Related papers: A fast direct solver for network matrices
Deep learning for distribution grid optimization can be advocated as a promising solution for near-optimal yet timely inverter dispatch. The principle is to train a deep neural network (DNN) to predict the solutions of an optimal power flow…
In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in $\mathbb{R}^d$, $d=2,3$. We consider the stochastic realizations using checkerboard configuration of the…
In numerical linear algebra, considerable effort has been devoted to obtaining faster algorithms for linear systems whose underlying matrices exhibit structural properties. A prominent success story is the method of generalized nested…
This manuscript presents a fast direct solution technique for solving two dimensional wave scattering problems from quasi-periodic multilayered structures. When the interface geometries are complex, the dominant term in the computational…
In the classical theory of cubic interpolation splines there exists an algorithm which works with only $O\left( n\right)$ arithmetic operations. Also, the smoothing cubic splines may be computed via the algorithm of Reinsch which reduces…
We address a linear fractional differential equation and develop effective solution methods using algorithms for inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed…
Fast linear transforms are ubiquitous in machine learning, including the discrete Fourier transform, discrete cosine transform, and other structured transformations such as convolutions. All of these transforms can be represented by dense…
Selected inversion is essential for applications such as Bayesian inference, electronic structure calculations, and inverse covariance estimation, where computing only specific elements of large sparse matrix inverses significantly reduces…
This paper discusses the explicit inverse of a class of seven-diagonal (near) Toeplitz matrices, which arises in the numerical solutions of nonlinear fourth-order differential equation with a finite difference method. A non-recurrence…
A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in R^3 is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a surface to a…
Numerically obtaining the inverse of a function is a common task for many scientific problems, often solved using a Newton iteration method. Here we describe an alternative scheme, based on switching variables followed by spline…
In this paper, a two-sided variable-coefficient space-fractional diffusion equation with fractional Neumann boundary condition is considered. To conquer the weak singularity caused by nonlocal space-fractional differential operators, a…
We propose a novel efficient algorithm to solve Poisson equation in irregular two dimensional domains for electrostatics. It can handle Dirichlet, Neumann or mixed boundary problems in which the filling media can be homogeneous or…
A linearly implicit conservative difference scheme is applied to discretize the attractive coupled nonlinear Schr\"odinger equations with fractional Laplacian. Complex symmetric linear systems can be obtained, and the system matrices are…
We consider scattered data approximation on product regions of equal and different dimensionality. On each of these regions, we assume quasi-uniform but unstructured data sites and construct optimal sparse grids for scattered data…
New implicit and implicit-explicit time-stepping methods for the wave equation in second-order form are described with application to two and three-dimensional problems discretized on overset grids. The implicit schemes are single step,…
A fast and accurate algorithm for solving a Bernstein-Vandermonde linear system is presented. The algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of Neville…
An efficient nonlinear contrast source inversion scheme for electromagnetic imaging of sparse two-dimensional investigation domains is proposed. To avoid generating a sequence of linear sparse optimization problems, the non-linearity is…
Efficient numerical solvers for sparse linear systems are crucial in science and engineering. One of the fastest methods for solving large-scale sparse linear systems is algebraic multigrid (AMG). The main challenge in the construction of…
Nonnegative matrix factorization (NMF) is a powerful technique for dimension reduction, extracting latent factors and learning part-based representation. For large datasets, NMF performance depends on some major issues: fast algorithms,…