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Matrices resulting from the discretization of a kernel function, e.g., in the context of integral equations or sampling probability distributions, can frequently be approximated by interpolation. In order to improve the efficiency, a…
We present a unified interpolation scheme that combines compactly-supported positive-definite kernels and multivariate polynomials. This unified framework generalizes interpolation with compactly-supported kernels and also classical…
The Empirical Interpolation Method (EIM) and its generalized version (GEIM) can be used to approximate a physical system by combining data measured from the system itself and a reduced model representing the underlying physics. In presence…
In this paper, we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations. The proposed method combines empirical interpolation techniques and local multiscale methods, such as the…
The combination of the Reduced Basis and the Empirical Interpolation Method (EIM) approaches have produced outstanding results in many disciplines. In particular, in gravitational wave (GW) science these results range from building…
We investigate new developments of the combined Reduced-Basis and Empirical Interpolation Methods (RB-EIM) for parametrized nonlinear parabolic problems. In many situations, the cost of the EIM in the offline stage turns out to be…
In this paper, we focus on the reduced basis methodology in the context of non-linear non-affinely parametrized partial differential equations in which affine decomposition necessary for the reduced basis methodology are not obtained [4,…
Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the inherent non-smoothness introduced by the…
Deep learning frameworks commonly implement convolution operators with GEMM-based algorithms. In these algorithms, convolution is implemented on top of matrix-matrix multiplication (GEMM) functions, provided by highly optimized BLAS…
The Fast Multipole Method (FMM) provides a highly efficient computational tool for solving constant coefficient partial differential equations (e.g. the Poisson equation) on infinite domains. The solution to such an equation is given as the…
Fast Incremental Expectation Maximization (FIEM) is a version of the EM framework for large datasets. In this paper, we first recast FIEM and other incremental EM type algorithms in the {\em Stochastic Approximation within EM} framework.…
We present a new rational approximation algorithm based on the empirical interpolation method for interpolating a family of parametrized functions to rational polynomials with invariant poles, leading to efficient numerical algorithms for…
Recently, a new framework to compute the photoionization rate in streamer discharges accurately and efficiently using the integral form and the fast multipole method (FMM) was presented. This paper further improves the efficiency of this…
In this paper we develop the Greedy Recombination Interpolation Method (GRIM) for finding sparse approximations of functions initially given as linear combinations of some (large) number of simpler functions. In a similar spirit to the…
This paper introduces a parallel directional fast multipole method (FMM) for solving N-body problems with highly oscillatory kernels, with a focus on the Helmholtz kernel in three dimensions. This class of oscillatory kernels requires a…
In this work we propose an efficient and accurate multi-scale optical simulation algorithm by applying a numerical version of slowly varying envelope approximation in FEM. Specifically, we employ the fast iterative method to quickly compute…
While direct statements for kernel based interpolation on regions $\Omega \subset \mathbb{R}^d$ are well researched, far less is known about corresponding inverse statements. The available inverse statements for kernel based interpolation…
In a number of problems in computational physics, a finite sum of kernel functions centered at $N$ particle locations located in a box in three dimensions must be extended by imposing periodic boundary conditions on box boundaries. Even…
A spectral mixture (SM) kernel is a flexible kernel used to model any stationary covariance function. Although it is useful in modeling data, the learning of the SM kernel is generally difficult because optimizing a large number of…
Computationally expensive Radiative Transfer Models (RTMs) are widely used} to realistically reproduce the light interaction with the Earth surface and atmosphere. Because these models take long processing time, the common practice is to…