Local Integral Operator Research Articles

Kernel-Based Methods for Solving Time-Dependent Advection-Diffusion Equations on Manifolds

In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and ghost point diffusion maps (GPDM), to solve the time-dependent advection-diffusion PDE on unknown smooth manifolds without and with boundaries. The core idea is to directly approximate the spatial components of the differential operator on the manifold with a local integral operator and combine it with the standard implicit time difference scheme. When the manifold has a boundary, a simplified version of the GPDM approach is used to overcome the bias of the integral approximation near the boundary. The Monte-Carlo discretization of the integral operator over the point cloud data gives rise to a mesh-free formulation that is natural for randomly distributed points, even when the manifold is embedded in high-dimensional ambient space. Here, we establish the convergence of the proposed solver on appropriate topologies, depending on the distribution of point cloud data and boundary type. We provide numerical results to validate the convergence results on various examples that involve simple geometry and an unknown manifold.

Stability for localized integral operators on weighted spaces of homogeneous type

AbstractLinear operators with off‐diagonal decay appear in many areas of mathematics including harmonic and numerical analysis, and their stability is one of the basic assumptions. In this paper, we consider a family of localized integral operators in the Beurling algebra with kernels having mild singularity near the diagonal and certain Hölder continuity property, and prove that their weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other on a space of homogeneous type with Ahlfors regular measure.

Stability of Localized Integral Operators on Normal Spaces of Homogeneous Type

The Lp-stability of linear operators is one of fundamental concepts in many research fields. In this article, we introduce a family of localized integral operators on a normal space of homogeneous type such that their kernels have some off-diagonal decay, mild singularity near the diagonal and certain Hölder regularity, and we show that localized integral operators in the family have their Lp-stability to be equivalent for different exponents .

Linear scaling Coulomb interaction in the multiwavelet basis, a parallel implementation

We present a parallel and linear scaling implementation of the calculation of the electrostatic potential arising from an arbitrary charge distribution. Our approach is making use of the multi-resolution basis of multiwavelets. The potential is obtained as the direct solution of the Poisson equation in its Green's function integral form. In the multiwavelet basis, the formally non local integral operator decays rapidly to negligible values away from the main diagonal, yielding an effectively banded structure where the bandwidth is only dictated by the requested accuracy. This sparse operator structure has been exploited to achieve linear scaling and parallel algorithms. Parallelization has been achieved both through the shared memory (OpenMP) and the message passing interface (MPI) paradigm. Our implementation has been tested by computing the electrostatic potential of the electronic density of long-chain alkanes and diamond fragments showing (sub)linear scaling with the system size and efficent parallelization.

Stability of Localized Integral Operators on Weighted L p Spaces

In this article, we consider localized integral operators whose kernels have mild singularity near the diagonal and certain Hölder regularity and decay off the diagonal. Our model example is the Bessel potential operator 𝒥γ, γ > 0. We show that if such a localized integral operator has stability on a weighted function space for some p ∈ [1, ∞) and Muckenhoupt A p -weight w, then it has stability on weighted function spaces and Muckenhoupt A p′-weights w′ for all p′ ∈ [1, ∞).

Wiener's lemma for localized integral operators

In this paper, we introduce two classes of localized integral operators on L 2 ( R d ) with the Wiener class W and the Kurbatov class K of integral operators as their models. We show that those two classes of localized integral operators are pseudo-inverse closed non-unital subalgebra of B 2 , the Banach algebra of all bounded operators on L 2 ( R d ) with usual operator norm.