text:
\section{Introduction}

This paper is concerned with the efficient solution of the
Lippmann-Schwinger equation, which describes time-harmonic scattering
from inhomogeneous media in acoustics and electromagnetics as well as
time-harmonic scattering from localized potentials in quantum
mechanics. The simplest form of this equation comes from inhomogeneous
acoustic scattering. Let $\omega$ be the frequency of the
time-harmonic wave and denote the index of refraction by $1-m(x)$. The
inhomogeneity $m(x)$ is a function supported in a compact domain
$\Omega\subset \mathrm{R}^d$ of size $O(1)$, so the index of refraction is $1$
outside $\Omega$. Given an incoming wave $u_I(x)$ that satisfies the
free space Helmholtz equation
\[
(\Delta + \omega^2) u_I(x) = 0,\quad x\in\mathrm{R}^d
\]
the problem of scattering from inhomogeneous media is to find the
scattered field $u(x)$ such that the total field $u(x)+u_I(x)$
satisfies
\begin{equation}
  ( \Delta + \omega^2(1-m(x)) ) (u(x)+u_I(x)) = 0, \quad x\in\mathrm{R}^d
  \label{eq:helm}
\end{equation}
and $u(x)$ obeys the Sommerfeld radiation condition:
\begin{equation}
  \lim_{r\rightarrow\infty} r^{(d-1)/2} \left(\frac{\partial u}{\partial r} - i \omega u\right) = 0.
  \label{eq:somm}
\end{equation}
For the free space Helmholtz operator $-(\Delta +
\omega^2)$, the Green's function is given
by
\[
G(x) =
\begin{cases}
  \frac{1}{2 i \omega} \exp(i \omega |x|), &d=1,\\
  \frac{i}{4} H^1_0(\omega |x|), & d=2,\\
  \frac{1}{4\pi |x|} \exp(i\omega |x|), & d=3.
\end{cases}
\]
Rewriting \eqref{eq:helm} as
\[
(-\Delta-\omega^2 + \omega^2 m(x)) u(x) = - \omega^2 m(x) u_I(x),\quad x\in\mathrm{R}^d
\]
and convolving it with $G(x)$ gives
\begin{equation}
  u + G \ast (\omega^2 m u) = G \ast (-\omega^2 m u_I),
  \label{eq:ls}
\end{equation}
which is the Lippmann-Schwinger equation written in terms of the
scattered field $u(x)$. One can also write the Lippmann-Schwinger in
terms of the total field $u(x)+u_I(x)$, but we shall stick to
\eqref{eq:ls} since its unknown $u(x)$ satisfies the Sommerfeld
radiation condition. For the inhomogeneous electromagnetic scattering
and quantum scattering, the same derivation results in integral
equations similar to \eqref{eq:ls}.

For high frequency wave fields (i.e., $\omega$ large), computing
numerical solution of scattering in inhomogeneous media remains a
challenging computational problem especially in 3D. For practical and
numerical purposes, working with \eqref{eq:ls} rather than
\eqref{eq:helm} offers several advantages.
\begin{itemize}
\item First, \eqref{eq:ls} is written in terms of $u(x)$ only for
  $x\in \Omega$ as $m(x)=0$ for $x\not\in\Omega$. Therefore, the
  unknown function in \eqref{eq:ls} is defined on the compact set
  $\Omega$ versus the whole space $\mathrm{R}^d$ as in \eqref{eq:helm}.
\item Second, once $u(y)$ for $y\in\Omega$ is computed, the whole
  scattered field in $\mathrm{R}^d$ defined via
  \[
  u(x) = - \int_\Omega G(x-y) \omega^2 m(y) (u(y)+u_I(y)) dy, \quad
  \forall x\in\mathrm{R}^d
  \]
  satisfies the Sommerfeld radiation condition \eqref{eq:somm}
  automatically. However, for \eqref{eq:helm} a special treatment such
  as perfectly matched layer (PML) \cite{berenger-1994,chew-1994} or
  absorbing boundary condition (ABC) \cite{engquist-77} is required to
  approximate \eqref{eq:somm} at a finite distance.
\item Third, the convolution structure of \eqref{eq:ls} allows for
  rapid application of the integral operator via the fast Fourier
  transform (FFT) \cite{cooley-1965}.
\item Finally, most treatments of \eqref{eq:helm} for high-frequency
  problems suffer from the pollution effect \cite{babuska-1997}, due
  to the error from the local finite difference or finite element
  stencils. This problem does not show up in \eqref{eq:ls} because of
  the explicit introduction of the Green's function.
\end{itemize}

However, working with \eqref{eq:ls} also brings a couple of numerical
issues. Discretizing the integral equation \eqref{eq:ls} results a
dense linear system, which renders typical direct solvers impractical
except for 1D problems. For this reason, most numerical solutions of
the Lippmann-Schwinger equation use iterative solvers. Since the
Lippmann-Schwinger equation is numerically ill-conditioned, standard
iterative methods without preconditioning requires a huge number of
iterations for high frequency problems. This is especially true when
the inhomogeneity $m(x)$ has sharp transitions (see \cite{duan-2009}
for example). Therefore, there is a clear need for developing good
preconditioners for the Lippmann-Schwinger equation.

There has been a substantial amount of work on computing numerical
solutions of the Lippmann-Schwinger in the recent literature, for
example
\cite{andersson-2005,bruno-2004,chen-2002,lanzara-2004,sifuentes-2010,vainikko-2000}.
Several of these methods are concerned with the design of efficient
preconditioners. In \cite{bruno-2004}, Bruno and Hyde constructed a
preconditioner by replacing the inhomogeneity $m(x)$ with a piecewise
constant and radially symmetric approximation and then inverting the
associated approximate integral operator with semi-analytic
methods. Such a preconditioner is effective when there exists a good
piecewise constant and radially symmetric approximation, and naturally
deteriorates when there is no such approximation.

In \cite{chen-2002}, Chen proposed a method for solving the 2D
Lippmann-Schwinger equation using a technique that is now often
referred as recursive interpolative decomposition or recursive
skeletonization. This method generates in $O(N^{3/2})$ steps an
approximate inverse which can be used either as a direct solver if the
accuracy is sufficient or as a preconditioner. More advanced and
efficient methods for general integral equations have been proposed in
\cite{corona-2014} for 2D problems and in \cite{ho-2013} for both 2D
and 3D problems. These recent methods achieve quasi-linear complexity
for integral equations with non-oscillatory kernels, but fall back to
the same $O(N^{3/2})$ cost for 2D high frequency Lippmann-Schwinger
equations. For all these methods, the prefactor of the complexity
depends on the required accuracy and can be very large when they are
used as direct solvers.

There have also been attempts to apply two-grid and multigrid
methods to precondition the Lippmann-Schwinger equation (see
\cite{sifuentes-2010} for example). However, due to the highly
oscillatory nature of the solution field, such methods tend to improve
the iteration number at most by a constant factor.

In this paper, we introduce a new preconditioner called the
sparsifying preconditioner for the Lippmann-Schwinger equation. The
main idea is to transform the discretized Lippmann-Schwinger equation
approximately into a sparse linear system. This is possible since the
scattered field indeed satisfies the Helmholtz equation in $\Omega$ and
approximately the absorbing-type boundary conditions on
$\partial\Omega$. Once the sparse linear system is ready, we invert it by
leveraging the efficiency of sparse linear algebra algorithms.

The rest of this paper is organized as follows. In Section 2 we
introduce the discretization scheme and describe the main idea.
Sections 3 and 4 explain the details of the sparsifying preconditioner
for rectangular domains and general domains, respectively. Section 5
extends our approach to the Laplace equation with
potential perturbation. Finally, discussions and future work are given in
Section 6.

\section{Discretization and main idea}

The sparsifying preconditioner to be presented is mostly independent
of the discretization scheme and the dimension. We assume that a
Nystr\"om method is used in order to keep the presentation simple.
Our discussion focuses on the 2D and 3D cases, since the 1D problem is
not computationally challenging.

