theta)
	+ 2 \nabla w({\bf x}, \theta) \cdot \frac{\nabla u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)} = p({\bf x})
	\label{3.3}
\end{equation}
for all $({\bf x}, \theta) \in \Omega \times [\underline \theta, \overline \theta].$
To eliminate the unknown $p$, we differentiate \eqref{3.3} with respect to $\theta$.
We obtain
\begin{multline}
	\Delta \partial_\theta w({\bf x}, \theta)
	+ \partial_\theta\Big[\frac{\Delta u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)} + k^2 c_0({\bf x})  \Big]w({\bf x}, \theta)
	\\
	+\Big[\frac{\Delta u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)} + k^2 c_0({\bf x})  \Big]\partial_\theta w({\bf x}, \theta)
	+ 2 \nabla \partial_\theta w({\bf x}, \theta) \cdot \frac{\nabla u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)}
	\\
	+ 2 \nabla w({\bf x}, \theta) \cdot  \partial_\theta \Big[\frac{\nabla u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)}\Big]
	 = 0
	\label{3.4}
\end{multline}
for all $({\bf x}, \theta) \in \Omega \times [\underline \theta, \overline \theta].$
Solving the third order equation \eqref{3.4} is extremely challenging.
An analytic and numerical approach to solve it is not developed yet.
Suggested by \cite{NguyenLiKlibanov:2019}, we only consider an approximation context due to a truncation of the Fourier series of the solution $w({\bf x}, \theta)$ with respect to the special basis.
This basis was first introduced in \cite{Klibanov:jiip2017}.
We briefly recall  the construction of this basis.
For each $m\geq 1$, define $\phi _{m}(\theta)=(\theta-\theta_{0})^{m-1}\exp (\theta-\theta_{0})$
where $\theta_{0}=(\underline{\theta}+\overline{\theta})/2$. The sequence $\{\phi
_{m}\}_{m=1}^{\infty }$ is complete in $L^{2}(\underline{\theta},\overline{\theta})$.
Applying the Gram-Schmidt orthonormalization procedure to the sequence $%
\{\phi _{m}\}_{m=1}^{\infty }$, we obtain an orthonormal basis in $L^{2}(%
\underline{\theta},\overline{\theta}),$ denoted by $\{\Psi _{m}\}_{m=1}^{\infty }$. It
is not hard to verify that for each $m,$ the function $\Psi _{m}(\theta)$ has the
form
$
\Psi _{m}(\theta)=P_{m-1}(\theta-\theta_{0})\exp (\theta-\theta_{0}),
$
where $P_{m-1}$ is a polynomial of the degree $(m-1)$. The
following result plays an important role in our analysis.

\begin{proposition}[see \protect\cite{Klibanov:jiip2017}]
For $m, n\geq 1$, we have
\begin{equation}
s_{mn}=\int_{\underline{\theta}}^{\overline{\theta}}\Psi _{m}(\theta)\Psi _{n}^{\prime
}(\theta)d\theta=\left\{
\begin{array}{ll}
1 & \mbox{if }n=m, \\
0 & \mbox{if }n<m.%
\end{array}%
\right.  \label{1}
\end{equation}%
Consequently, let $N>1$ be an integer. Then, the $N\times N$ matrix
\begin{equation}
S=(s_{mn})_{m,n=1}^{N}  \label{matrix D}
\end{equation}
has determinant $1$ and is invertible.
\label{prop MK}
\end{proposition}

Recall the function $w$ defined in \eqref{w}.
For each $({\bf x}, \theta) \in \Omega \times [\underline \theta, \overline \theta]$, we write
\begin{equation}
	w({\bf x}, \theta) = \sum_{n = 1}^\infty w_n({\bf x}) \Psi_n(\theta)
	\simeq \sum_{n = 1}^N w_n({\bf x}) \Psi_n(\theta)
	\label{3.7}
\end{equation}
where
\begin{equation}
	w_n({\bf x}) = \int_{\underline \theta}^{\overline \theta} w({\bf x}, \theta) \Psi_n(\theta)d\theta,
	\quad n = 1, 2, \dots
	\label{3.8}
\end{equation}
\begin{remark}[The choice of $N$]
The cut-off number $N$ in \eqref{3.7} is chosen numerically as follows.
For each $N$, we define
\[
	\varphi(N) = \Big\|w({\bf x}, \theta) - \sum_{n = 1}^N w_n({\bf x}) \Psi_n(\theta)\Big\|_{L^{\infty}(\Gamma \times [\underline \theta, \overline \theta])}
\]
where $\Gamma$ is a subset of $\partial \Omega.$
Since the function $u$ is given on $\partial \Omega \times [\underline \theta, \overline \theta])$, using \eqref{w} and \eqref{3.8}, we can compute $\varphi(N)$ directly. We then choose $N$ such that $\varphi(N)$ is sufficiently small. In our computation, $N = 35$. With this choice of $N$, we have $\varphi(N) < 5\times10^{-3}.$
See Figure \ref{figchooseN} for an illustration.
\label{rem_N}
\end{remark}

From now on, we assume that the approximation in \eqref{3.7} is valid.
Plugging the truncating formula \eqref{3.7} into \eqref{3.4}, we obtain
\begin{multline}
	\sum_{n = 1}^N \Delta  w_n({\bf x}) \Psi_n'(\theta)
	+ \partial_\theta\Big[\frac{\Delta u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)} + k^2 c_0({\bf x})  \Big] \sum_{n = 1}^N w_n({\bf x}) \Psi_n(\theta)
	\\
	+\Big[\frac{\Delta u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)} + k^2 c_0({\bf x})  \Big]\sum_{n = 1}^N w_n({\bf x}) \Psi_n'(\theta)
	+ 2 \sum_{n = 1}^N \nabla w_n({\bf x}) \Psi_n'(\theta) \cdot \frac{\nabla u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)}
	\\
	+ 2 \sum_{n = 1}^N \nabla w_n({\bf x}) \Psi_n(\theta) \cdot  \partial_\theta \Big[\frac{\nabla u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)}\Big]
	 = 0
	\label{3.9}
\end{multline}
for all $({\bf x}, \theta) \in \Omega \times [\underline \theta, \overline \theta]$.
For each $m \in \{1, \dots, N\},$ we multiply $\Psi_m(\theta)$ to both sides of \eqref{3.9}. Then, we integrate the resulting equation with respect to $\theta$ on $[\underline \theta, \overline \theta]$.
We get
\begin{equation}
	\sum_{n = 1}^N s_{mn}\Delta  w_n({\bf x})
	+ \sum_{n = 1}^N a_{mn}({\bf x})  w_n({\bf x})
	+ \sum_{n = 1}^N {\bf b}_{mn}({\bf x}) \cdot \nabla  w_n({\bf x})  = 0
	\label{3.10}
\end{equation}
for all ${\bf x} \in \Omega$.
Here,
\begin{align*}
	s_{mn} &= \int_{\underline \theta}^{\overline \theta} \Psi_n'(\theta) \Psi_m(\theta)d\theta,	\\
a_{mn}({\bf x}) &= \int_{\underline \theta}^{\overline \theta} \Big[
		\partial_\theta\Big(\frac{\Delta u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)} + k^2 c_0({\bf x})  \Big)\Psi_n(\theta)
		\\
		&\hspace{3cm}+ \Big(\frac{\Delta u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)} + k^2 c_0({\bf x})  \Big)\Psi_n'(\theta)
	\Big]\Psi_m(\theta)d\theta,
	\\
	{\bf b}_{mn}({\bf x}) &= 2 \int_{\underline \theta}^{\overline \theta}  \Big[ \frac{\nabla u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)} \Psi_n'(\theta) +   \partial_\theta \Big( \frac{\nabla u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)} \Big)\Psi_n(\theta)\Big] \Psi_m(\theta)d\theta
\end{align*}
for all $m, n \in \{1, \dots, N\}.$
Equation \eqref{3.10}, with $m \in \{1, \dots, N\}$, forms a linear system of linear elliptic equations for the vector $W = (w_n)_{n = 1}^N.$

