Related papers: An O(N) algorithm for constructing the solution op…
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans…
We propose a graph-based sweep algorithm for solving the steady state, mono-energetic discrete ordinates on meshes of high-order curved mesh elements. Our spatial discretization consists of arbitrarily high-order discontinuous Galerkin…
Many applications rely on time-intensive matrix operations, such as factorization, which can be sped up significantly for large sparse matrices by interpreting the matrix as a sparse graph and computing a node ordering that minimizes the…
Linear-scaling electronic-structure techniques, also called O(N) techniques, rely heavily on the multiplication of sparse matrices, where the sparsity arises from spatial cut-offs. In order to treat very large systems, the calculations must…
The minimum degree algorithm is one of the most widely-used heuristics for reducing the cost of solving large sparse systems of linear equations. It has been studied for nearly half a century and has a rich history of bridging techniques…
This paper introduces a novel approach to algebraic multigrid methods for large systems of linear equations coming from finite element discretizations of certain elliptic second order partial differential equations. Based on a discrete…
Numerical continuation calculations for ordinary differential equations (ODEs) are, by now, an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences. Although…
We propose an algorithm for solution of high-dimensional evolutionary equations (ODEs and discretized time-dependent PDEs) in the Tensor Train (TT) decomposition, assuming that the solution and the right-hand side of the ODE admit such a…
We develop numerical methods for elliptic systems governed by partial segregation constraints, in which three nonnegative components are required to have a vanishing pointwise product throughout the domain. This constraint enforces that at…
Perturbing a deterministic $n$-dimensional matrix with small Gaussian noise is a cornerstone of smoothed analysis of algorithms [Spielman and Teng, JACM 2004], as it reduces the condition number of the input to $O(n)$, and with it the…
The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE…
The obstacle problem is a class of free boundary problems which finds applications in many disciplines such as porous media, financial mathematics and optimal control. In this paper, we propose two operator-splitting methods to solve the…
We consider approximations to the solutions of differential Riccati equations in the context of linear quadratic regulator problems, where the state equation is governed by a multiscale operator. Similarly to elliptic and parabolic…
This article deals with the computation of guaranteed lower bounds of the error in the framework of finite element (FE) and domain decomposition (DD) methods. In addition to a fully parallel computation, the proposed lower bounds separate…
Traditional solution approaches for problems in quantum mechanics scale as $\mathcal O(M^3)$, where $M$ is the number of electrons. Various methods have been proposed to address this issue and obtain linear scaling $\mathcal O(M)$. One…
Finite element codes typically use data structures that represent unstructured meshes as collections of cells, faces, and edges, each of which require associated coordinate systems. One then needs to store how the coordinate system of each…
We consider the numerical solution of the scattering of time-harmonic plane waves from an infinite periodic array of reflection or transmission obstacles in a homogeneous background medium, in two dimensions. Boundary integral formulations…
Motivated by a certain molecular reconstruction methodology in cryo-electron microscopy, we consider the problem of solving a linear system with two unknown orthogonal matrices, which is a generalization of the well-known orthogonal…
Neural networks are powerful tools for approximating high dimensional data that have been used in many contexts, including solution of partial differential equations (PDEs). We describe a solver for multiscale fully nonlinear elliptic…
We consider fast deterministic algorithms to identify the "best" linearly independent terms in multivariate mixtures and use them to compute, up to a user-selected accuracy, an equivalent representation with fewer terms. One algorithm…