At any point $x\in \Omega$, the local wave velocity is given by $c(x) =
(1-m(x))^{-1/2}$. At frequency $\omega$, the local wavelength at $x$
is equal to $\lambda(x) = (2\pi/\omega) (1-m(x))^{-1/2}$.

\subsection{Discretization}

Let us first consider the 2D case. The domain $\Omega$ is discretized
with a uniform grid of step size $h$, where $h$ is chosen such that
there are at least a few grid points per wavelength across the
domain. Such a uniform grid is convenient when the variation in $c(x)$
is not too large. For problems with large variations in $c(x)$, an
adaptively-refined grid should be used instead. The grid points are
indexed by integer pairs $i=(i_1,i_2)$ with location given by $x_i =
ih =(i_1h,i_2h)$.

We assume that $m(x)$ is zero within two layers of grid points from
the boundary $\partial\Omega$, which can be easily satisfied by slightly
enlarging $\Omega$ if necessary. The discretization uses $m_i =
m(x_i)$ and let $u_i$ be the numerical approximation to
$u(x_i)$. Writing the integral on the left hand side of \eqref{eq:ls}
explicitly at the point $x_i$ gives
\begin{equation}
  (G\ast (\omega^2 m u))(x_i) = \int_\Omega G(x_i-y) (\omega^2 m(y) u(y)) dy
  \label{eq:Gmuint}
\end{equation}
and this integral is approximated with numerical quadrature using the
grid points in $\Omega$.

When $m(x)$ is discontinuous, discretizing \eqref{eq:Gmuint}
efficiently and accurately is a non-trivial task and we refer to
\cite{beylkin-2008,beylkin-2009} for recent developments. For
simplicity, we assume here that $m(x)$ is smooth (but still with
possible sharp transitions) and that the grid size $h$ is chosen to
sufficiently resolve the transitions in $m(x)$. For such $m(x)$, the
discretization of \eqref{eq:Gmuint} is more straightforward and the
only difficulty is the singularity of $G(x)$ at the origin. A simple
quadrature rule is
\[
\int_\Omega G(x_i-y) (\omega^2 m(y) u(y)) dy \approx
\sum_{j: x_j\in\Omega} k_{i-j} \omega^2 m_j u_j,
\]
where
\[
k_t = G(ht) h^2, \quad t\not=(0,0)
\]
and $k_{(0,0)}$ is given by a quadrature correction near the
origin. This is a one-point correction of the trapezoidal rule and
gives an accuracy of O($h^4 \log(1/h))$ (see \cite{duan-2009} for
example). There are also higher order versions that correct the grid
points in the neighborhood of the origin and they do not affect the
following discussion of the sparsifying preconditioner significantly.

After quadrature approximation, the discrete form of the
Lippmann-Schwinger equation \eqref{eq:ls} takes the form, for $i$ with
$x_i\in \Omega$,
\begin{equation}
  u_i + \sum_{j:x_j\in\Omega} k_{i-j} \omega^2 m_j u_j = g_i,
  \label{eq:lsd}
\end{equation}
where $g_i$ is the discretized value of the right hand side of
\eqref{eq:ls}. In a slight abuse of notation, we define $u$ to be the
vector with entries given by $u_i$ and similarly for $g$ and
$m$. Then, \eqref{eq:lsd} can be written as
\begin{equation}
  (I + K\omega^2 m) u = g,
  \label{eq:lso}
\end{equation}
where $K$ is the matrix with entries defined via $K(i,j) = k_{i-j}$ and
$m$ is understood as operator of multiplying with the vector $m$
entry-wise.

The situation for the 3D case is almost the same except that $i$ is
now an integer triple $(i_1,i_2,i_3)$ and $k_t = G(th) h^3$ for
$t\not=(0,0,0)$.

\subsection{Main idea}

The main idea of our approach is to find a sparse and local operator
$Q$ such that in
\[
Q (I+ K \omega^2 m) u = Qg
\]
the operator $Q (I+ K \omega^2 m)$ is also numerically sparse and
localized. This is possible since the scattered field $u$ satisfies
\begin{itemize}
\item the Helmholtz equation \eqref{eq:helm} in the interior of $\Omega$, and
\item approximately the absorbing-type boundary conditions on the
  boundary of $\Omega$.
\end{itemize}

For a grid point indexed by $i$, we define its neighborhood $\mu(i)$
to be
\[
\mu(i) = \{j: \|i-j\|_\infty \le 1, x_j \in \Omega \}.
\]
A grid point $i$ is called an {\em interior} point if all the grid
points in $\mu(i)$ are also in $\Omega$, and otherwise a {\em
  boundary} point. The set $\mu(i)$ contains 9 points in 2D and 27
points in 3D for an interior point, but has fewer points for a
boundary point. Let $N_I$ and $N_B$ denote the numbers of interior
and boundary grid points, respectively. The total number of grid
points $N$ is equal to $N_I + N_B$.

For the interior points, we shall design a matrix $A$ of size $N_I
\times N$, with the rows indexed by the interior grid points. For a
fixed interior index $i$, the row $A(i,:)$ satisfies two conditions:
\begin{itemize}
\item $A(i,:)$ has support in $\mu(i)$, and
\item $A(i,:) K(:,j) \equiv A(i,\mu(i)) K(\mu(i),j) \approx 0$ for any
  $j\not\in \mu(i)$, or, equivalently $A(i,:) K(:,\mu(i)^c) \approx
  0$.
\end{itemize}
Once $A$ has been calculated, we can define a sparse matrix $C$ of the
same size such that
\begin{itemize}
\item $C(i,:)$ has support in $\mu(i)$, and
\item $C(i,\mu(i)) := A(i,\mu(i)) K(\mu(i),\mu(i))$.
\end{itemize}
The above conditions imply that
\begin{itemize}
  \item $C$ and $A$ are sparse operators that represent a local
    stencil, and
  \item $A K \approx C$.
\end{itemize}
Therefore,
\begin{equation}
  A(I + K \omega^2 m) \approx (A + C \omega^2 m).
  \label{eq:A}
\end{equation}

For the boundary points, we shall design a matrix $B$ of size $N_B
\times N$, with the rows indexed by the boundary grid points. For a
fixed boundary index $i$, the row $B(i,:)$ satisfies two conditions:
\begin{itemize}
\item $B(i,:)$ has support in $\mu(i)$, and
\item $B(i,:) K(:,j) \equiv B(i,\mu(i)) K(\mu(i),j) \approx 0$ for any
  $j\not\in \mu(i)$.
\end{itemize}
Since $m_j=0$ for any grid point $j$ within two layers from the
boundary and $i$ is a boundary point, $m_j=0$ for $j\in\mu(i)$.
Therefore, the second condition implies that
\[
B(i,:) (K \omega^2 m)(:,:) \approx 0.
\]
Hence,
\begin{equation}
  B(I+ K\omega^2 m) \approx B.
  \label{eq:B}
\end{equation}

Applying matrices $A$ and $B$ to both sides of \eqref{eq:lso} yields
\begin{equation}
  \begin{bmatrix}
    A\\B
  \end{bmatrix}
  (I+ K \omega^2 m) u =
  \begin{bmatrix}
    A\\B
  \end{bmatrix}
  g.
  \label{eq:AB}
\end{equation}
Combining \eqref{eq:A} and \eqref{eq:B} with \eqref{eq:AB} gives
\begin{equation}
  \begin{bmatrix}
    A + C\omega^2 m \\ B
  \end{bmatrix}
  u \approx
  \begin{bmatrix}
    A \\ B
  \end{bmatrix}
  g.
  \label{eq:PCapp}
\end{equation}
This suggests defining the preconditioner as the mapping from $g$ to
$u$ given by
\begin{equation}
  u \Leftarrow
  \begin{bmatrix}
    A + C\omega^2 m \\ B
  \end{bmatrix}^{-1}
  \begin{bmatrix}
    A \\ B
  \end{bmatrix}
  g.
  \label{eq:PC}
\end{equation}
Since $A$ and $B$ are sparse, applying these operators on the right
hand side is fast. In addition, since $A$, $B$, and $C$ are all
designed to be local operators, the linear system solve in
\eqref{eq:PC} can be done efficiently with, for example, the nested
dissection method and the multifrontal method
\cite{duff-1983,george-1973}.