We next derive the boundary conditions for $W$.
Due to \eqref{1.2} and \eqref{3.8}, for all ${\bf x} \in \partial \Omega$ and $n \in \{1, \dots, N\},$ we have
\begin{equation}
	w_n({\bf x}) = \int_{\underline \theta}^{\overline \theta} \frac{u({\bf x}, \theta)}{-k^2u_0({\bf x}, \theta)} \Psi_n(\theta) d\theta =
	\int_{\underline \theta}^{\overline \theta} \frac{f({\bf x}, \theta)}{-k^2u_0({\bf x}, \theta)} \Psi_n(\theta) d\theta
	\label{3.11}
\end{equation}
and
\begin{align}
	\partial_{\nu} w_n({\bf x})  &= \int_{\underline \theta}^{\overline \theta} \partial_{\nu}\Big(\frac{u({\bf x}, \theta)}{-k^2 u_0({\bf x}, \theta)}  \Big) \Psi_n(\theta)d\theta \notag \\
	&= \int_{\underline \theta}^{\overline \theta} \frac{\partial_{\nu} u({\bf x}, \theta) u_0({\bf x}, \theta) - u({\bf x}, \theta)\partial_{\nu}u_0({\bf x}, \theta) }{-k^2 u_0^2({\bf x}, \theta)}\Psi_n(\theta) d\theta \notag\\
	&= \int_{\underline \theta}^{\overline \theta} \frac{g({\bf x}, \theta) u_0({\bf x}, \theta) - f({\bf x}, \theta)\partial_{\nu}u_0({\bf x}, \theta) }{-k^2u_0^2({\bf x}, \theta)} \Psi_n(\theta) d\theta. \label{3.12}
\end{align}
Due to \eqref{3.10}, \eqref{3.11} and \eqref{3.12},
the vector $W$ satisfies
\begin{equation}
\left\{
\begin{array}{ll}
	 \Delta W({\bf x}) + S^{-1}A({\bf x}) W({\bf x}) + S^{-1}{\bf B}({\bf x}) \cdot \nabla W({\bf x}) = 0 & {\bf x} \in \Omega,
	\\
	 W({\bf x})  = F({\bf x}) & {\bf x} \in \partial \Omega,\\
	 \partial_{\nu} W({\bf x}) = G({\bf x})
	 & {\bf x} \in \partial \Omega
\end{array}
\right.
\label{3.13}
\end{equation}
where $S = (s_{mn})_{m, n = 1}^N$, $A({\bf x}) = (a_{mn}({\bf x}))_{m, n = 1}^N$, ${\bf B}({\bf x}) = ({\bf b}_{mn}({\bf x}))_{m, n = 1}^N$ and
\begin{align}
	F({\bf x}) &= \displaystyle \Big(\int_{\underline \theta}^{\overline \theta} \frac{f({\bf x}, \theta)}{-k^2 u_0({\bf x}, \theta)} \Psi_n(\theta) d\theta\Big)_{n = 1}^N, \label{3.14}\\
	G({\bf x}) &= \displaystyle \Big(\int_{\underline \theta}^{\overline \theta} \frac{g({\bf x}, \theta) u_0({\bf x}, \theta) - f({\bf x}, \theta)\partial_{\nu}u_0({\bf x}, k) }{-k^2 u_0^2({\bf x}, \theta)} \Psi_n(\theta) d\theta\Big)_{n = 1}^N. \label{3.15}
\end{align}
The invertibility of $S$ is an important property of the basis $\{\Psi_m\}_{m \geq 1}.$ See Proposition \ref{prop MK} and the proof in \cite{Klibanov:jiip2017}.
Using other basis, for e.g the popular trigonometric basis, is not suitable because the corresponding $S$ is not invertible.
Solving Problem \ref{isp} becomes the problem of finding a function $W$ satisfying \eqref{3.13}.

\begin{remark}
	Using the truncation in \eqref{3.7} to derive \eqref{3.13} is inspired by the Garlekin approximation in the frequency domain.
	Studying the behavior of this approximation context as $N \to \infty$ is extremely challenging.
This is out of the scope of the paper.
Although \eqref{3.13} is not exact, the approximation is good enough for us to obtain out of expectation numerical results.
This phenomenon is true for a long list of inverse problems. Here is an incomplete list of inverse problems that were solved numerically by using similar truncation approaches.
We refer the reader to \cite{KlibanovNguyen:ip2019} for the $X$-ray tomography problem,
\cite{KlibanovAlexeyNguyen:SISC2019} for an inverse source problem for the full radiative transfer equation, \cite{VoKlibanovNguyen:IP2020, KhoaKlibanovLoc:SIAMImaging2020,  LeNguyen:preprint2021} for the inverse scattering problem,
\cite{Nguyen:CAMWA2020, Nguyens:jiip2020} for a coefficient inverse problem for parabolic equations, \cite{LeNguyen:2020} for an inverse source problem for nonlinear parabolic equations; \cite{LeNguyenNguyenPowell:JOSC2021} for an inverse source problem for hyperbolic equations. Especially, we refer the reader to \cite{NguyenLiKlibanov:2019} for an algorithm to solve an inverse source problem that is a particular case of Problem \ref{isp}. The inverse source problem in \cite{NguyenLiKlibanov:2019} only for the case when $u_0$ does  only on the wave number $k$ while in this paper, $u_0$ is allowed to depend on the spatial variable ${\bf x}$.
\end{remark}

Solving Problem \ref{isp} becomes finding a vector valued function $W$ that satisfies \eqref{3.13}. Since \eqref{3.13} is over-determined, we apply the quasi-reversibility method to solve it.
It is worth mentioning that the quasi-reversibility method was first introduced in \cite{LattesLions:e1969}. Then, it was used very often in solving over-determined boundary value problems, see e.g., \cite{Becacheelal:AIMS2015, Bourgeois:ip2006, Klibanov:jiipp2013, LeNguyenNguyenPowell:JOSC2021, LocNguyen:ip2019, NguyenLiKlibanov:2019}.
Having the solution $W = (w_1, \dots, w_N) \in H^2(\Omega)^N$ of \eqref{3.13} in hand, we can compute $w({\bf x}, \theta)$ for all $({\bf x}, \theta) \in \overline \Omega \times [\underline \theta, \overline \theta]$ via \eqref{3.7}. Then, we can compute the source function $p$ by using \eqref{3.3}.
This procedure is summarized in Algorithm \ref{alg 2}.

\begin{algorithm}[h!]
	\caption{\label{alg 2} A numerical method to solve Problem \ref{isp}}
	\begin{algorithmic}[1]
	\State Define the basis $\{\Psi\}_{n \geq 1}$ (see \cite{Klibanov:jiip2017}) and choose a cut-off number $N$ as in Remark \ref{rem_N}.
	\State\label{state_quasi} Solve \eqref{3.13} by the quasi-reversibility method (see Section \ref{s5}) for a vector value function $W^{\rm comp} \in H^2(\Omega)^N.$
	\State Compute
	\[
	    w^{\rm comp}({\bf x}, \theta) = \sum_{n = 1}^N w_n({\bf x}) \Psi_n(\theta)
	    \quad \mbox{for all } ({\bf x}, \theta) \in \overline \Omega \times [\underline \theta, \overline \theta].
	\]
	\State The source function is given by
	\begin{multline*}
	    p^{\rm comp}({\bf x}) = \frac{-1}{\overline \theta - \underline \theta} \int_{\underline \theta}^{\overline \theta}
	    \Big[
	        \Delta w^{\rm comp}({\bf x}, \theta)
	        + \Big[\frac{\Delta u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)} + k^2 c_0({\bf x})  \Big]w^{\rm comp}({\bf x}, \theta)
	        \\
	        + 2 \nabla w^{\rm comp}({\bf x}, \theta) \cdot \frac{\nabla u_0({\bf x}, \theta)}{u_0({\bf x}, \theta)}
	    \Big]d\theta
	\end{multline*}
	for all ${\bf x} \in \Omega.$
 	\end{algorithmic}
\end{algorithm}

The quasi-reversibility method used in Step \ref{state_quasi} of Algorithm \ref{alg 2} will be presented in Section \ref{s5}.

\section{The quasi-reversibility method} \label{s5}

For the convenience in the analysis, we consider in this section the boundary data $W = F$ and $\partial_{\nu} W = G$ respectively as the ``indirect data" of Problem \ref{isp}.
In fact, having the data of the inverse problem, see \eqref{1.2}, in hand, we can use the explicit formulas in \eqref{3.14} and \eqref{3.15} to compute $F$ and $G$.
Let $\delta > 0$ be a noise level.
Let $F^{\delta}$ and $G^{\delta}$ and $F^*$ and $G^*$ be the noisy and noiseless versions of $F$ and $G$ respectively.
The corresponding noiseless versions are denoted by $F^*$ and $G^*$.
By noise, we mean that there exists an ``error" function $\mathcal E \in H^2(\Omega)^N$ such that
\begin{equation}
    \left\{
    \begin{array}{l}
         \mathcal E|_{\partial \Omega} = F^\delta - F^*,\\
         \partial_{\nu} \mathcal E|_{\partial \Omega} = G^{\delta} - G^*,\\
         \|\mathcal E\|_{H^2(\Omega)} \leq \delta.
    \end{array}
    \right.
    \label{errorfn}
\end{equation}
\begin{remark}
    The existence of the error function in \eqref{errorfn} imply that the noise can be regularized; for e.g. \cite[Section 5]{AmmariWenjiaLoc:jde2013} for the noise model such that it can be smooth out. This condition is significant only for the proof of the convergence theorem (see Theorem \ref{thm_main} and its proof in \cite{NguyenLiKlibanov:2019}). However, in numerical study, we do not have to smooth the noise.
   In fact, we compute the noisy data $F^\delta$ and $G^{\delta}$ using formulas \eqref{3.14} and\eqref{3.15} with $f$ and $g$ replaced by $f^{\delta}$ and $f^{\delta}$ respectively. Here,
    \begin{equation}
        f^{\delta} = f^*(1 + \delta \mbox{rand}),
        \quad
        g^{\delta} = g^*(1 + \delta \mbox{rand})
        \label{noiserand}
    \end{equation}
    where $\mbox{rand}$ is a function taking uniformly distributed random numbers in $[-1, 1]$ and $f^*$ and $g^*$ are the noiseless versions of $f$ and $g$ respectively.

