5 research outputs found
A convergent Born series for solving the inhomogeneous Helmholtz equation in arbitrarily large media
We present a fast method for numerically solving the inhomogeneous Helmholtz
equation. Our iterative method is based on the Born series, which we modified
to achieve convergence for scattering media of arbitrary size and scattering
strength. Compared to pseudospectral time-domain simulations, our modified Born
approach is two orders of magnitude faster and nine orders of magnitude more
accurate in benchmark tests in 1-dimensional and 2-dimensional systems
Shifted Laplacian multigrid for the elastic Helmholtz equation
The shifted Laplacian multigrid method is a well known approach for
preconditioning the indefinite linear system arising from the discretization of
the acoustic Helmholtz equation. This equation is used to model wave
propagation in the frequency domain. However, in some cases the acoustic
equation is not sufficient for modeling the physics of the wave propagation,
and one has to consider the elastic Helmholtz equation. Such a case arises in
geophysical seismic imaging applications, where the earth's subsurface is the
elastic medium. The elastic Helmholtz equation is much harder to solve than its
acoustic counterpart, partially because it is three times larger, and partially
because it models more complicated physics. Despite this, there are very few
solvers available for the elastic equation compared to the array of solvers
that are available for the acoustic one. In this work we extend the shifted
Laplacian approach to the elastic Helmholtz equation, by combining the complex
shift idea with approaches for linear elasticity. We demonstrate the efficiency
and properties of our solver using numerical experiments for problems with
heterogeneous media in two and three dimensions
A penalty method for PDE-constrained optimization in inverse problems
Many inverse and parameter estimation problems can be written as
PDE-constrained optimization problems. The goal, then, is to infer the
parameters, typically coefficients of the PDE, from partial measurements of the
solutions of the PDE for several right-hand-sides. Such PDE-constrained
problems can be solved by finding a stationary point of the Lagrangian, which
entails simultaneously updating the paramaters and the (adjoint) state
variables. For large-scale problems, such an all-at-once approach is not
feasible as it requires storing all the state variables. In this case one
usually resorts to a reduced approach where the constraints are explicitly
eliminated (at each iteration) by solving the PDEs. These two approaches, and
variations thereof, are the main workhorses for solving PDE-constrained
optimization problems arising from inverse problems. In this paper, we present
an alternative method that aims to combine the advantages of both approaches.
Our method is based on a quadratic penalty formulation of the constrained
optimization problem. By eliminating the state variable, we develop an
efficient algorithm that has roughly the same computational complexity as the
conventional reduced approach while exploiting a larger search space. Numerical
results show that this method indeed reduces some of the non-linearity of the
problem and is less sensitive the initial iterate
High order methods for acoustic scattering: Coupling Farfield Expansions ABC with Deferred-Correction methods
Arbitrary high order numerical methods for time-harmonic acoustic scattering
problems originally defined on unbounded domains are constructed. This is done
by coupling recently developed high order local absorbing boundary conditions
(ABCs) with finite difference methods for the Helmholtz equation. These ABCs
are based on exact representations of the outgoing waves by means of farfield
expansions. The finite difference methods, which are constructed from a
deferred-correction (DC) technique, approximate the Helmholtz equation and the
ABCs, with the appropriate number of terms, to any desired order. As a result,
high order numerical methods with an overall order of convergence equal to the
order of the DC schemes are obtained. A detailed construction of these DC
finite difference schemes is presented. Additionally, a rigorous proof of the
consistency of the DC schemes with the Helmholtz equation and the ABCs in polar
coordinates is also given. The results of several numerical experiments
corroborate the high order convergence of the novel method.Comment: 36 pages, 20 figure
PHIST: a Pipelined, Hybrid-parallel Iterative Solver Toolkit
The increasing complexity of hardware and software environments in high-performance computing poses big challenges on the
development of sustainable and hardware-efcient numerical software. This paper addresses these challenges in the context of sparse
solvers. Existing solutions typically target sustainability, flexibility or performance, but rarely all of them.
Our new library PHIST provides implementations of solvers for sparse linear systems and eigenvalue problems. It is a productivity
platform for performance-aware developers of algorithms and application software with abstractions that do not obscure the view on
hardware-software interaction.
The PHIST software architecture and the PHIST development process were designed to overcome shortcomings of existing packages.
An interface layer for basic sparse linear algebra functionality that can be provided by multiple backends ensures sustainability, and
PHIST supports common techniques for improving scalability and performance of algorithms such as blocking and kernel fusion.
We showcase these concepts using the PHIST implementation of a block Jacobi-Davidson solver for non-Hermitian and generalized
eigenproblems. We study its performance on a multi-core CPU, a GPU and a large-scale many-core system. Furthermore, we show
how an existing implementation of a block Krylov-Schur method in the Trilinos package Anasazi can beneft from the performance
engineering techniques used in PHIST