In the following sections, we shall apply this procedure to several
different setting. In each case, we explicitly give the construction
for $A$ and $B$, analyze the complexity of constructing and applying
the preconditioner, and provide numerical results.

\section{Rectangular domains}
\label{sec:R}

This section considers the case of rectangular $\Omega$.

\subsection{Algorithm}
Let us consider the 2D case first. For simplicity, let $\Omega =
(0,1)^2$ and define $n=1/h$ to be the number of points in each
dimension. Clearly, $N=n^2$, $N_B = O(n)$, and $N_I = O(n^2)$.  In
order for the operator $B$ to serve as a sufficiently accurate
approximation to the Sommerfeld radiation condition, we assume that
$m(x)$ vanishes in a buffer region of a constant, but small, width
near the boundary and the constant $b$ denotes the ratio between the
width and the step size.

For each interior point $i$, recall that we require
\[
A(i,\mu(i)) K(\mu(i),j) \approx 0, \quad \forall j\not\in \mu(i).
\]
Since $K$ is translational invariant, it is convenient to require $A$
be so as well. By translating the point $i$ to the origin, it is
sufficient to consider the problem of finding a vector $\alpha$ such
that
\[
\alpha\cdot K(\mu(0),j) \approx 0, \quad \forall j \in I_n := \{j: -n<j_1,j_2<n,
j\not\in \mu(0)\}.
\]
To solve for $\alpha$, we formulate the following optimization problem
\begin{equation}
  \min_{\alpha: \|\alpha\|=1} \|\alpha\cdot K(\mu(0), I_n)\|^2.
  \label{eq:Fmin}
\end{equation}
Through the singular value decomposition $ K(\mu(0),I_n) = U S V^*$,
the solution is given by
\[
\alpha = U(:,|\mu(0)|)^*
\]
and we set $A(i,\mu(i)) = \alpha$ for all interior $i$.

For the boundary, there are two separate cases: edge points and corner
points. For a boundary point $i$ on an edge (suppose the right one),
we require
\[
B(i,\mu(i)) K(\mu(i),j) \approx 0, \quad \forall j\not\in \mu(i).
\]
Translating the point $i$ to the origin gives the sufficient problem
of finding $\alpha$ such that
\[
\alpha\cdot K(\mu_E(0),j) \approx 0, \quad \forall j \in E_n := \{j: -n<j_1<-b, -n<j_2<n\},
\]
where $\mu_E(0)$ is the translated copy of $\mu(i)$. To solve for
$\alpha$, we consider
\begin{equation}
  \min_{\alpha: \|\alpha\|=1} \|\alpha\cdot K(\mu_E(0), E_n)\|^2.
  \label{eq:Emin}
\end{equation}
Once again, the singular value decomposition $ K(\mu_E(0),E_n) = U S
V^*$ gives the solution
\[
\alpha = U(:,|\mu_E(0)|)^*
\]
and we set $B(i,\mu(i)) = \alpha$ for all boundary $i$ on the edges.

For a boundary point $i$ at a corner (suppose the top-right one), we require
\[
B(i,\mu(i)) K(\mu(i),j) \approx 0, \quad \forall j\not\in \mu(i).
\]
Translating the point $i$ to the origin gives the problem of finding
$\alpha$ such that
\[
\alpha\cdot K(\mu_C(0),j) \approx 0, \quad \forall j \in C_n := \{j: -n<j_1,j_2<-b\},
\]
where $\mu_C(0)$ is the translated copy of $\mu(i)$. To solve for
$\alpha$, we consider
\begin{equation}
  \min_{\alpha: \|\alpha\|=1} \|\alpha\cdot K(\mu_C(0), C_n)\|^2.
  \label{eq:Cmin}
\end{equation}
The singular value decomposition $ K(\mu_C(0),C_n) = U S V^*$ gives the solution
\[
\alpha = U(:,|\mu_C(0)|)^*
\]
and we set $B(i,\mu(i)) = \alpha$ for all corner points $i$.

Once $A$ and $B$ are known, we compute $C$ and form
\begin{equation}
  \begin{bmatrix}
    A + C\omega^2 m \\ B
  \end{bmatrix},
  \label{eq:ABop}
\end{equation}
which is built from local compact stencils on a uniform rectangular
grid. As we mentioned earlier, the nested dissection algorithm is used
to compute an factorization of this operator.

The extension to 3D is quite straightforward, except there are three
types for boundary points: face points, edge points, corner
points. For the translated optimization problem, the point sets are
defined via
\begin{itemize}
\item interior point
  \[
  I_n := \{j: -n<j_1,j_2,j_3<n, j\not\in \mu(0)\},
  \]
\item surface point (suppose on $x_1=1$)
  \[
  F_n := \{j: -n<j_1<-b, -n<j_2,j_3<n\},
  \]
\item edge point (suppose on $x_1=1, x_2=1$)
  \[
  E_n := \{j: -n<j_1,j_2<-b, -n<j_3<n\},
  \]
\item corner point (suppose at $x_1=2,x_2=1,x_3=1$)
  \[
  C_n := \{j: -n<j_1,j_2,j_3<-b\}.
  \]
\end{itemize}

\subsection{Complexity}
Here we analyze the cost of constructing and applying the sparsifying
preconditioner.

In 2D, the setup algorithm consists of two parts: computing the
singular value decompositions and factorizing \eqref{eq:ABop} with the
nested dissection algorithm. The former has a $O(n^2)=O(N)$ cost while
the latter takes $O(n^3) = O(N^{3/2})$ steps. Therefore, the total
setup cost is $O(N^{3/2})$. Applying the preconditioner is essentially
a solve with the nested dissection algorithm, which has $O(n^2 \log n)
= O(N\log N)$ complexity.

In 3D, factorizing \eqref{eq:ABop} with the nested dissection
algorithm has an $O(n^6) = O(N^2)$ cost and this dominates the setup
cost. The cost of applying the preconditioner is equal to $O(n^4) =
O(N^{4/3})$, which is the cost of a nested dissection solve.

\subsection{Numerical results}

This preconditioner and the necessary nested dissection algorithm are
implemented in Matlab. The numerical results below are obtained on a
desktop computer with CPU speed at 2.0Hz. For the iterative solver
the GMRES algorithm is used with relative tolerance equal to
$10^{-6}$. The inhomogeneity $m(x)$ is confined in the unit cube and
the velocity field $c(x)$ for each example is between $2/3$ and
$1$. The grid step size $h$ is chosen such that there are $6$ points
per wavelength for the homogeneous region. This ensures a minimum of 4
points per wavelength across the whole domain.

The 2D case is tested with two examples: a Gaussian bump and a
smoothed square cavity. The incoming wave is a plane wave pointing
downward at frequency $\omega$ and the results are summarized in
Figures \ref{fig:R21} and \ref{fig:R22}. The columns of the tables
are:
\begin{itemize}
\item $\omega$ is the frequency,
\item $N$ is the number of unknowns,
\item $T_s$ is the setup time of the preconditioner in seconds,
\item $T_a$ is the application time of the preconditioner in seconds,
\item $n_p$ is the iteration number of the preconditioned iteration, and
\item $T_p$ is the solution time of the preconditioned iteration in
  seconds.
\end{itemize}

The 3D case is also tested with two examples: a Gaussian bump and a
smoothed cubic cavity. The incoming wave is again a plane wave
pointing downward at frequency $\omega$ and the results are given in
Figures \ref{fig:R31} and \ref{fig:R32}.