    \label{remnoise}
\end{remark}
Define the set of admissible solution
\begin{equation}
    H = \big\{
        V \in H^2(\Omega):
        V|_{\partial \Omega} = F^\delta \mbox{and }
        \partial_{\nu} V|_{\partial \Omega} = G^{\delta}
    \big\}.
    \label{H}
\end{equation}
We assume that the set $H \not = \emptyset.$
Due to the presence of noise in the given data, problem \eqref{3.13} is over-determined. It is natural to solve it using the quasi-reversibility method, which is similar to the least-squares optimization together with a Tikhonov regularization term. That means, we minimize the functional
\[
    J(W) =
    \int_{\Omega} \big|\Delta W + S^{-1} A({\bf x}) W({\bf x}) + S^{-1} {\bf B}({\bf x}) \cdot \nabla W({\bf x})\big|^2d{\bf x}
    + \epsilon \|W\|_{H^2(\Omega)^N}^2
\]
subject to the boundary conditions $W|_{\partial \Omega} = F^\delta$ and $\partial_{\nu} W|_{\partial \Omega} = G^{\delta}$.
\begin{remark}
    The presence of the Tikhonov regularization term is significant. In the theoretical part, it makes the functional $J$ coercive, which is important for the existence of a minimizer of $J$.
    In practice, we cannot obtain good numerical result without the presence of this Tikhonov regularization term.
    In our numerical study in Section \ref{s5}, we choose $\epsilon = 10^{-5}$. This value of $\epsilon$ was chosen by a trial and error process.
    We manually try many values of $\epsilon$ for test 1 until we obtain good numerical result. Then, we use this value for all other tests and for several noise levels $\delta \in [5\%, 50\%]$.
\end{remark}

We have the theorem.
\begin{theorem}
 Let $F^{\delta}$ and $G^{\delta}$ be the noisy boundary data for $W$. Here, $\delta > 0$ is the noise level in the sense of \eqref{errorfn}. Assume that the set $H$ defined in \eqref{H} is nonempty. Then, for any $\epsilon > 0$, the functional $J$ has a unique minimizer in $H$.
    Denote by $W_{\rm min}$ the obtained minimizer.
     Moreover, let $W^*$ be true solution to \eqref{3.13}. Then, the following estimate holds true
    \begin{equation}
        \|W^{\delta} - W^*\|_{H^1(\Omega)^N}^2 \leq C \Big(\delta^2 + \epsilon \|W^*\|_{H^2(\Omega)^N}^2\Big).
    \end{equation}
 \label{thm_main}
\end{theorem}

For brevity, we do not present the proof of Theorem \ref{thm_main}. We refer the reader to \cite[Theorem 3.1 and Theorem 5.1]{NguyenLiKlibanov:2019} for the proof of this theorem. We also refer to
\cite[Proposition 4.1 and Theorem 4.1]{Nguyen:CAMWA2020} for the proof of a similar theorem when $J$ involves the boundary integrals of the data.

\begin{remark}
	The quasi-reversibility method we employ here is based on least squares optimization with a special Tikhonov regularization term.
	In general, to solve the over-determined boundary value problem \eqref{3.13}, one can use the least squares optimization method with many other choices for the regularization term. It is worth to study the convergence of the method with each of such choice.
\end{remark}

\section{Numerical study}\label{sec_num}

For simplicity in implementation, we numerically solve Problem \ref{isp} in 2D.
Let $\Omega$ be the square  $(-1, 1)^2 \subset \mathbb{R}^2$.
Since solving the Helmholtz equation in \eqref{Heqn} on the whole space, to generate the simulated data, is challenging, we solve its approximation on $\Omega$, say
\begin{equation}
	\left\{
		\begin{array}{ll}
			\Delta u({\bf x}, \theta) + k^2 c_0({\bf x})u({\bf x}, \theta) = -k^2 p({\bf x}) u_0({\bf x}, \theta) &{\bf x} \in \Omega,\\
			\partial_{nu} u({\bf x}, \theta) - \ii k u({\bf x}, \theta) = 0 &{\bf x} \in \partial \Omega.
		\end{array}
	\right.
	\label{Heqn_appro}
\end{equation}
This change is acceptable in the sense that it does not effect the analysis in Section \ref{sec_Ana} because our arguments depend only on the form of the governing partial differential equation while the boundary values of $u$ and the flux $\partial_{\nu} u$ serve as the given data.
Due to the motivation of Problem \ref{isp} in Section \ref{motivation}, we choose $u_0$ as the incident wave $v_0$, defined in \eqref{incwave} where $k = 3\pi.$ For simplicity, we  choose $c_0 \equiv 1$
The range of the angle $[\underline \theta, \overline \theta] = [0, 2\pi].$
We solve \eqref{Heqn_appro} by the finite difference method.
We arrange a uniform $N_x \times N_x \times N_\theta$ grid of points in $\Omega \times [\underline \theta, \overline \theta]$ as
\begin{multline*}
    \Big\{
        (x_i, y_j, \theta_l): x_i = -1 + (i-1)d_x,  y_j = -1 + (j - 1)d_x,
        \\
        \theta_l = \underline \theta + (l - 1)d_\theta,
        1 \leq i, j \leq N_x
        1 \leq l \leq N_\theta
    \Big\}
\end{multline*}
where $d_x = 2/(N_x - 1)$ and $d_\theta = (\overline \theta - \underline \theta)/(N_\theta - 1)$. In our computational program $N_x = 80$ and $N_{\theta} = 250.$
Since solving \eqref{Heqn_appro} by the finite difference method to compute the solution $u({\bf x}, \theta)$ is standard in the scientific community, we do not describe the procedure here.
Having $u|_{\partial \Omega}$ and $\partial_{\nu} u$ in hand, we can compute the indirect data  $F|_{\partial \Omega}$ and $G|_{\partial \Omega}$ \eqref{3.14} and \eqref{3.15} respectively. The noisy data are as in Remark \ref{remnoise} and \eqref{noiserand}.

We now present an example of the choice of the cut-off number $N$ in Remark \ref{rem_N}. We choose $\Gamma = \big\{(x = 1, y): |y| \leq 1\big\} \subset \partial \Omega$. We numerically examine how the function $\sum_{n = 1}^N w({\bf x}, \theta)$ approximates the function $w({\bf x}, \theta)$ on $\Gamma \times [\underline \theta, \overline \theta]$ by testing their $L^{\infty}$ difference
\[
    \varphi(N) = \Big\| \sum_{n = 1}^N w_n({\bf x}) \Psi_n(\theta) - w({\bf x})\Big\|_{L^{\infty}(\Gamma \times [\underline \theta, \overline \theta])}.
\]
The graphs of $\varphi(N)$ for $N \in\{ 15, 25, 35\}$ are displayed in Figure \ref{figchooseN}.
It is evidence that $\varphi(N = 35)$ is sufficiently small. We choose $N = 35$ in all of our numerical tests.

\begin{figure}[h!]
	\subfloat[]{\includegraphics[width=.3\textwidth]{ChooseN15}}
	\quad
	\subfloat[]{\includegraphics[width=.3\textwidth]{ChooseN25}}
	\quad
	\subfloat[]{\includegraphics[width=.3\textwidth]{ChooseN35}}
	\caption{\label{figchooseN} The graph of the function $\varphi(N) = \|w({\bf x}, \theta) - \sum_{n = 1}^N w_n({\bf x}) \Psi_n(\theta)\|_{L^{\infty}(\Gamma \times [\underline \theta, \overline \theta])}$ where  $\Gamma = \{(x = R, y): |y| < R\}$ is a subset of $\partial \Omega$.
	In (A), $N = 15$ and $\varphi(15) = 21.3 \times 10^{-3}$
	In (B), $N = 25$, $\varphi(N) = 8.34 \times 10^{-3}$
	 In (C), $N = 35$, $\varphi(N) = 3.26 \times 10^{-3}$.
	 The data is taken from Test 4 below. In these graphs, the horizontal axis indicates the range of $y \in [-R, R]$ and the vertical axis represents the angle $\theta \in [0, 2\pi].$}
\end{figure}

In Step \ref{state_quasi} of Algorithm \ref{alg 2}, rather than solving \eqref{3.13}, we solve the equivalent problem
\begin{equation}
\left\{
\begin{array}{ll}
	 S \Delta W({\bf x}) + A({\bf x}) W({\bf x}) + {\bf B}({\bf x}) \cdot \nabla W({\bf x}) = 0 & {\bf x} \in \Omega,
	\\
	 W({\bf x})  = F({\bf x}) & {\bf x} \in \partial \Omega,\\
	 \partial_{\nu} W({\bf x}) = G({\bf x})
	 & {\bf x} \in \partial \Omega.
\end{array}
\right.
\label{4.2}
\end{equation}
We suggest this change to remove the inverse of the matrix $S$ in the system of PDE. The main reason is that the Gram-Schmidt procedure is unstable, so is $S^{-1}.$
The unstability of $S^{-1}$ might lead to some unnecessary difficulties in numerical experiments.
 The implementation of the quasi-reversibility method to solve \eqref{4.2} is very similar to that in \cite{Nguyen:CAMWA2020, Nguyens:jiip2020}.
 We do not repeat it here.
The implementation of other Steps of Algorithm \ref{alg 2} are straightforward.

We next show four numerical tests.

