Related papers: A Fast Direct Solver for Boundary Integral Equatio…
We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the…
The objective of this study is to present a novel, efficient, and fast direct method for solving linear systems of equations whose coefficient matrix is a tridiagonal Quasi-Toeplitz matrix. Such matrices are frequently encountered in the…
This paper describes a fast direct boundary element method for elastodynamic transmission problems in two dimensions, which can be used for analyzing elastic wave scattering by an inclusion. We develop an efficient solver based on a…
Efficient solutions of large-scale, ill-conditioned and indefinite algebraic equations are ubiquitously needed in numerous computational fields, including multiphysics simulations, machine learning, and data science. Because of their…
A generalization of a recently introduced recursive numerical method for the exact evaluation of integrals of regular solid harmonics and their normal derivatives over simplex elements in $\mathbb{R}^3$ is presented. The original Quadrature…
We present factorization and solution phases for a new linear complexity direct solver designed for concurrent batch operations on fine-grained parallel architectures, for matrices amenable to hierarchical representation. We focus on the…
A numerical method for variable coefficient elliptic problems on two dimensional domains is described. The method is based on high-order spectral approximations and is designed for problems with smooth solutions. The resulting system of…
We introduce a two-level direct solver for the Hierarchical Poincar\'e-Steklov (HPS) method for solving linear elliptic PDEs. HPS combines multidomain spectral collocation with a direct solver, enabling high-order discretizations for highly…
Inversion of sparse matrices with standard direct solve schemes is robust, but computationally expensive. Iterative solvers, on the other hand, demonstrate better scalability; but, need to be used with an appropriate preconditioner (e.g.,…
This paper proposes a direct-indirect mixed Burton-Miller boundary integral equation for solving Helmholtz scattering problems with transmissive scatterers. The proposed formulation has three unknowns, one more than the number of unknowns…
We develop a triangular formulation of the hierarchical Poincar\'e-Steklov (HPS) method for elliptic partial differential equations on surfaces, allowing high-order discretizations on unstructured meshes and complex geometries. Classical…
The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order…
The recently developed quadrature by expansion (QBX) technique accurately evaluates the layer potentials with singular, weakly or nearly singular, or even hyper singular kernels in the integral equation reformulations of partial…
This paper proposes an $O(N)$ fast direct solver for two-dimensional elastic wave scattering problems. The proxy surface method is extended to elastodynamics to obtain shared coefficients for low-rank approximations from discretized…
Integral-equation-based fast direct solvers for electromagnetic scattering can substantially reduce computational costs, especially in the presence of multiple excitations. We recently proposed a new high-frequency fast direct solver…
An H-matrix accelerated direct solver employing the high-order Chebyshev-based Boundary Integral Equation (CBIE) method has been formulated, tested, and profiled for performance on high contrast dielectric materials and electrically large…
This paper introduces a new approximation scheme for solving high-dimensional semilinear partial differential equations (PDEs) and backward stochastic differential equations (BSDEs). First, we decompose a target semilinear PDE (BSDE) into…
In this paper, we present a parallel higher-order boundary integral method to solve the linear Poisson-Boltzmann (PB) equation. In our method, a well-posed boundary integral formulation is used to ensure the fast convergence of Krylov…
A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in three dimensions is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a…
A numerical method for solving elliptic PDEs with variable coefficients on two-dimensional domains is presented. The method is based on high-order composite spectral approximations and is designed for problems with smooth solutions. The…