The results show that the setup and application costs of the
preconditioner scale with $\omega$ and $N$ according to the complexity
analysis given above. The preconditioner reduces the iteration number
dramatically and in fact it becomes essentially frequency-independent.

\begin{figure}[h!]
  \begin{center}
    \begin{tabular}{|cc|cc|cc|cc|}
      \hline
      $\omega$ & $N$ & $T_s$(sec) & $T_a$(sec) & $n_p$ & $T_p$(sec) \\
      \hline
      1.0e+02 & 9.0e+03 & 1.3e+00 & 6.6e-02 & 5 & 4.2e-01\\
      2.0e+02 & 3.6e+04 & 6.4e+00 & 2.6e-01 & 5 & 1.4e+00\\
      4.0e+02 & 1.5e+05 & 3.9e+01 & 1.0e+00 & 6 & 7.8e+00\\
      \hline
    \end{tabular}
    \includegraphics[height=1.8in]{R21c.pdf} \hspace{0.25in} \includegraphics[height=1.8in]{R21u.pdf}\\
  \end{center}
  \caption{Example 1 of 2D rectangular case. Top: numerical results.
    Bottom: $c(x)$ (left) and $u(x)+u_I(x)$ (right) for the largest
    $\omega$ value.}
  \label{fig:R21}
\end{figure}

\begin{figure}[h!]
  \begin{center}
    \begin{tabular}{|cc|cc|cc|cc|}
      \hline
      $\omega$ & $N$ & $T_s$(sec) & $T_a$(sec) & $n_p$ & $T_p$(sec) \\
      \hline
      1.0e+02 & 9.0e+03 & 1.3e+00 & 7.7e-02 & 6 & 4.3e-01\\
      2.0e+02 & 3.6e+04 & 7.0e+00 & 4.1e-01 & 8 & 4.1e+00\\
      4.0e+02 & 1.5e+05 & 4.2e+01 & 1.3e+00 & 8 & 1.3e+01\\
      \hline
    \end{tabular}
    \includegraphics[height=1.8in]{R22c.pdf} \hspace{0.25in} \includegraphics[height=1.8in]{R22u.pdf}\\
  \end{center}
  \caption{Example 2 of 2D rectangular case. Top: numerical
    results. Bottom: $c(x)$ (left) and $u(x)+u_I(x)$ (right) for the
    largest $\omega$ value.}
  \label{fig:R22}
\end{figure}

\begin{figure}[h!]
  \begin{center}
    \begin{tabular}{|cc|cc|cc|}
      \hline
      $\omega$ & $N$ & $T_s$(sec) & $T_a$(sec) & $n_p$ & $T_p$(sec) \\
      \hline
      2.5e+01 & 1.2e+04 & 5.2e+00 & 6.3e-02 & 4 & 2.8e-01\\
      5.0e+01 & 1.0e+05 & 1.0e+02 & 6.4e-01 & 4 & 2.9e+00\\
      1.0e+02 & 8.6e+05 & 3.2e+03 & 7.8e+00 & 4 & 3.4e+01\\
      \hline
    \end{tabular}
    \includegraphics[height=1.8in]{R31c.pdf} \hspace{0.25in} \includegraphics[height=1.8in]{R31u.pdf}\\
  \end{center}
  \caption{Example 1 of 3D rectangular case. Top: numerical
    results. Bottom: cross-sections of $c(x)$ (left) and $u(x)+u_I(x)$
    (right) in the middle of the domain for the largest $\omega$
    value.}
  \label{fig:R31}
\end{figure}

\begin{figure}[h!]
  \begin{center}
    \begin{tabular}{|cc|cc|cc|}
      \hline
      $\omega$ & $N$ & $T_s$(sec) & $T_a$(sec) & $n_p$ & $T_p$(sec) \\
      \hline
      2.5e+01 & 1.2e+04 & 5.6e+00 & 7.3e-02 & 6 & 4.7e-01\\
      5.0e+01 & 1.0e+05 & 1.0e+02 & 5.9e-01 & 8 & 5.7e+00\\
      1.0e+02 & 8.6e+05 & 2.9e+03 & 7.7e+00 & 9 & 7.5e+01\\
      \hline
    \end{tabular}
    \includegraphics[height=1.8in]{R32c.pdf} \hspace{0.25in} \includegraphics[height=1.8in]{R32u.pdf}\\
  \end{center}i
  \caption{Example 2 of 3D rectangular case. Top: numerical
    results. Bottom: cross-sections of $c(x)$ (left) and $u(x)+u_I(x)$
    (right) in the middle of the domain for the largest $\omega$
    value.}
  \label{fig:R32}
\end{figure}

\section{General domains}
\label{sec:G}

For most problems, the support $\Omega$ of $m(x)$ is not necessarily
rectangular. We can apply the approach in Section \ref{sec:R} by
embedding $\Omega$ into a rectangular domain. However, this approach
increases the number of unknowns significantly, especially in 3D. This
section proposes a different approach that does not suffer from this.

\subsection{Algorithm}

The construction of the operator $A$ for the interior grid points is
exactly the same as the one in Section \ref{sec:R}. The main
difference is in the construction of $B$. For a boundary point $i$, we
require
\[
B(i,\mu(i)) (K\omega^2 m)(\mu(i),:) \approx 0.
\]
Since the local geometry at $i$ can be quite different from point to
point, it is impossible to find a uniform stencil $B(i,\mu(i))$ as was
done in Section \ref{sec:R}. Therefore, one needs to consider $i$
point by point. However, applying the approach of Section \ref{sec:R}
to each $i$ individually is too costly since each singular value
decomposition costs $O(N)$ steps. To keep the complexity under
control, we propose the following randomized approach.

Let $R$ be a Gaussian random matrix of size $N\times r$ where $r=O(1)$
is a constant multiple of the maximum stencil size (i.e., $9$ in 2D
and $27$ in 3D) and define the $N\times r$ matrix $T$ via
\[
T = (K \omega^2 m) R.
\]
For each boundary $i$, we look for $B(i,\mu(i))$ such that
\[
B(i,\mu(i))  T(\mu(i),:) \approx 0.
\]
This is done by solving the optimization problem
\[
\min_{\alpha: \|\alpha\|=1} \|\alpha\cdot T(\mu(i), :)\|^2.
\]
With the singular value decomposition $T(\mu(i),:) = USV^*$, the optimal $\alpha$ is given by
\[
\alpha = U(:,|\mu(i)|)^*.
\]
Finally, we set $B(i,\mu(i)) = \alpha$. This process is repeated for
each boundary point $i$. Since $T(\mu(i),:)$ is of size $O(1)\times
O(1)$, the computational cost of the singular value decomposition setup for each $i$ is also of
order $O(1)$.

The effectiveness of this preconditioner depends on the assumption
that there exists a good local stencil in $\mu(i)$ for a boundary
point $i$. When the domain is convex or nearly convex, such a stencil
is guaranteed due to the absorbing-type boundary conditions.  However,
when the domain is highly non-convex the preconditioner is less
effective.

\subsection{Complexity}
To analyze the cost of this approach, we assume that the size of the
support of $m(x)$ is more than a constant fraction of the size of its
bounding box, which implies that $N=O(n^d)$.

In 2D, the setup algorithm consists of three parts: (i) forming $T =
(K \omega^2 m) R$, (ii) computing an singular value decomposition for
each boundary point $i$, and (iii) factorizing \eqref{eq:ABop} with
the nested dissection algorithm.
\begin{itemize}
\item The first part can be accomplished in $O(N\log N)$ steps by
  using the fast Fourier transform.
\item The second part takes $O(\sqrt{N})$ steps since there are at
  most $O(\sqrt{N})$ boundary points and computing singular value
  decomposition for each one takes $O(1)$ steps.
\item Finally, factorizing with the nested dissection algorithm takes
  $O(N^{3/2})$ steps.
\end{itemize}
Adding these together shows that the construction cost is
$O(N^{3/2})$. The application cost of the preconditioner is $O(N\log
N)$ since it is essentially a solve with an existing 2D nested
dissection factorization.