\noindent{\bf Test 1.} We test the case when the support of the true source function takes the form
\[
    p_{\rm true}(x, y) = \left\{
        \begin{array}{ll}
             2&\mbox{if }\max(|x - y|/0.8,|x + y|/0.35)  < 1, \\
             0& \mbox{otherwise}
        \end{array}
    \right.
    \quad \mbox{for all } (x, y) \in \Omega.
\]
The graphs of the true source function and its reconstructions are displayed in Figure \ref{fig_test1}.
\begin{figure}[h!]
    \centering
    \subfloat[]{\includegraphics[width=.3\textwidth]{p_true_Model1}}
    \quad
    \subfloat[]{\includegraphics[width=.3\textwidth]{p_comp_Model1_Noise5}}
    \quad
    \subfloat[]{\includegraphics[width=.3\textwidth]{p_comp_Model1_Noise10}}
    \caption{Test 1. The true and reconstruction of the source functions $p$. (A) The function $p_{\rm true}$. (B) The function $p_{\rm comp}$ computed from data containing $\delta = 5\%$ noise.
    (C) The function $p_{\rm comp}$ computed from data containing $\delta = 10\%$ noise.}
    \label{fig_test1}
\end{figure}
 In this test, the support of the true source function $p$ looks like a rectangle. It is evident that this rectangle is detected successfully.
 On the other hand, the maximum values of the source
are computed quite accurate. When $\delta = 5\%$,
$\frac{|\max_{{\bf x} \in \overline \Omega} p_{\rm comp} - \max_{{\bf x} \in \overline \Omega} p_{\rm true}|}{|\max_{{\bf x} \in \overline \Omega} p_{\rm true}|}  = 6.72\%$.
When $\delta = 10\%$, this error is $6.95\%$.

\noindent {\bf Test 2.} We next test the case when the value and the support of the true source are larger than those in Test 1.
The true source function is given by
\[
    p_{\rm true}(x, y) = \left\{
        \begin{array}{ll}
             4 & \mbox{if } \max\{|x + y|, |x - y|\} < 0.6\\
             0 & \mbox{otherwise}
        \end{array}
    \right.
    \quad \mbox{for all } (x, y) \in \Omega.
\]
The graphs of the true source function and its reconstructions are displayed in Figure \ref{fig_test2}.
\begin{figure}[h!]
    \centering
    \subfloat[]{\includegraphics[width=.3\textwidth]{p_true_Model2}}
    \quad
    \subfloat[]{\includegraphics[width=.3\textwidth]{p_comp_Model2_Noise10}}
    \quad
    \subfloat[]{\includegraphics[width=.3\textwidth]{p_comp_Model2_Noise30}}
    \caption{Test 2. The true and reconstruction of the source functions $p$. (A) The function $p_{\rm true}$. (B) The function $p_{\rm comp}$ computed from data containing $\delta = 10\%$ noise.
    (C) The function $p_{\rm comp}$ computed from data containing $\delta = 30\%$ noise.}
    \label{fig_test2}
\end{figure}
The support of true source function $p_{\rm true}$ is the square centered at the origin, rotated $45^\circ$. We can see that reconstruction this square is out of expectation. The maximum values the source function inside the square is quite correctly computed.
When $\delta = 10\%$, $
\frac{|\max_{{\bf x} \in \overline \Omega} p_{\rm comp} - \max_{{\bf x} \in \overline \Omega} p_{\rm true}|}{|\max_{{\bf x} \in \overline \Omega} p_{\rm true}|}= 5.13\%$.
When $\delta = 30\%$, this error is $5.44\%$.

\noindent{\bf Test 3.} We next test the case when the graph of the true source has a void.
The true source function in this test is
\[
    p_{\rm true}(x, y) = \left\{
        \begin{array}{ll}
             4 &  \mbox{if } 0.4^2 < x^2 + y^2 < 0.8^2\\
             0 & \mbox{otherwise}
        \end{array}
    \right.
    \quad \mbox{for all } (x, y) \in \Omega.
\]
The graphs of the true source function and its reconstructions are displayed in Figure \ref{fig_test3}.
\begin{figure}[h!]
    \centering
    \subfloat[]{\includegraphics[width=.3\textwidth]{p_true_Model3}}
    \quad
    \subfloat[]{\includegraphics[width=.3\textwidth]{p_comp_Model3_Noise5}}
    \quad
    \subfloat[]{\includegraphics[width=.3\textwidth]{p_comp_Model3_Noise50}}
    \caption{Test 3. The true and reconstruction of the source functions $p$. (A) The function $p_{\rm true}$. (B) The function $p_{\rm comp}$ computed from data containing $\delta = 5\%$ noise.
    (C) The function $p_{\rm comp}$ computed from data containing $\delta = 50\%$ noise.}
    \label{fig_test3}
\end{figure}
The support of true source function $p_{\rm true}$ is the ring centered at the origin with outer radius $0.8$ and inner radius $0.4$.
This test is interesting due to the presence of the void.
We can see that reconstruction this ring is acceptable. The maximum values the source function inside the square is quite correctly computed.
When $\delta = 5\%$, $\frac{|\max_{{\bf x} \in \overline \Omega} p_{\rm comp} - \max_{{\bf x} \in \overline \Omega} p_{\rm true}|}{|\max_{{\bf x} \in \overline \Omega} p_{\rm true}|}  = 6.35\%$.
When $\delta = 50\%$, this error is $8.13\%$.

\noindent{\bf Test 4.}
We next consider a more interesting case. The function $p_{\rm true}$ is the characteristic function of the letter $Y$. Although this true source function has complicated structure, we are able to well compute it.
The graphs of the true source function and its reconstructions are displayed in Figure \ref{fig_test4}.
\begin{figure}[h!]
    \centering
    \subfloat[]{\includegraphics[width=.3\textwidth]{p_true_Model4}}
    \quad
    \subfloat[]{\includegraphics[width=.3\textwidth]{p_comp_Model4_Noise10}}
    \quad
    \subfloat[]{\includegraphics[width=.3\textwidth]{p_comp_Model4_Noise50}}
    \caption{Test 4. The true and reconstruction of the source functions $p$. (A) The function $p_{\rm true}$. (B) The function $p_{\rm comp}$ computed from data containing $\delta = 10\%$ noise.
    (C) The function $p_{\rm comp}$ computed from data containing $\delta = 50\%$ noise.}
    \label{fig_test4}
\end{figure}
We can see that reconstruction the letter $Y$ is acceptable. The maximum values the source function inside the square is quite correctly computed.
When $\delta = 10\%$, $\frac{|\max_{{\bf x} \in \overline \Omega} p_{\rm comp} - \max_{{\bf x} \in \overline \Omega} p_{\rm true}|}{|\max_{{\bf x} \in \overline \Omega} p_{\rm true}|}  = 0.83\%$.
When $\delta = 50\%$, this error is $1.92\%$.

\begin{remark}
These numerical examples numerically show that Algorithm \ref{alg 2} is robust. Especially, it is  stable with respect to the noise. We can obtain good numerical results even when the noise level is up to $50\%$.
\end{remark}

\section{Concluding remarks}\label{sec_concl}
In this paper, we solve an inverse source problem.
This inverse source problem is the linearization of the highly nonlinear and severely ill-posed inverse scattering problem.
In order to solve the inverse source problem, in the first step, we derive a system of linear elliptic PDEs in which the source function is absent.
We solve this system by the quasi-reversibility method.
We choose the quasi-reversibility method because its convergence as noise tends to $0$ was proved.
The efficiency of our method is confirmed by some interesting numerical examples.

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text:
Author: Andrew Mak
Disciplinary and Regulatory Proceedings in Hong Kong was the first book of its kind, and is a leading reference for anyone wishing to research and understand Hong Kong's law relating to disciplinary proceedings.
by Andrew Mak (Author)
Book (Hardback)
Published: 26 March, 2019
Since the last edition in 2015, the law in disciplinary and regulatory action has developed some new trends and some new grounds, notably in the area of preliminary investigation and decision to prosecute, misconduct in public office, changes in modus in operations such as forming of limited companies to operate group practice, the more emphasis in a gradual shifting in standards of conduct due to changing background such as the #metoo movement, and new developments elsewhere in particular in the UK.
Andrew Mak
Andrew Mak Andrew practises mainly in commercial and property litigation. He has appeared regularly in trials and appeals in the High Court. Before 1997 he was led to appear in the Privy Council case of Silver Mountain Investments Ltd v Attorney General of Hong Kong [1994] 1 WLR 925. He is also active in international commercial arbitration both in Hong Kong and in Mainland PRC. He has sat as Chief Arbitrator in the Shenzhen Court of International Arbitration. He is a member of panels of arbitrators, including the Hong Kong International Arbitration Centre (HKIAC), Shenzhen (SAIC), Panel of Arbitrators of SCIA Financial Arbitration Centre, Shanghai (SHIAC), Jinan Arbitration Commission, Tianjin Arbitration Commission, Nanjing Arbitration Commission and Malaysia (KLRCA). He is also a Chinese legal consultant of the PRC in Shanghai. His legal opinion has been accepted by the PRC courts. He gives lectures at the Peking University, the East China University of Politics and Law (formerly the St John University), and the Shanghai University of Finance and Economics. His major professional works include "The Law of Injunction in Hong Kong" 2nd Ed, (2018), Conveyancing & Property Litigation in Hong Kong 2nd Ed (2019), Disciplinary & Regulatory Proceedings in Hong Kong, 3rd Ed, (2019). He is also contributor to Halsbury's Laws of Hong Kong Vol 1 Administrative Law, 2nd Ed (2014), Atkin's Court Forms Hong Kong (looseleaf) Administrative Law, Emden's Construction Law in Hong Kong Chapter 10, Butterworths Hong Kong Building Law Handbook 4th Ed, Butterworths Hong Kong Statutory Interpretation Handbook (3rd Ed).
View all books by Andrew Mak (5)
Andrew`s Bestseller
Law of Injunctions in Hong Kong – Third Edition
Publication Date: 2019-12-16 00:00:00
The Professional Conduct of Lawyers in Hong Kong
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Dr Pourghadiri has been practising orthodontics exclusively since 1995. He has published in the British Dental Journal and presented research at scientific meetings. He is a referee and reviewer for papers and books submitted to the European Journal of Orthodontics.
He is on the Specialist Register of the General Dental Council; is a member of the British, European and American Orthodontic Societies; and is a Fellow of the World Federation of Orthodontists.


