Related papers: Evolving Generalizable Multigrid-Based Helmholtz P…
The Helmholtz equation is a prototypical model for time-harmonic wave propagation. Numerical solutions become increasingly challenging as the wave number $k$ grows, due to the equation's elliptic yet noncoercive character and the highly…
We develop and analyze a new approach for simultaneously computing multiple solutions to the Helmholtz equation for different frequencies and different forcing functions. The new Multi-Frequency WaveHoltz (MFWH) algorithm is an extension of…
This paper describes a massively parallel algebraic multigrid method based on non-smoothed aggregation. It is especially suited for solving heterogeneous elliptic problems as it uses a greedy heuristic algorithm for the aggregation that…
To solve Math Word Problems, human students leverage diverse reasoning logic that reaches different possible equation solutions. However, the mainstream sequence-to-sequence approach of automatic solvers aims to decode a fixed solution…
In this paper, we present a structured solver based on the preconditioned conjugate gradient method (PCGM) for solving the linear quadratic (LQ) optimal control problem for $K \times N$ sub-systems connected in a two-dimensional (2D) grid…
We present an improved multigrid preconditioner for the acoustic Helmholtz equation with enhanced scalability. Standard multigrid fails to converge for the Helmholtz equation, and the well-known complex shifted Laplacian method overcomes it…
This paper introduces a new pseudodifferential preconditioner for the Helmholtz equation in variable media with absorption. The pseudodifferential operator is associated with the multiplicative inverse to the symbol of the Helmholtz…
This manuscript presents an efficient solver for the linear system that arises from the Hierarchical Poincar\'e-Steklov (HPS) discretization of three dimensional variable coefficient Helmholtz problems. Previous work on the HPS method has…
We introduce an algorithm to solve linear inverse problems regularized with the total (gradient) variation in a gridless manner. Contrary to most existing methods, that produce an approximate solution which is piecewise constant on a fixed…
We present a preconditioning method for the linear systems arising from the boundary element discretization of the Laplace hypersingular equation on a $2$-dimensional triangulated surface $\Gamma$ in $\mathbb{R}^3$. We allow $\Gamma$ to…
In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence…
Solution of Helmholtz equation with impedance boundary condition on finite interval is equivalently reformulated as steady state of initial boundary value problem for first order hyperbolic system of partial differential equations.…
The purpose of this research is to describe an efficient iterative method suitable for obtaining high accuracy solutions to high frequency time-harmonic scattering problems. The method allows for both refinement of local polynomial degree…
The discontinuous Galerkin time-stepping method has many advantageous properties for solving parabolic equations. However, it requires the solution of a large nonsymmetric system at each time-step. This work develops a fully robust and…
In materials science, the challenge of rapid prototyping materials with desired properties often involves extensive experimentation to find suitable microstructures. Additionally, finding microstructures for given properties is typically an…
A new idea for iterative solution of the Helmholtz equation is presented. We show that the iteration which we denote WaveHoltz and which filters the solution to the wave equation with harmonic data evolved over one period, corresponds to a…
Topology optimization for large scale problems continues to be a computational challenge. Several works exist in the literature to address this topic, and all make use of iterative solvers to handle the linear system arising from the Finite…
A parallelization of a sweeping preconditioner for 3D Helmholtz equations without large cavities is introduced and benchmarked for several challenging velocity models. The setup and application costs of the sequential preconditioner are…
Fully realizing the potential of multigrid solvers often requires custom algorithms for a given application model, discretizations and even regimes of interest, despite considerable effort from the applied math community to develop fully…
Implicit solvers present strong limitations when used on supercomputing facilities and in particular for adaptive mesh-refinement codes. We present a new method for implicit adaptive time-stepping on adaptive mesh refinement-grids. We…