In 3D, among the three parts of the setup algorithm, only the cost of
computing a nested dissection factorization increases to $O(N^2)$. This
now becomes the dominant part of the setup cost. The application
algorithm is again a solve with the existing nested dissection
factorization with a cost of $O(N^{4/3})$.

Though the asymptotic complexity obtained for this case is similar to
the one for the rectangular case, the actual running time is often
lower because the sizes of the supernodes in the nested dissection
algorithm is often much smaller.

\subsection{Numerical results}

The 2D case is tested with two examples: a smoothed cavity of an
$\ell_2$ ball and a smoothed cavity of an $\ell_1$ ball. The incoming
wave is a plane wave pointing downward at frequency $\omega$ and the
numerical results are given in Figures \ref{fig:G21} and
\ref{fig:G22}.

\begin{figure}[h!]
  \begin{center}
    \begin{tabular}{|cc|cc|cc|}
      \hline
      $\omega$ & $N$ & $T_s$(sec) & $T_a$(sec) & $n_p$ & $T_p$(sec) \\
      \hline
      1.0e+02 & 7.6e+03 & 9.8e-01 & 4.1e-02 & 6 & 2.4e-01\\
      2.0e+02 & 3.0e+04 & 3.7e+00 & 1.3e-01 & 7 & 1.2e+00\\
      4.0e+02 & 1.2e+05 & 2.1e+01 & 5.1e-01 & 8 & 6.5e+00\\
      \hline
    \end{tabular}
    \includegraphics[height=1.8in]{G21c.pdf} \hspace{0.25in} \includegraphics[height=1.8in]{G21u.pdf}\\
  \end{center}
  \caption{Example 1 of 2D general case. Top: numerical results.
    Bottom: $c(x)$ (left) and $u(x)+u_I(x)$ (right) for the largest
    $\omega$ value.}
  \label{fig:G21}
\end{figure}

\begin{figure}[h!]
  \begin{center}
    \begin{tabular}{|cc|cc|cc|}
      \hline
      $\omega$ & $N$ & $T_s$(sec) & $T_a$(sec) & $n_p$ & $T_p$(sec) \\
      \hline
      1.0e+02 & 5.7e+03 & 7.4e-01 & 3.3e-02 & 5 & 2.0e-01\\
      2.0e+02 & 2.1e+04 & 2.8e+00 & 7.6e-02 & 8 & 1.1e+00\\
      4.0e+02 & 7.9e+04 & 1.8e+01 & 2.9e-01 & 8 & 5.3e+00\\
      \hline
    \end{tabular}
    \includegraphics[height=1.8in]{G22c.pdf} \hspace{0.25in} \includegraphics[height=1.8in]{G22u.pdf}\\
  \end{center}
  \caption{Example 2 of 2D general case. Top: numerical results.
    Bottom: $c(x)$ (left) and $u(x)+u_I(x)$ (right) for the largest
    $\omega$ value.}
  \label{fig:G22}
\end{figure}

Two examples are tested for the 3D case: a smoothed cavity of an
$\ell_2$ ball and a smoothed cavity of an $\ell_1$ ball. The incoming
wave is a plane wave pointing downward at frequency $\omega$ and the
numerical results are given in Figures \ref{fig:G31} and
\ref{fig:G32}.

\begin{figure}[h!]
  \begin{center}
    \begin{tabular}{|cc|cc|cc|}
      \hline
      $\omega$ & $N$ & $T_s$(sec) & $T_a$(sec) & $n_p$ & $T_p$(sec) \\
      \hline
      2.5e+01 & 6.5e+03 & 7.1e+00 & 2.7e-02 & 4 & 1.6e-01\\
      5.0e+01 & 5.5e+04 & 7.9e+01 & 3.4e-01 & 5 & 2.1e+00\\
      1.0e+02 & 4.5e+05 & 1.4e+03 & 4.1e+00 & 5 & 2.3e+01\\
      \hline
    \end{tabular}
    \includegraphics[height=1.8in]{G31c.pdf} \hspace{0.25in} \includegraphics[height=1.8in]{G31u.pdf}\\
  \end{center}
  \caption{Example 1 of 3D general case. Top: numerical
    results. Bottom: cross-sections of $c(x)$ (left) and $u(x)+u_I(x)$
    (right) in the middle of the domain for the largest $\omega$
    value.}
  \label{fig:G31}
\end{figure}

\begin{figure}[h!]
  \begin{center}
    \begin{tabular}{|cc|cc|cc|}
      \hline
      $\omega$ & $N$ & $T_s$(sec) & $T_a$(sec) & $n_p$ & $T_p$(sec) \\
      \hline
      2.5e+01 & 3.7e+03 & 4.9e+00 & 3.0e-02 & 4 & 1.0e-01\\
      5.0e+01 & 3.1e+04 & 4.5e+01 & 1.8e-01 & 6 & 1.4e+00\\
      1.0e+02 & 2.3e+05 & 7.5e+02 & 2.1e+00 & 9 & 2.3e+01\\
      \hline
    \end{tabular}
    \includegraphics[height=1.8in]{G32c.pdf} \hspace{0.25in} \includegraphics[height=1.8in]{G32u.pdf}\\
  \end{center}
  \caption{Example 2 of 3D general case. Top: numerical
    results. Bottom: cross-sections of $c(x)$ (left) and $u(x)+u_I(x)$
    (right) in the middle of the domain for the largest $\omega$
    value.}
  \label{fig:G32}
\end{figure}

The results show that the setup and application costs of the
preconditioner are lower compared to the ones of the rectangular
case. The number of iterations remains essentially independent of the
frequency, indicating that the boundary stencils computed randomly
provide a sufficiently accurate boundary condition for the solution.

\section{Laplace equation}

In this section, we consider the 3D Laplace equation with compact
potential perturbation, i.e.,
\begin{equation}
  (-\Delta + V(x))u(x) = f(x),\quad x\in\mathrm{R}^3
  \label{eq:lap}
\end{equation}
where $V(x)$ is supported in $\Omega = (0,1)^3$ and $u(x)$ decays like
$1/|x|$ as $x$ goes to infinity. This can be regarded as a limiting
case as $\omega$ goes to zero. The Green's function is given by
\[
G(x) = \frac{1}{4 \pi |x|}.
\]
Convolving $G(x)$ with \eqref{eq:lap} gives the Lippmann-Schwinger
equation
\begin{equation}
  u + G\ast(Vu) = G\ast f.
  \label{eq:lslap}
\end{equation}
For a potential $V$ that is not too negative, this problem remains
elliptic and standard iteration methods such as GMRES converge within
a small number of iterations. However, when $V(x)$ is sufficiently
negative, the problem loses ellipticity and the iteration number
increases dramatically.

As we mentioned, when $\omega$ is finite, PML and ABC offer reasonable
approximations to the Sommerfeld radiation condition. When $\omega$
goes to zero, it is not clear how to define analytically a similar
local boundary condition at a finite distance for the $1/|x|$ decaying
condition. However, the sparsifying preconditioner proposed in
Sections \ref{sec:R} and \ref{sec:G} can be applied to
\eqref{eq:lslap} without any modification and the resulting operator
$B$ offers a good numerical approximation to the $1/|x|$ decaying
condition.