text:
Home > Travel > Hill Stations in India > Matheran
Matheran, Maharashtra, Hill Stations in India
Matheran, located at a distance of 105 kilometres from Mumbai, is one of the finest hill resorts of Maharashtra. Surrounded with a number of other tourist spots it is one of the most picturesque sights in Maharashtra.
Matheran is one of the splendid hill stations of Maharashtra located at the top of the Sahyadri hills in Western Ghats. Surrounded with greenery and pollution free atmosphere it is one of the most attractive sights in Maharashtra. Set in the lap of nature this is probably one of the smallest hill stations of India which is known for maintaining its pastoral beauty in a pollution free atmosphere. This Hill station is surrounded with some popular sights that provide great opportunity for the visitors to have a splendid view of the surroundings.
Location of Matheran
Matheran is located at a height of 803 meters which is 2636 ft (approx) which covers an area of 8 sq. kilometres above the Sahyadri hills. Surrounded with a number of sights this is one of the most celebrated sights of Maharashtra.
History of Matheran
The lonely sight of Matheran was discovered in 1850 by Hugh Poyntz Malet, the collector of Thane district. While he was busy in campaigning around the hill station he once halted on the hill top to discover one of the most picturesque sights in Maharashtra. His temptation forces to select the area and develop it as a hill station where he and others can come for relaxation. Matheran grew more accessible and popular under the patronage of Lord Elphinstone, the governor of Mumbai. He visited the place in 1855 and got mesmerised with its scenic beauty. He selected a beautiful sight for building a bunglow which was named after him as Elphinstone Logde. Under his patronage connecting roadways were also created. Along with a serpentine railway track this place has been connected to Mumbai and other important places. The splendid ride on the toy train passes through the spectacular sights of grassy hills sides, plains, sheer cliffs, plateau and thick forest cover. This excites the visitors towards the picturesque sights of Matheran.
Climate of Matheran
Matheran enjoys one of the most pleasant atmospheres. Temperature in Matheran varies from 16 degree Celsius in winters to 32 degree Celsius in summer with an average rainfall of 524 cm. The best season to visit Matheran is between October to May. As the Sahyadri hills receive the brunt of Monsoon clouds this keeps the hill station covered with greenery. The hill station is at its best during the end of monsoons. The long rainy season of Matheran enriches the region with breathtaking waterfalls, mist covered valleys and dew dripped greenery along with floating clouds which make it one of the picturesque sights in Maharashtra.
Culture of Matheran
People of Matheran wear some suitable cloths which are comfortable according to the atmosphere with long summers and light winters. With some common languages like Marathi, Hindi and English the people continue with some local business on Sweet Chikki, local honey, some beautifully carved footwear with unique cane goods which also support the economy of the region.
Attractions of Matheran
Matheran is all about the views. It is Asia's smallest hill station. Following are the popular places of Matheran:
Louisa point: Louisa Point in the middle of Matheran, is the oldest spot with an English name and lies on the southernmost tip of Matheran's westward range. The sight of the lake gardens and the beautiful waterfalls from this point are amazing. The view of the grand Lion's Head is a popular site that resembles the head of a lion.
Panorama point: This picturesque point offers 360-degree panoramic views of the Western Ghats and the green plains spread with villages far below. Panorama Point offers beautiful sunset and sunrise views.
Monkey point: This unique destination overlooks beautiful scenery of the Western Ghats and its intimidating mountains and deep gorges. It is famous for its mesmerising sunsets and many monkeys.
Charlotte Lake: It is also known as Sharlott Lake. It cleanly provides a pleasing view with flourishing green surroundings. Echo point and Louisa point are other attractions around this place. Pisarnath temple nearby is another attraction of Matheran.
Shivaji's ladder: Downwards from one tree hill, the steps are in the shape of a ladder that mange to face up to even the most enthusiastic of trekkers.
Porcupine point: Porcupine point presents a great sight. Close to the point is Prabalgad fort which can be seen from here.
Echo point: This photogenic point is famous for the reverberations and echoes it generates. The peak also gives a bird's eye view of the entire region which is a sight for sore eyes. Carpeted with a blanket of green grass, the mountains of Sahyadri are a sight to observe.
Alexander Point: The Alexander Point is a beautiful valley flourishing with natural scenic beauty. This place offers captivating panoramic views of the surrounding destinations such as Palasdari Lake, Rambaug Point, Garbut Point and Chowk Point.
Honeymoon Hill: A celebratory destination to a Britishers and the Parsis, back in history. It is another attraction of Matheran.
Prabalgad Fort: Now only the remains and ruins of the construction can be seen. One can find himself wrapped in a historic attraction. The fort was itself captured by Shivaji Maharaj from the Mughals.
Other popular places of Matheran includes Olympia, Rambagh point, Little Chowk Point and Hart point which make Matheran a popular hill station of India.
Visiting information of Matheran
Matheran is well connected to Mumbai and Pune by rail and road, while Neral is the nearest rail station. Inside Matheran, the available transport facilities are horses and hand-pulled rickshaws. The nearest airport is Chhatrapati Shivaji International Airport, Mumbai that is located around 100 kilometres away from Matheran.


text:
Croteam | February 2017 - Croteam
New update for SSVR: The First Encounter is live
Achievements, languages, Linux and "blink" teleport mode for VR are now fully supported in the game, as well as keyboard + mouse and gamepad support (as promised).
The full changelog for the latest update is available on Steam forums, here.
Oh, and while we're at it, we also updated the prices on our merch store and everything is at 50% off until February 21. Check it out!
Hallelujah! Another free update is out now!
Serious Wednesday update!
A free update for all Serious Sam owners, Serious Fusion 2017, is coming. Also: SteamOS/Linux/OSX support, more VR games and The Last Hope "Shanti" update are on the way!
New planet, new guns, new enemy type, new features AND a new boss. Shaanti update for SSVR: The Last Hope is coming!
Good news everybody! A great PR team could milk this list for an entire month or more, but since our PR team (and in reality it's just me, the bald middle aged guy writing this. Hi!) never claimed to be any good we're just going to drop this right here.
It's been an incredibly exciting time for us as we launched our first VR title and our first game in Early Access. We were amazed with the amount of positive comments, great ideas and new features that our fans suggested. We were also amazed by our own stupidity and naivety. We tried to implement, well, almost all of those ideas.
While co-op was and remains one of the often requested features, the truth is that a fairly small percentage of players actually plays in co-op. But we said we'd listen, so we rushed to implement this feature and were very proud that we actually pulled it off. Serious Sam VR: The Last Hope in co-op requires serious skill and teamwork at harder difficulty settings and the game is even more fun when played with a friend. The thing is, the decision to implement co-op straight away made a joke of our timetable for regular content updates and this left us with more and more fans requesting a status update on the game (we DO have one today).
Note to self: stick to the original GDD and ship planned features first, collect Early Access feedback and implement those ideas later.
Case in point: one of the most requested features from VR fans was actually moving around the map and exploring. Since The Last Hope movement was on a room-scale level and the whole game is essentially an arcade-like experience and a love letter to old machines, adding full locomotion to the game meant that we'd have to start from scratch, ditch or re-make most of the levels and basically abandon the original idea altogether. Instead, we started experimenting with VR and prototyping movement in The First Encounter and made surprisingly good progress. So good, in fact, that we felt confident that the game was *almost* ready for VR. The decision was made to bring this game to Early Access as well.
The good: we couldn't have possibly predicted all of the preferred ways of locomotion, but once more we got amazing feedback and added even more ways to move around the virtual world. And comments like this kept us moving.
The bad: We now have two games in Early Access, both set in Serious Sam universe, which seems to create some confusion among fans and it's our fault. To make things right and to make it perfectly clear that those are two very different games, we came up with a brilliant plan (if we do say so ourselves): we'll just add two or three more games into the VR mix! That way there can be no doubt: we remain committed to VR and having fun with this amazing new tech! Of course, we'll first ship updates and work faster to complete the existing games in Early Access. Our VR teams were recently expanded and today we're announcing a tech update that will enable us to easily ship updated code more regularly!
Finally, it's been a really long time since somebody asked about Serious Sam 4 in the comments section, but we'll go ahead and say it anyway: yes, it's still happening.
Oh, right, here's the news:
15th anniversary of Serious Sam: The Second Encounter
So… almost a year after the release of The First Encounter, on February 5th, 2002 (though some shops in the UK started selling the game two weeks earlier!), we released Serious Sam: The Second Encounter worldwide. We worked even harder on that one, as TFE was received with such critical acclaim that we didn't want to disappoint our fans, so the sequel had to be even better than the original.
Naturally, in order to fill shoes that big, we had to bring more people on board. The first two to join the party and increase our number to a hearty dozen were Ivan Mika and Nikola Mosettig. Ivan was hired as a level designer, while Nikola ended up in the programming department, though he wanted to be a designer as well… Fifteen years later, and they are both still with us, we just can't get rid of them (and Nikola is still not a level designer) 🙂
Here is what they have to say:
"The first level I built was inspired by Teotihuacan, an ancient Mesoamerican city located in a huge flat valley. I was so excited that I was given a chance to work on it, that I just worked and worked and worked on it, day and night until it ended up being so big, we had to split it into three levels in the end (which was exactly 1/4 of the full game)! It also contained some experiments I did with jump pads and gravity that turned out to be some of the most memorable experiences in the game (and probably the most difficult and hated ones)."
Ivan Mika, Level designer for TSE (today: Level design lead)
"During the development of TSE, we had some issues with framerate, mostly in our newly built particle system. I managed to single-handedly DOUBLE our framerate just by deleting a duplicate line of code that was making everything render twice (which I introduced into code in the first place)!"
Nikola Mosettig, Content programmer (today: CIO)
By the end of the development cycle, we had to make decisions regarding retail box. Our publisher sent us box cover art, and boy, we had so much to say. Turned out we were short on time and most of our remarks ended up in "can't be done now" category.
Sometime in that period, we managed to rent a much bigger office, where we had so much space that we didn't know what to do with it, but we knew we liked it! So we brought even more people aboard, to prepare working on XBOX edition that will include both TFE and TSE. But that is another story…
Here are few pics and concept art from that era, hope you'll enjoy them
Serious Sam 4
Genre: Action, Indie,
Publisher: Devolver Digital
Release Date: Sep 24, 2020
Serious Sam 4 reignites the classic FPS series in a high-powered prequel loaded with an explosive arsenal, intergalactic carnage, and perfectly timed one-liners.
Official Steam Group
Official Twitter Profile
BUY NOW ON STEAM


