Related papers: Highly accurate operator factorization methods for…

We present a fast direct algorithm for computing symmetric factorizations, i.e. $A = WW^T$, of symmetric positive-definite hierarchical matrices with weak-admissibility conditions. The computational cost for the symmetric factorization…

This paper provides a finite difference discretization for the backward Feynman-Kac equation, governing the distribution of functionals of the path for a particle undergoing both reaction and diffusion [Hou and Deng, J. Phys. A: Math.…

This paper reviews the method of surface Laplacian differentiation to study EEG. We focus on topics that are helpful for a clear understanding of the underlying concepts and its efficient implementation, which is especially important for…

In this paper, we are concerned with the numerical solution for the two-dimensional time fractional Fokker-Planck equation with tempered fractional derivative of order $\alpha$. Although some of its variants are considered in many recent…

We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation and general hyperbolic equations. The problem is to evaluate numerically…

We introduce the sparse operator compression to compress a self-adjoint higher-order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which…

The logarithmic Laplacian on the (whole) N-dimensional Euclidean space is defined as the first variation of the fractional Laplacian of order 2s at s=0 or, alternatively, as a singular Fourier integral operator with logarithmic symbol.…

Triangular factorizations are an important tool for solving integral equations and partial differential equations with hierarchical matrices ($\mathcal{H}$-matrices). Experiments show that using an $\mathcal{H}$-matrix LR factorization to…

We present a novel numerical scheme to approximate the solution map $s\mapsto u(s) := \mathcal{L}^{-s}f$ to partial differential equations involving fractional elliptic operators. Reinterpreting $\mathcal{L}^{-s}$ as interpolation operator…

We introduce a geometric stencil selection algorithm for Laplacian in 3D that significantly improves octant-based selection considered earlier. The goal of the algorithm is to choose a small subset from a set of irregular points surrounding…

We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain $\Omega$, where $\Omega$ is either in $\mathbb{R}^n$ or in a Riemannian manifold. For linear systems of equations arising from low-order…

This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…

Let $\Delta_{\Lambda}\le \lambda_{\Lambda}$ be a semi-bounded self-adjoint realization of the Laplace operator with boundary conditions (Dirichlet, Neumann, semi-transparent) assigned on the Lipschitz boundary of a bounded obstacle…

In this paper, we consider the following nonlinear system involving the fractional Laplacian \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. (1) \end{equation}…

It is established $L^{p}$ estimates for the fractional $\Phi$-Laplacian operator defined in bounded domains where the nonlinearity is subcritical or critical in a suitable sense. Furthermore, using some fine estimates together with the…

We adopt the integral definition of the fractional Laplace operator and analyze solution techniques for fractional, semilinear, and elliptic optimal control problems posed on Lipschitz polytopes. We consider two strategies of…

We extend previously developed two-level coarsening procedures for graph Laplacian problems written in a mixed saddle point form to the fully recursive multilevel case. The resulting hierarchy of discretizations gives rise to a hierarchy of…

The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity.…

In this paper we consider approximations of Neumann problems for the integral fractional Laplacian by continuous, piecewise linear finite elements. We analyze the weak formulation of such problems, including their well-posedness and…

Factorization method is developed for a family of discretely spiked harmonic oscillators. Two sets of intertwining and ladder operators are presented to algebraically generate eigenstates with energies isomorphic to those of the ordinary…