\subsection{Numerical results}
The tests are performed for two examples of $V(x)$: a negative
Gaussian function and a smoothed characteristic function of an
$\ell_2$ ball. In each example, the right hand side $f$ is chosen to
be a delta source in $\Omega$ located near the top left corner of the
middle cross-section. The potential $V(x)$ is chosen to be negative
and proportional to $1/h^2$ in order to make sure that the problem
loses ellipticity. The numerical results are summarized in Figures
\ref{fig:L31} and \ref{fig:L32}.

\begin{figure}[h!]
  \begin{center}
    \begin{tabular}{|cc|cc|cc|}
      \hline
      $\max(|V|)$ & $N$ & $T_s$(sec) & $T_a$(sec) & $n_p$ & $T_p$(sec) \\
      \hline
      5.8e+02 & 1.2e+04 & 5.0e+00 & 4.9e-02 & 5 & 2.1e-01\\
      2.3e+03 & 1.0e+05 & 9.8e+01 & 2.8e-01 & 5 & 1.8e+00\\
      9.2e+03 & 8.6e+05 & 2.1e+03 & 3.0e+00 & 6 & 2.1e+01\\
      \hline
    \end{tabular}
    \includegraphics[height=1.8in]{L31c.pdf} \hspace{0.25in} \includegraphics[height=1.8in]{L31u.pdf}\\
  \end{center}
  \caption{Example 1 of 3D Laplace equation with potential
    perturbation. Top: numerical results. Bottom: cross-sections of
    $V(x)$ (left) and $\log_{10}|u(x)|$ (right) in the middle of the
    domain for the highest resolution test.}
  \label{fig:L31}
\end{figure}

\begin{figure}[h!]
  \begin{center}
    \begin{tabular}{|cc|cc|cc|}
      \hline
      $\max(|V|)$ & $N$ & $T_s$(sec) & $T_a$(sec) & $n_p$ & $T_p$(sec) \\
      \hline
      5.8e+02 & 1.2e+04 & 5.1e+00 & 3.8e-02 & 5 & 2.9e-01\\
      2.3e+03 & 1.0e+05 & 9.8e+01 & 2.6e-01 & 7 & 2.3e+00\\
      9.2e+03 & 8.6e+05 & 2.0e+03 & 2.8e+00 & 7 & 2.4e+01\\
      \hline
    \end{tabular}
    \includegraphics[height=1.8in]{L32c.pdf} \hspace{0.25in} \includegraphics[height=1.8in]{L32u.pdf}\\
  \end{center}
  \caption{Example 2 of 3D Laplace equation with potential
    perturbation. Top: numerical results. Bottom: cross-sections of
    $V(x)$ (left) and $\log_{10}|u(x)|$ (right) in the middle of the
    domain for the highest resolution test.}
  \label{fig:L32}
\end{figure}

The results demonstrate that the number of iterations grows at most
logarithmically with the problem size. This implies that the
numerically computed boundary condition operator $B$ is a good
approximation to the decaying condition of $u$ at infinity.

\section{Conclusion}

This paper introduces the sparsifying preconditioners for the
numerical solution of the Lippmann-Schwinger equation. The main idea
is to numerically transform this integral equation into a sparse and
localized linear system and then leverage existing efficient sparse
linear algebra algorithms. This approach combines the appealing
features of the integral equation formulation with the efficiency of
the PDE formulations. We discuss the algorithmic details for
rectangular and general domains, and extend this approach to the 3D
Laplace equation with a potential perturbation.

From the numerical results, we observe that the most of the
construction time of the preconditioners is spent on the construction
of the nested dissection factorization for the operator in
\eqref{eq:PC}. It is possible to replace this step with other solvers such as
the sweeping preconditioners \cite{engquist-2011a,engquist-2011b},
which are asymptotically more efficient.

Most of the discussion here is in the setting of inhomogeneous
acoustic scattering. An immediate task is to use this preconditioner
to study problems from electromagnetic and quantum scattering.

This approach is not restricted only to scattering in free space. A
future direction is to extend it to fully periodic systems or systems
that are periodic in certain directions. This should have direct
applications in the study of photonic crystal.

For most numerical methods, the stencil is defined analytically based
on the PDE. In the current approach, however, the stencils in the
preconditioning operators $A$ and $B$ are defined using the typical
behavior of the solution, either deterministically as in Section
\ref{sec:R} or randomly as in Section \ref{sec:G}. In general, such a
practice seems to be fairly unexplored in the field of numerical
analysis.

\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv


If you have a technical issue or question, please contact our support team. We are here to help!
Browse our library of helpful resources and tools, including video tutorials, answers to frequently asked questions, and downloadable PDF instructions for your Great Plains Communications services.
Q: How do I program my remote using the three-digit code included with my new remote?
A: Getting your new remote to work with your cable box will be a snap after watching this tutorial.
Q: How do I program my remote if I don't know the three digit code?
A: Don't have the three digit code to program your remote? The Great Plains Communications' "Auto Search Option" tutorial will walk you through the process.
Q: How do I use my remote with multiple devices?
A: Using your remote with multiple devices doesn't have to be hard. The Great Plains Communications' "Switching Devices on Your Remote" tutorial shows you how, with the push of a button, you can use one remote for many of your entertainment devices.
Q: Why is my remote not working like it normally does?
A: Sometimes the simplest solution is also the right one. If your remote is slow or unresponsive, the Great Plains Communications "Trouble Shooting" tutorial can help.
Q: How do I find out what HD channels I have in my area?
A: It's easy to find out what channels are available with the crystal-clear picture of High-Definition (HD). The Great Plains Communications "Searching for HD Channels" tutorial shows you how to find a full listing of HD channels in your area.
Q: How do I find a program using the guide?
A: Find the program you want to watch in seconds with the Great Plains Communications "Search for Programming" tutorial.
Q: How do I search for a program if I only know the show's title?
A: Find out how to search for a program by the show's title with the Great Plains Communications "Search for a Show by Title" tutorial.
Q: How do I record a program?
A: It's easy with the Great Plains Communications "Recording a Program" tutorial.
eMail Guardian protects your email from spam.
Click below for instructions on setting up and maximizing voice mail features.
Click below to learn about call forwarding and stay connected wherever you are.