text:
Martin Ferris To Attend Siamsa Event In Honour Of Royal Couple
Deputy Martin Ferris.
MARTIN Ferris will attend the event in honour of the Prince of Wales and Duchess of Cornwall in Tralee this Friday.
The Kerry Sinn Féin TD has said that he has accepted an invitation to attend the event in Siamsa Tíre which is being hosted by Kerry County Council.
Accepting the invitation, Deputy Ferris said: "While this decision may not meet everyone's approval, I came to the conclusion, that as we move closer and closer to the eventual re-unification of our country, Republicans must continue the process of reconciliation between the different traditions on our island and with our nearest neighbours.
"When Martin McGuinness took the historic decision to meet with Queen Elizabeth it was rightly praised by most as an important step towards national reconciliation and I hope that my decision is seen in the same light."
#Royal visit #siamsa tire
Siamsa Tíre To Put On A Performance For Prince Charles And Duchess Of Cornwall
PHOTOS: Great Turnout At Siamsa For Deirdre's Album Launch
Bosco Is Coming To Siamsa Tíre In January
Siamsa Tíre To Host Exhibition Of Local Artists Work This Friday
Well Martin Ferris as "we move closer and closer to the eventual re-unification of our country" or are we moving closer and closer to United Kingdom as in a united Ireland there will be a sizable unionist community both north and south.
Ireland is no longer so mono cultural – Ireland is more and more multi cultural and as such less and less are patriotic to Ireland and it may be Ireland fate to be voted back into a United Kingdom in the future!
All it take is a strong unionist party in Dáil Éireann with a referendum to be put to the new multi cultural people of Ireland.
Sinn Féin have become confused to the difference between liberal left democracy and national republicanism and that will one day seal the Irish republics fate.
Nat that it matters as we are only an EU state at present, so why not a British one!


After being in a car accident, Michelle woke up in a bunker underground with two men. She learned that there was a chemical attack, and that the best thing for them is to stay put. However, because of Howard's menacing nature, she escapes the bunker and finds out the truth about the outside world.


text:
How Historically Accurate is season 1 of Versailles?
150 bytes added, 20:03, 26 June 2017
This article contains spoilers
[[File:The-BBC-confirmed-the-news-to-fans-tonight-619209.jpeg|thumbnail|left|Versailles Season 1 image.]]
[https://www.amazon.com/gp/product/B01LZJ13QN/ref=as_li_tl?ie=UTF8&camp=1789&creative=9325&creativeASIN=B01LZJ13QN&linkCode=as2&tag=dailyh0c-20&linkId=556c34932c533cb589b4a4f1a60e0d64 Versailles] This article contains spoilers. Check out our article on the historical accuracy of the 1st season of [[How Historically Accurate is season 1 of Versailles?|Versailles]]. Season 1 follows the French king Louis XIV, the so-called Sun King, during the early years of his reign when he commissioned the construction of the palace of Versailles, a former hunting lodge. Construction of the palace began in 1661 and lasted, on and off, until 1715. Despite the objection of much of the nobility and even his court, Louis was determined to make the palace the greatest in the world and solidify his rule. Although later Louis was known as a strong and respected ruler in Europe, the early years of his reign were fraught with problems with the nobility, wars with the low countries, conspiracies, and ongoing conflict between Catholics and Protestants. It is this background that sets up the story in the first season.
==History and Key Events==
Automoderated users, Bots, Bureaucrats, checkuser, Moderators, Administrators
Retrieved from "https://dailyhistory.org/Special:MobileDiff/8553"


Read OANDA forex broker review, rating and detail information for their services below. OANDA is a forex broker based in San Francisco, California that provides forex trading services around the globe, established in the mid 1990s. The company is said to be founded by Dr Michael Stumm and Dr. Richard Olsen. When the online trading platform FXTrade was launched in 2001, it was the first fully automated forex trading platform. OANDA offers trading on commodities, bonds and stock indices in addition to forex. It uses its own fxTrade platform and offers a companion app. The program lacks some of the features we look for but is still an above-average trading platform. OANDA has no commissions and uses the bid/ask spread as a broker markup.
OANDA is a trustworthy company with strong regulation. To open an account with OANDA doesn't require a minimum deposit and has no minimum lot size. Because of this, OANDA becomes too popular among small retail traders. OANDA has many powerful charting tools and research material for their clients. Most noteworthy, they also have a great user friendly web trading platform. As a result, OANDA has almost 20% share of the retail forex trading market and continue to grow.


Jodi Lynnae, Psychic Medium and Clairvoyant, discusses how spirits communicate with us including how to know or recognize when a spirit is trying to communicate.
Host: How do I know or recognize when a spirit is trying to communicate? Jodi Lynnae: I would say the most important thing for a client to recognize how and when a spirit might be communicating with them, is to first open themselves up to realm of possibility that, hey maybe that light bulb thats burned out three times in two days has something more to it rather than just the logical explanation. Maybe that radio that comes on by itself isnt just a wiring default; maybe its actually a spirit trying to communicate and manipulate those devices to get our attention. The most important thing is learning how to quiet the mind, so that you are open and receptive; so that you will not explain these little oddities away with logical reason.


The hubby is healing well, from the broken humorous he sustained last week. It looks like we will be taking it easy for several weeks, as it could be 8-12 weeks before he has full use of his arm back. It will definitely, still be in the sling and requiring care, while in New York the beginning of June. The rest of the week went well and we are getting use to a new schedule, with only having one fully functional adult.
I am really happy with these pre-planned, pre-cooked meals, that I can do on the weekend to help during the week. This week Cuban beans and rice, Potato Quiche, and the meatball pasta bake were made ahead of time. I will be pre-preparing the lunches for the hubby so he can eat during the day when I am not around.
My son is very helpful and caring. He also does amazingly entertaining himself in the backyard.
In keeping with making things simpler this week, here's the meal plan.
This entry was posted in Uncategorized and tagged dinner, food, meal plan, recipes, vegetarian on April 28, 2013 by mlwilkie.
Pinning this now as how I hate to meal plan! 🙂 Thanks!


text:
package judge

import (
	"sort"

	"github.com/chapsuk/frissgo/config"
	"github.com/chapsuk/frissgo/github"
)

type byWeight []*topItem

func (w byWeight) Len() int           { return len(w) }
func (w byWeight) Swap(i, j int)      { w[i], w[j] = w[j], w[i] }
func (w byWeight) Less(i, j int) bool { return w[i].weight > w[j].weight }

type estimatorFunc func(iss *github.Issue) *topItem

type Category struct {
	Name      string
	Coeff     int
	estimator estimatorFunc
	top       *top
}

func newCategory(strategy *config.Strategy, cfg config.Category) *Category {
	return &Category{
		Name:      cfg.Name,
		Coeff:     cfg.Coeff,
		top:       newTopChart(cfg.Size),
		estimator: createEstimatorFunc(strategy, cfg),
	}
}

func (c *Category) GetIssues() []*github.Issue {
	result := make([]*github.Issue, 0, len(c.top.items))
	sort.Sort(byWeight(c.top.items))
	for _, t := range c.top.items {
		result = append(result, t.issue)
	}
	return result
}

func createEstimatorFunc(strategy *config.Strategy, cfg config.Category) estimatorFunc {
	return func(iss *github.Issue) *topItem {
		item := &topItem{
			issue: iss,
		}

		if cfg.Issues.Activity != 0 {
			item.weight += cfg.Issues.Activity * *iss.Iss.Comments
		}

		if cfg.Issues.Author != 0 {
			if strategy.IsPriorAuthor(*iss.Iss.User.Login) {
				item.weight += cfg.Issues.Author
			}
		}

