Related papers: A kernel-based least-squares collocation method fo…
Modern large-scale kernel-based tests such as maximum mean discrepancy (MMD) and kernelized Stein discrepancy (KSD) optimize kernel hyperparameters on a held-out sample via data splitting to obtain the most powerful test statistics. While…
Nonparametric tests via kernel embedding of distributions have witnessed a great deal of practical successes in recent years. However, statistical properties of these tests are largely unknown beyond consistency against a fixed alternative.…
This paper considers a class of nonparametric autoregressive models with nonstationarity. We propose a nonparametric kernel test for the conditional mean and then establish an asymptotic distribution of the proposed test. Both the setting…
In this paper, we propose a general meshless structure-preserving Galerkin method for solving dissipative PDEs on surfaces. By posing the PDE in the variational formulation and simulating the solution in the finite-dimensional approximation…
Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work,…
We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial…
This paper introduces an approach for detecting differences in the first-order structures of spatial point patterns. The proposed approach leverages the kernel mean embedding in a novel way by introducing its approximate version tailored to…
Kernel-based hypothesis tests offer a flexible, non-parametric tool to detect high-order interactions in multivariate data, beyond pairwise relationships. Yet the scalability of such tests is limited by the computationally demanding…
We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework…
We numerically solve two-dimensional heat diffusion problems by using a simple variant of the meshfree local radial-basis function (RBF) collocation method. The main idea is to include an additional set of sample nodes outside the problem…
This paper introduces a novel meshfree methodology based on Radial Basis Function-Finite Difference (RBF-FD) approximations for the numerical solution of partial differential equations (PDEs) on surfaces of codimension 1 embedded in…
We introduce a novel kernel-based framework for learning differential equations and their solution maps that is efficient in data requirements, in terms of solution examples and amount of measurements from each example, and computational…
A kernel based procedure for correcting experimental data for distortions due to the finite resolution and limited detector acceptance is presented. The unfolding problem is known to be an ill-posed problem that can not be solved without…
Stochastic sampling methods are arguably the most direct and least intrusive means of incorporating parametric uncertainty into numerical simulations of partial differential equations with random inputs. However, to achieve an overall error…
This paper considers flow problems in multiscale heterogeneous porous media. The multiscale nature of the modeled process significantly complicates numerical simulations due to the need to compute huge and ill-conditioned sparse matrices,…
We provide a comprehensive overview of meshfree collocation methods for numerically approximating differential operators on continuously labeled unstructured point clouds. Meshfree collocation methods do not require a computational grid or…
We derive a new discretisation method for first order PDEs of arbitrary spatial dimension, which is based upon a meshfree spatial approximation. This spatial approximation is similar to the SPH (smoothed particle hydrodynamics) technique…
We suggest a method for simultaneously generating high order quadrature weights for integrals over Lipschitz domains and their boundaries that requires neither meshing nor moment computation. The weights are determined on pre-defined…
In this paper, exponential Runge-Kutta methods of collocation type (ERKC) which were originally proposed in (Appl Numer Math 53:323-339, 2005) are extended to semilinear parabolic problems with time-dependent delay. Two classes of the ERKC…
We propose a new conditional dependence measure and a statistical test for conditional independence. The measure is based on the difference between analytic kernel embeddings of two well-suited distributions evaluated at a finite set of…