You can always get here using the "Calendar Logins" menu item in the Calendar menu on every creativetango.com page.
Who makes my updates for me?
4) by requesting John to make changes. If you ask John, you have to wait on his schedule which may take hours, days or weeks. We strongly suggest you share your calendar duties with others in your entourage if possible.
Bottom line: teachers and sponsors are responsible for their own updates.
Do you make exceptions for those who don't understand computers?
We suggest you share your calendar duties with others in your entourage. If you are a teacher or host, you are the calendar feed owner, but you can have others manage the calendar feeds you own.
Oh, no! That is so sad 🙁 Unfortunately John does not have access to your passwords. If you can't remember it, you'll need to reset it following the readily available "forgot password" link next to where you are trying to sign in.
How do I share my calendars to my students and followers?
If you use this system, students and followers can subscribe to your calendars, and you will be able to make simultaneous updates to as many websites as subscribe to your calendar. Sharing codes can be found for your calendars under Settings. Further details are available under the Google documentation help button.
Is it OK to copy calendar(s) onto my site?
The overall site is copyrighted, but the calendar feeds are public. Teachers and hosts are very welcome to share their own feeds on their own sites or with whomever else they choose. Additional permissions tend to be freely given when sought, especially by anyone who helps defray the costs of running this site and providing aggregated feeds with a donation.
As for those who copy big gobs to their sites without asking, representing it as their own work, I don't know who they are, especially when I receive inquiries regarding opportunities for performances, events, television commercials, and so on.
Creativetango.com consolidates the distinct calendars maintained by each teacher and host into what you see on this website.
You still haven't answered my question!
Help on using the Google calendar system in general is available by clicking the gear and selecting "Help" at the top right corner of any calendar account screen. Details on using the system as it relates to creativetango.com specifically are under the Get Started…. and Advanced User popouts on this page.
How to login and make your updates. This is the important stuff that makes the system work.
Add new events to existing calendars with the Create Events link on the upper left.
If you make a change to a repeated event, a popup will ask for more information. This is handy for changing one event only out of a continuing series, such as when a particular milonga moves to a different week of the month for just one month and then returns to its regular schedule.
Save is your best friend. I have found the system to be incredibly reliable over many years of use with several organizations, but I do know some people who forget to save.
The calendar fields are yours to change as you like, but we do have a few guidelines to play nice with each other.
Please be sure to ALWAYS select the correct calendar. Otherwise your events may end up in some other calendar you have access to or your own personal default calendar and thus not shared on creativetango.com at all. This is the most common mistake beginners to this calendar system make. If you save an event on a calendar that John doesn't control, it will not show up on creativetango.com, and there is nothing John can do to fix this.
The dropdown default is your personal calendar. E.g. if your name is Albus Dumbledore and your email is wizard@gmail.com, the dropdown next to the label "Calendar" will say, Albus-wizard and you just click the dropdown next to Calendar and select AD Classes. It is so easy! You can do it! Or you can find someone who can do it for you. Or you can wait for John to get to it, possibly until well after the event has passed.
Please keep the "What" simple." What" shows on the creativetango.com calendar. Lengthy items may displace others, and by doing so find themselves displaced in turn. How much of the"What" field shows varies by the viewer's computer and browser, so rest assured that no matter how much you tweak to get the "What" to show exactly what you want on your computer, it will look completely different on almost anyone else's computer, tablet, phone or other device.
I hope you agree that level or description (e.g. Intermediate or Salon/Pista or Nuevo) is the most appropriate "What" item to put for classes.
Please be sure to include your contact information in the Description. Just ask yourself whether you really want your prospective students or attendees trying to contact John to ask questions about your events, because, though John is especially nice to the newly interested, he does not have all your stuff memorized.
Please enter accurate address including zip code in the "Where" field . However, if your address contains a fraction such as 2714-1/2 Greenville, please enter only the whole number (2714) in the "Where" field, or it will not map properly. (Using "1/2″ throws the map off.) You can give the fraction detail in the Description.
Please enter accurate date and time in the "When" field.
You may also add your website address.
Please be sure to review that things appear as you wish on the calendar pages on creativetango.com. If it does not appear, refresh, and if that doesn't fix it, see above.
I can see it on my calendar, but it's not showing up on creativetango.com!
Are you sure the date is correct including the year? Be especially careful around New Years.
Take a deep breath and select the correct calendar from the Calendar drop down in the 'create event' area. Go to where you edit your event. Scroll down or up or side to side until you find the dropdown label that says, "Calendar." The dropdown default is your personal calendar.
E.g. if your name is Albus Dumbledore and your initials are AD and your email is wizard@gmail.com, the dropdown next to the label "Calendar" will say, Albus-wizard. Please just click the dropdown next to Calendar and select AD Classes or whatever the appropriate calendar is. It is so easy! You can do it!


Find an allergy specialist in Royal Oak, MI or an asthma specialist to help with food allergies, hay fever, nasal congestion, dust and pollen allergies, asthma, dermatitis, and more.
An Asthma Specialist specializes in facilitating breathing and alleviating heart burn and nasal congestion. An allergy specialist specializes in treating allergies. You might see an allergy specialist to find out what you're allergic to. An allergy specialist can perform a variety of tests such as skin or blood tests. Some doctors are both allergy specialists and asthma specialists.
Asthma Specialists can be a doctor, allergist or pulmonologist who treats asthma.
An Allergy Specialist treats ear, nose, and throat allergies. An Allergy Specialist can perform skin tests, blood tests, x-rays, and CT scans. If you are sneezing and have allergy symptoms, you should see an allergy specialist.
Why should I see an Asthma Specialist?
If you are having problems breathing, frequent sinus infections or heartburn or nasal congestion then you should see an Asthma Specialist.
Where do Asthma - Allergy Specialistss work?
Asthma - Allergy Specialistss can work in a hospital, clinic, or private practice.
Can I just see one doctor instead of going to both an Asthma - Allergy Specialists?
Yes, sometimes physicians are both Asthma - Allergy Specialistss.
How do I find an Asthma - Allergy Specialists in my city and state?
The Wellness.com directory will help you locate an Asthma - Allergy Specialists in your state. Select Asthma - Allergy Specialists from the professionals menu and select the state that you are looking to locate an Asthma - Allergy Specialists in. After you have located your state, find the city that you will need an Asthma - Allergy Specialists in. Select the state and city and you will see a list of Asthma - Allergy Specialistss in your city and state.


Ideal for businesses who require increased flexibility and online options to manage their banking transactions and cash deposit levels. Avoid monthly maintenance fees by maintaining an average monthly balance of $25,000 OR total relationship balances*1 of $100,000. Enjoy 500 item transactions included per month with no fee*6.
Get your new business account today. A business banker will be happy to assist you.


text:
By Erin Arvedlund Philadelphia Inquirer
December 12, 2020 — 10:00am
Most Americans now teleworking from home want to keep doing so, with more than half saying they would work remotely after the pandemic, a new Pew Research Center report finds.
The national survey of U.S. adults reveals that while the coronavirus may have changed the location of our jobs — whether in an office or from home — it hasn't significantly reshaped our work duties and culture for a majority of employed adults.
"Another third said they'd want to work from home some of the time. A very small share want to go back to the office full-time," said Juliana Horowitz, associate director of social trends research at Pew Research Center and one of the co-authors.
As for meeting on web platforms, "a majority see that as a good substitute for in-person contact," Horowitz said. "We don't see 'Zoom fatigue' in our survey." Videoconferencing and webinar fatigue showed up among an estimated 37% of those surveyed, she said.
Among workers who remained in the same job but shifted to remote work, more than 60% are as satisfied as before the pandemic and saw no change in productivity or job security.
Those working from home all or most of the time now see clear upsides with telework. About half (49%) of Americans now have more flexibility to choose when they put in their hours, and 38% say it's easier to balance work with family responsibilities.
Still, for most Americans (62%), jobs can't be done from home. And that's exposed a clear class divide between workers who can and cannot telework.
Those working from home mostly have college degrees, about two-thirds with a bachelor's degree or beyond. That compares with only 23% of those without a four-year college degree working from home.
Similarly, while a majority of upper-income workers work from home, most lower- and middle-income workers cannot. The Pew Research Center survey of 10,332 U.S. adults was conducted between Oct. 13 and 19.
The analysis was based on 5,858 adults among the respondents who are working part-time or full-time and who may have multiple jobs but have only one position they consider their primary employment.
Workers' ability to do their job from home varies by industry. Americans in information and technology sectors (84%); banking, finance, accounting, real estate, or insurance (84%); education (59%); and professional, scientific, and technical services (59%) can mostly work from home.
Public-facing workers and front-line employees such as restaurant staff, nursing and health care, police and fire, and waste management aren't so lucky.
About three-quarters or more of those employed in retail, trade, or transportation (84%); manufacturing, mining, construction, agriculture, forestry, fishing, and hunting (78%); and hospitality, service, arts, entertainment, and recreation (77%) say their jobs can't be done from home. Two-thirds of those in the health care and social-assistance sector say the same.
Among those in government, public administration, or the military, the divide is roughly even: 46% say their job can be done from home and 54% say it cannot.
Also, the pandemic has exposed an age gap in virtual work.
Among those working from home, Americans 50 or younger are significantly more likely to say it's been difficult for them to get their work done without interruptions (38% for workers ages 18-49 vs. 18% for workers 50-plus and older). The youngest workers in the survey are among those most likely to say they lack motivation: 53% of those ages 18-29 years.
Men and women who can work from home are about equally likely to say they'd want to continue after the pandemic. Women are more likely than men to want to work from home all of the time (31% vs. 23%).
That's the case whether they have minor children or not. In fact, the shares of workers with or without children below 18 who say they would work from home permanently are nearly identical.
"Women are more likely to want to work from home. Period," she said.
When it came to balancing work and family, parents with young children found it harder to do so, "and mothers, in particular, much more than fathers."
Among employed adults with some college or less education who say they can do their job from home, 60% say they would want to work from home all or most of the time post-pandemic, compared with half of those with at least a bachelor's degree.
"For those who can't work from home, there are big differences about concerns about exposure to the virus and unknowingly exposing others and their families," she said.
Wild Gaudreau's goal gives Wild 3-2 victory in the Big Apple
US judge won't rein in federal wild horse roundup in Nevada
A federal judge decided Friday not to rein in the roundup and capture of wild horses in eastern Nevada, rejecting advocates' claims the federal government was "needlessly and recklessly" killing free-roaming mustangs in violation of U.S. law.
Arriving just hours after a Pennsylvania bridge collapse, President Joe Biden stared into the cratered muddy earth where the aging span fell early Friday, striking evidence supporting the $1 trillion infrastructure law he already had planned to tout on his trip to Pittsburgh.
A 50-year-old bridge collapsed in Pittsburgh early Friday, requiring rescuers to rappel down a ravine and form a human chain to reach a few occupants of a municipal bus that plummeted along with the span. No deaths were reported.
Exurbs rise to the top in ranking of Twin Cities area's hottest housing markets • Business
3M ordered to pay $110 million in latest earplug suit • Business
Some Vegas Strip casinos to get new facades, familiar names • Business
As omicron surged, Minnesotans adjusted quickly, kept going out • Business
With Halloween over, Target and others roll out Black Friday deals • Business