		if cfg.Issues.Reactions.HasCoeff() {
			item.weight += cfg.Issues.Reactions.Total * *iss.Iss.Reactions.TotalCount
			item.weight += cfg.Issues.Reactions.Plus * *iss.Iss.Reactions.PlusOne
			item.weight += cfg.Issues.Reactions.Laugh * *iss.Iss.Reactions.Laugh
			item.weight += cfg.Issues.Reactions.Hooray * *iss.Iss.Reactions.Hooray
			item.weight += cfg.Issues.Reactions.Heart * *iss.Iss.Reactions.Heart
			item.weight += cfg.Issues.Reactions.Minus * *iss.Iss.Reactions.MinusOne
			item.weight += cfg.Issues.Reactions.Confused * *iss.Iss.Reactions.Confused
		}

		if cfg.Comments.Activity != 0 {
			item.weight += cfg.Comments.Activity * len(iss.Comments)
		}

		if cfg.Comments.Author != 0 || cfg.Comments.Reactions.HasCoeff() {
			for _, comm := range iss.Comments {
				if cfg.Comments.Author != 0 {
					if strategy.IsPriorAuthor(*comm.User.Login) {
						item.weight += cfg.Issues.Author
					}
				}

				if cfg.Comments.Reactions.HasCoeff() {
					item.weight += cfg.Comments.Reactions.Total * *comm.Reactions.TotalCount
					item.weight += cfg.Comments.Reactions.Plus * *comm.Reactions.PlusOne
					item.weight += cfg.Comments.Reactions.Laugh * *comm.Reactions.Laugh
					item.weight += cfg.Comments.Reactions.Hooray * *comm.Reactions.Hooray
					item.weight += cfg.Comments.Reactions.Heart * *comm.Reactions.Heart
					item.weight += cfg.Comments.Reactions.Minus * *comm.Reactions.MinusOne
					item.weight += cfg.Comments.Reactions.Confused * *comm.Reactions.Confused
				}
			}
		}

		return item
	}
}

redpajama_set_name: RedPajamaGithub


Soccer fans around the world have caught World Cup fever. The bug has even bit us here at Community Health Network. Many of our employees gather at Indy Eleven watch parties around Indianapolis to cheer on Team USA.
The high-intensity games played at the World Cup are a thrill to watch, but can make players susceptible to common soccer injuries. Despite the training these professional athletes undergo, injuries still occur, as we witnessed in the United States first match against Ghana. Jozy Altidore suffered a hamstring injury and Clint Dempsey left the game with a broken nose.
So, we asked Dr. Michael DaRosa, sports medicine physician and team physician to the Indy Eleven for Community, to share the most common injuries in soccer.
"A hamstring injury like the one Altidore experienced is actually quite common," said DaRosa. "Also seen often in the sport of soccer are ankle sprains, knee sprains and concussions."
Perform the proper strengthening, conditioning, and flexibility exercises before playing.
Use the correct heading technique.
Wear ankle braces if you have previous ankle sprains.
Undergo the proper rehab or physical therapy if you have an injury.
Wear equipment that fits correctly.
To learn more about Community Sports Medicine, visit our website or call 800-777-7775.