redpajama_set_name: RedPajamaCommonCrawl


text:
Houston Archive
Spidey Spin Offs: Scarlet Spider #3
April 2, 2012Spidey Spin OffsNo comments
When Marvel first announced they were planning on launching a new comic book series dedicated to Peter Parker's deranged clone Kaine, I immediately thought the only way the series would work is if the creative team gave Kaine his very own identity, in his very own environment, separate from
March 5, 2012Spidey Spin OffsNo comments
Geographic location in a comic book series seems to be such an afterthought these days, especially when the vast bulk of Marvel series take place in or around New York City. So when it was revealed that the new Scarlet Spider series was going to be set in Houston,


Tonight Shannon chose a pose that for one reason or another just didn't work out. I came up with a secondary concept and we both really liked it. I shot a couple different versions and we agreed this was our favorite of the two. I realize that her foot is cut off in the photo but it was intentional and gave something to the composition in my minds eye. We also have a version with her foot included and I really liked it but it just wasn't as strong as this one.


First, the good news: USAA ranks extremely high in customer satisfaction surveys from J.D. Power and Consumer Reports, usually knocking every competitor out of the water. However, USAA home insurance is only available to current and former members of the military and their families. USAA is our top pick for military members, veterans, and their families in our review of the best homeowners insurance companies, thanks to its outstanding customer service and financial strength. But even if you qualify, it's worth comparing quotes and coverage options with a few other providers — USAA may not always provide you with the best policies at the best price.
Founded in 1922 by Army officers who were denied auto insurance because they were deemed too risky, USAA prides itself on being the best choice for military members and their families today when it comes to insurance, banking, and investment services. With home insurance, USAA claims that its policies are specifically tailored for military members.
Yes. If you qualify for USAA homeowners insurance, you won't find a better option when it comes to customer service or financial strength. Plus, its policies really do cater to military members — USAA even offers uniform and military equipment coverage in the event of a loss. USAA also provides a few innovative add-ons that some of its competitors lack, such as home-sharing coverage for homeowners who rent out a room (or their entire house).
The only reason USAA wouldn't be the best option for a military family is if another company offered the same coverage for a lower price. While bundling your USAA homeowners insurance with your auto insurance can save you up to 10% on your premium, USAA doesn't offer nearly as many discounts as providers like Allstate or Amica, so it's possible that you'll find a better deal somewhere else. That's why it's so important to compare home insurance quotes from multiple companies.
USAA doesn't just boast the best customer service ratings in the industry — it's the best by a mile. In J.D. Power's annual home insurance survey, it places first at 891 out of 1,000, beating the next-best option (Amica) by a whopping 30 points. In Consumer Reports' survey of more than 7,000 homeowners, USAA was rated "Excellent" in every possible category. It goes above and beyond when it comes to customer support, with resources like an online forum called the USAA Community and a great mobile app with claims and support functionality.
A few insurance companies like Allstate provide excellent online resources to augment their customer service experience, but no provider does it better than USAA. The USAA Community portal is a combination forum, resource center, newsroom, and support system all rolled into one. You can ask detailed questions about your home insurance options, seek financial advice, search for relevant conversations, and keep up with news and events that are relevant for military members, veterans, and homeowners.
An insurance provider's financial strength ratings from A.M. Best, Moody's, and S&P Global are an estimate of how financially secure the company will be in the near-to-long-term. Why does that matter? Because you don't want all of your insured assets in the hands of a company that can't afford to pay your claims in the event of a recession like the 2008 financial crisis — and you definitely don't want your insurer to file for bankruptcy or close down completely. USAA's financial strength ratings are consistently among the highest in the insurance industry, thanks in part to its stable base of customers; there will always be plenty of military members and veterans in the United States, regardless of what's happening on Wall Street. With USAA, financial strength isn't a concern at all.
Many home insurance providers, including Allstate, Amica, and State Farm, will automatically discount your premium if your home is protected by a security system. While USAA doesn't do that, it does have a partnership with ADT that allows members to save 50% off installation and 10% off monthly monitoring if you've served in the military for 20 years or more. Plus, installing a home security system is one of the most common ways to lower the cost of your home insurance premium, because it makes your home less risky to insure — so while USAA won't automatically provide a discount, it may well take a security system into account the next time it re-evaluates your premium.
Great customer service usually comes at a cost. While the price of your USAA premium will vary depending on a host of different factors, it's possible that you might find similar coverage for less money at an alternate provider — especially if you qualify for multiple discounts. If you're willing to risk a less-than-stellar customer service experience with another homeowners insurance company in exchange for a lower rate, compare quotes from several companies, starting with our picks for the best homeowners insurance companies.
Because Allstate offers more discounts and more coverage options than the average home insurance provider, it's entirely possible that Allstate may provide some military members and their families with more affordable and more flexible home insurance policies. However, USAA's customer service and financial strength rankings are consistently higher than Allstate's, so as the saying goes, you get what you pay for. We recommend getting quotes from both companies and weighing your priorities before making a decision.
While USAA sells its own home insurance, Geico outsources policies to third-party underwriters. As we noted in our Geico home insurance review, that means you won't get to choose which home insurance provider Geico matches you with. It also means your experience as a Geico customer is really a roll of the dice, depending on which company you're paired with. The advantage of USAA is that you'll know exactly what you're getting — the highest-ranked customer satisfaction and the best financial strength in the industry. Geico may offer you a cheaper premium, though, so it's always worth getting quotes and comparing your options.
The only insurance provider that rivals USAA's customer service is Amica. The two are often at the very top of J.D. Power's and Consumer Reports' rankings, sometimes alternating for first place depending on the year and the survey. Since Amica offers insurance to civilians and military members alike, it's our pick for best customer service in our roundup of the best home insurance companies, and we recommend getting a quote from Amica in addition to USAA. However, keep in mind that Amica doesn't offer many online resources, so the USAA Community may be a better resource if you're still learning about insurance.
Do I have to be in the military to qualify for USAA homeowners insurance?
What does USAA homeowners insurance cover?
USAA's standard home insurance policies cover your home and related structures, liability and medical payments when someone is injured on your property, and "loss of use" if your home is rendered uninhabitable. It also covers identity theft and military uniforms and equipment.
Does USAA offer flood insurance?
No. Like most insurance companies, USAA sells flood insurance policies through the National Flood Insurance Program (NFIP). If you live in a flood-prone area, make sure to ask your agent about adding flood insurance. For what it's worth, flood insurance is included in USAA's standard renters insurance policies.
If you qualify for membership, USAA is most likely your best choice for homeowners insurance, thanks to its stellar customer service and financial strength. The only reason to consider alternative providers is if they offer more affordable or more customizable coverage — which may well be the case for many customers. We recommend that all USAA-eligible homeowners compare quotes at several different providers, beginning with our picks for the best homeowners insurance.