text:
Publisher Animal and Plant Health Agency Arts Council England Ashford Borough Council Barrow Borough Council British Geological Survey (BGS) Broxtowe Borough Council Cabinet Office Calderdale Metropolitan Borough Council Cambridgeshire Insight Centre for Environment, Fisheries & Aquaculture Science Centre for Environmental Data Analysis Cheltenham Borough Council Cherwell District Council Chesterfield Borough Council City of Bradford Metropolitan District Council City of London City of York Council Commission for Architecture and the Built Environment Cornwall Council Craven District Council Dartmoor National Park Authority Department for Business, Energy and Industrial Strategy Department for Children, Schools and Families Department for Education Department for Environment, Food and Rural Affairs Department for Social Development Department for Transport Department for Work and Pensions Department of Agriculture and Rural Development (Northern Ireland) Department of Energy and Climate Change Department of Health and Social Care Department of Justice (Northern Ireland) Driving Standards Agency Education and Skills Funding Agency EngagedX Environment Agency Environmental Information Data Centre Exmoor National Park Authority Financial Conduct Authority (FCA) Food Standards Agency Foreign and Commonwealth Office Foreign, Commonwealth and Development Office (FCDO) Forestry Commission Government Digital Service Greater London Authority Greater Manchester Ecology Unit Guildford Borough Council HM Courts Service HM Land Registry HM Treasury Healthcare Quality Improvement Partnership Her Majesty's Revenue and Customs High Peak Borough Council High Speed 2 Limited Higher Education Statistics Agency Highways England Home Office Horsham District Council ISD Scotland Innovate UK Ipswich Borough Council Joint Nature Conservation Committee Lake District National Park Authority Leeds City Council Lincolnshire County Council London Borough of Barnet London Borough of Camden London Borough of Hackney London Borough of Lambeth London Borough of Sutton Malvern Hills District Council Marine Environmental Data & Information Network Marine Management Organisation Met Office Mid Sussex District Council Ministry of Defence Ministry of Housing, Communities and Local Government Ministry of Justice NHS Digital NHS England National Records of Scotland Natural England Natural Resources Wales New Forest National Park Authority Newcastle City Council North Hertfordshire District Council North Somerset Council North York Moors National Park Authority North Yorkshire County Council Northern Ireland Spatial Data Infrastructure Northern Ireland Statistics and Research Agency Northumberland County Council Nottingham City Council Ofcom Office for National Statistics Office of Rail and Road Oil and Gas Authority OpenDataNI Ordnance Survey Peak District National Park Authority Peterborough City Council Plymouth City Council Public Health England Rochdale Borough Council Royal Borough of Kensington and Chelsea Runnymede Borough Council Rural Payments Agency Scottish Government Scottish Government SpatialData.gov.scot Service Personnel and Veterans Agency Skills for Care Snowdonia National Park Authority South Downs National Park Authority Teignbridge District Council The Food and Environment Research Agency The Social Investment Business Group Thurrock Council Transport Scotland Transport for Greater Manchester Tunbridge Wells Borough Council United Kingdom Hydrographic Office University of Edinburgh Valuation Office Agency Veterinary Laboratories Agency Warwickshire County Council Waverley Borough Council West Dorset District Council Wiltshire Council Wycombe District Council Yorkshire Dales National Park Authority
Format CSV ESRI REST GEOJSON HTML KML OGC WMS WFS WMS ZIP
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Mean house price
Average house price (with regular updates) Source: Land Registry Publisher: Communities and Local Government (CLG) Geographies: Local Authority District (LAD), Government Office Region (GOR),...
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Quarterly survey of volume and value of sales to non-domestic consumers, by sizeband, for major domestic electricity and gas suppliers.
Quarterly Energy Prices
Quarterly data and statistics on energy prices to domestic and industrial consumers for all the major fuels. Source agency: Business, Energy and Industrial Strategy Designation: National...
Weekly prices of unleaded petrol and diesel Source agency: Business, Energy and Industrial Strategy Designation: National Statistics Language: English Alternative title: road fuel prices
Stamp duty land tax: Property prices
Estimated numbers of transactions, value of transactions and total yield attributable to residential and non residential land and property Source: HM Customs and Revenue Publisher: HM Customs and...
Timber Price Indices: Data to September 2018
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HM Land Registry Price Paid Data tracks land and property sales in England and Wales submitted to us for registration.
High quality IT delivery Performance Report
Performance Report on High quality IT delivery. HMRC's objective is to transform the performance of the Department through the exploitation of information and technology services.
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There are a large number of Housing spreadsheets that provide the latest, most useful or most popular data, presented by type and other variables, including by geographical area or on a temporal...
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We'll market your business and equipment using the KWIPPED website, email database and social media presence.
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As a member of the KWIPPED Supplier Network, you will optimize rental profitability, tap into leasing opportunities, reduce your overhead and grow your business!
KWIPPED RentalWatch offers a preview of live opportunities on the KWIPPED platform.
As a supplier partner, you would have an opportunity to acquire these customers.
We are a very established tattoo parlor in Arizona. We are interested in either a long term rental or PURCHASING (via lease through KWIPPED) a Laser tattoo removal system for our business. We are researching options at this stage and would appreciate recommendations and prices so we can determine if we can do this. Thank you!
I am a figure skating club in Westborough, MA, and am looking to RENT an electric 3 Tray Oven to heat frozen trays of food for a large event I am hosting over a four day period in October. If you have a Alto-Shaam ASC-2E or similar model that can heat frozen food to consumable temperature, that will be fine. Please submit quotes through your KWIPPED Supplier Account. Please message customers with questions. Thank you.
Needing to get some pricing for an event where we would conduct complimentary blood pressure screenings.
I am with a university research center. I'm looking into the costs of renting a CT scanner for some preclinical studies I'm involved in. Would require a scanner for 3 weeks, although at this time I'm unsure of the exact dates (estimated to be in October). In addition to a quote, I'd like to see if I can get some additional information, such as electrical needs to hook up the unit and size of the unit so we can determine if we have a space available to park it. Also, our project involves experimental animals that would be receiving an oncolytic virus (biosafety level 2). Would this be a problem for renting this unit?
I am with the Royal College of General Practitioners. I am looking to rent 12 smaller sizeed -80C freezers that can fit through a standard door frame. I am looking to rent them for 3-6 months. The freezers will need to be at 12 different locations in England. Would you be able to accommodate me? Thank you!
We are a healthcare center and are hosting an event in Waco, Texas to provide free healthcare to the citizens of our community. This equipment would provide valuable vision services for this event.
I have a trade show in Houston TX and am looking to rent an electric under-counter single phased low-temp high speed dishwasher for trade show booth from Sept 13-21. Please submit rental quotes through your KWIPPED Supplier Accounts. Please message customer with questions. Thank you.
We are a snack food manufacturer, and need to RENT the following equipment Sept 10-15 for the Olympia Body Building Event at the Las Vegas Conv. Center: (1) Electric 1-Stack Full Size Standard Depth Convection Oven w/ 5-Racks [for standard sheet pans, similar to a Blodgett MarkV-100, in excellent working order], (1) 2-Door Standing Freezer [any style], (1) Food Warmer; Large Countertop -OR- Standing [large enough for standard sheets pans], (3) 6' Stainless Steel Prep Tables [no refrigeration], (1) 2-Compartment Hand Washing Sink [no electricity or waterlines needed; only water tanks for fresh and grey water], (1) Free-Standing Sheet Pan Rack [for standard sheet trays]. Need quotes ASAP! All equipment quoted must be current models (not too all). Please email Photos + Spec Sheets for all equipment being quoted to James Tritten w/ KWIPPED - [email protected] Thank you!
We are looking to RENT a cable spooler to assist with swapping out 1-3/4" wire rope off of a crane in Kittery, ME. There will be approx. 4,600' of cable that will weigh approx. 5.67 lbs. / ft. for an estimated total weight of approx. 30,000 lbs. Please submit quotes through your KWIPPED Supplier Account. Thank you.
We are a lab services company. One of our clients is going through a renovation for about 1 year starting in September this year. They need a BSC Class II. This lab will be aliquoting out body fluid samples to split to do their testing in house and externally at another lab. Since they are working with diseases that can be transmissible, they are required to use a hood that is BSL2 compliant. I do have a vendor locally that can install and certify.
I am with a large hospital in Miami, FL. We need (qty 15 - 20) hi lo treatment chairs for a 3 day course in Miami Beach. Thank you!
The Ottawa Section of the Mercedes-Benz Club of America is planning to put on a Driving Skills event, and we are looking to rent 100 traffic cones. Please sign-up to become a KWIPPED Supplier if you wish to quote this item out. Thank you!
We are a non-profit senior living community. We are interested in a rental quote for a portable wastewater treatment system. Our subdivision will house 220 bathrooms. We anticipate a flow of 18,000 gallons per day.
I am in Lisle, IL looking to RENT a Middleby Marshall - Part number PS520 double counter-top conveyor 240V Three Phase or a comparable piece of equipment for a 4 day event. Please submit quotes through your KWIPPED Supplier Account. Thank you.
I am in Lisle, IL looking to rent a floor model mixer either a Stephan UM 60 universal machine or a Hobart HCM450 for a week in August. If you have something comparable that will work as well. Please submit rental quotes through your KWIPPED Supplier Account. Thank you.
I am with a national training program for rheumatologists. We will need to rent (qty 3) ultrasounds for a class in August and want to go ahead and get them scheduled. Looking specifically for three GE Logiq P9 with ML6-15 and 9L probes / transducers. Would also be interested in renting three Logiq E in addition to Logiq P9.
Send me a quote for the Siemens Healthineers CLINITEK Advantus Urine Chemistry Analyzer or comparable equipment.
I am in Rosemont, IL looking to RENT a Merchandising refrigerator for displaying Apple Cider Vinegar bottles for a 3-day tradeshow. This needs to be a countertop or standing glass merchandiser with about 25CU FT of space. Please submit RENTAL quotes through your KWIPPED Supplier Account. Thank you.
We are a dental office Located at Aurora, Colorado and our office want to rent a mobile van for a church event ( community outreach ) in Athens, Ohio for the month of July- a week event.
I am the medical aesthetics instructor at a college. Please send a quote for the Sharplight Technoliges Omnimax or comparable equipment. We require a second device for short term, as we own an Omnimax s4 already. Thank you.
I am planning on opening a business this year that focuses on laser hair removal. I have financial backing and am interested in a purchasing option (via lease through KWIPPED). I would really be interested in the Candela GentleMax Pro with 755 and/or 1064 wavelengths. I am open to both new and refurbished options. If you have any questions, please reach out via KWIPPED messaging or to Kate Scarborough ([email protected]). Please submit quotes on your KWIPPED supplier account. Thank you.
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The Japanese House: "I've never felt so healthy about releasing music and touring"
In 2019, we're finally getting The Japanese House's debut album.
Published: 12:00 pm, December 07, 2018Words: Jamie Muir.
That sort of exploration and freedom has rung through The Japanese House over the past year - a feeling that if you want to try something, then why not muck about and see what happens? It's how the very best of albums are made and with 2019 now careering into view, now is the time The Japanese House steps up to the plate with something special. More so, with something that sounds unlike anything you will have heard from her before.
Flash back, and the journey remains quite stunning. One that started with mysterious tracks and questions about who on earth The Japanese House is blossomed into an artist full of exciting development, where each EP was different to the last, and those larger rooms became home. There were arenas, headline tours around the globe and that feeling that anything could be possible. Now feels different, though.
"I've never felt so healthy about releasing music and touring. The past year, my life has changed, and I've changed more than I've ever done before. I've broken myself down and then built myself up, and now I'm excited about sharing music and excited to go on tour, and I'm not anxious about anything. That's greatly helped by making an album that I'm proud of and I have no doubts within myself," Amber explains. "At the moment, anyway," she cracks.
Taking that change in her own life, she made a conscious decision that to make the album she wanted she needed to step away.
"I've become a lot more open and honest, and I think that's just part of growing up and getting older"
Moving to Wisconsin and teaming up with producer BJ Burton, Amber found herself in the same cabin and rural hideout that Bon Iver first crafted the start of his raw-ripped era, and there's something about that place that left its mark on Amber.
Free of any distractions ("I have probably the worst attention span in the entire world," she laughs) it was all about Amber and BJ working away and opening up to what she wanted to say and do. With similarities and differences abound, it was exactly what Amber needed.
The result is stunning, with the tracks heard so far signalling a bolder and more ambitious Japanese House that isn't just set on serenading, but owning the moment. They're songs "probably inspired by playing live a lot", coupled with the unique surroundings where the foundations of the record joined into place.
It's an openness you can hear radiating from the resultant tracks, all of these elements morphing into the perfect moment for Amber to lay out the thoughts and feelings flowing through her own life.
Coupled with recording time in Brussels and Oxford (and teaming up with longtime collaborator George Daniel of The 1975 fame), those tracks thrive off a freshness that develops on the magically raw sound that captivated from the very first moment The Japanese House stepped into the world. Yet now, there's no holding back. Bouncing from synth-pop bops to deep glitchy grooves, almost War On Drugs-esque warmth and blending electronica, this could very well be one of the best albums of 2019 in waiting - not that we'd want to lay all our cards on the table at this stage, right?!
As the countdown clock ticks down, know one thing. 2019 is the year The Japanese House calls the shots.
Taken from the December 2018 / January 2019 issue of Dork. The Japanese House's debut album 'Good at Falling' is out 1st March.


text:
Q: Android WebView seems to behave different for each device I'm building an app that is dependend on WebViews to render HTML-files in the assets folder. These files contain Javascript. The app works perfectly on my own device, including all the Javascript. But the Android emulator from Android Studio executes only some of the Javascript, on some of the pages. And the device from one of the testers shows blank pages instead of the HTML-files. And the device from another one of our testers also doesn't execute all the Javascript properly.
So, I'm not really sure what to do here. All pages are loaded with the exact same WebView-method and most of the Javascript is also similar. I can't debug or release an app that behaves differently on each device.
The code to load the WebViews:
    WebView myWebView = (WebView) findViewById(R.id.webview);
    myWebView.addJavascriptInterface(new WebAppInterface(this), "Android");
    WebSettings webSettings = myWebView.getSettings();
    CookieManager.getInstance().setAcceptCookie(true);
    webSettings.setJavaScriptEnabled(true);
    webSettings.setAllowFileAccessFromFileURLs(true);
    webSettings.setAllowUniversalAccessFromFileURLs(true);
    webSettings.setAllowFileAccess(true);
    webSettings.setAllowContentAccess(true);
    webSettings.setDomStorageEnabled(true);
    webSettings.setLoadWithOverviewMode(true);
    myWebView.setWebChromeClient(new WebChromeClient() {
        public void onGeolocationPermissionsShowPrompt(String origin, GeolocationPermissions.Callback callback) {
            callback.invoke(origin, true, false);
        }
    });
    myWebView.loadUrl("file:///android_asset/www/file.html");

    myWebView.setWebViewClient(new WebViewClient() {
        @Override
        public boolean shouldOverrideUrlLoading(WebView view, WebResourceRequest request) {
            view.loadUrl(request.toString());
            return true;
        }
    });

Once again, the app works perfectly on my own Android device. Even after I reinstalled it. I just don't see why it wouldn't work on other devices. Does someone have a solution or explaination?

A: I advice you to use the WebView Remote Debugging,so that you can see the source code that running in the WebView of your html file,and then you can click the web page in you Chrome Browser to see how the page react.
That is the way I took when I met some problem in webview.
all you need to do is just run the following code
if (Build.VERSION.SDK_INT >= 19){
     WebView.setWebContentsDebuggingEnabled(true);
}

or you can find more detail about remote-debugging

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