Related papers: Polynomial (chaos) approximation of maximum eigenv…
This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are…
The approximation of smooth functions with a spectral basis typically leads to rapidly decaying coefficients where the rate of decay depends on the smoothness of the function and vice-versa. The optimal number of degrees of freedom in the…
We continue the study of random matrix universality in two-dimensional conformal field theories. This is facilitated by expanding the spectral form factor in a basis of modular invariant eigenfunctions of the Laplacian on the fundamental…
Most approximation methods in high dimensions exploit smoothness of the function being approximated. These methods provide poor convergence results for non-smooth functions with kinks. For example, such kinks can arise in the uncertainty…
This is a survey on best polynomial approximation on the unit sphere and the unit ball. The central problem is to describe the approximation behavior of a function by polynomials via smoothness of the function. A major effort is to identify…
Numerical solutions of differential equations are usually not smooth functions. However, they should resemble the smoothness of the corresponding real solutions in one way or another. In two of our recent papers, a kind of spacial…
We study two inexact methods for solutions of random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric matrix operator, the methods solve for eigenvalues and…
We study efficient solution methods for stochastic eigenvalue problems arising from discretization of self-adjoint partial differential equations with random data. With the stochastic Galerkin approach, the solutions are represented as…
A rational approximation by a ratio of polynomial functions is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non- Lipschitz functions, where polynomial…
We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed:…
The estimation of probability density functions (PDF) using approximate maps is a fundamental building block in computational probability. We consider forward problems in uncertainty quantification: the inputs or the parameters of a…
In many machine learning and data related applications, it is required to have the knowledge of approximate ranks of large data matrices at hand. In this paper, we present two computationally inexpensive techniques to estimate the…
A key quantity that occurs in the error analysis of several numerical methods for eigenvalue problems is the distance between the eigenvalue of interest and the next nearest eigenvalue. When we are interested in the smallest or fundamental…
This paper discusses a methodology for determining a functional representation of a random process from a collection of scattered pointwise samples. The present work specifically focuses onto random quantities lying in a high dimensional…
We give a new fast method for evaluating sprectral approximations of nonlinear polynomial functionals. We prove that the new algorithm is convergent if the functions considered are smooth enough, under a general assumption on the spectral…
A central challenge in machine learning is to understand how noise or measurement errors affect low-rank approximations, particularly in the spectral norm. This question is especially important in differentially private low-rank…
Polynomial chaos expansion is a popular way to develop surrogate models for stochastic systems with arbitrary random variables. Standard techniques such as Galerkin projection, stochastic collocation, and least squares approximation, are…
Multivariate global polynomial approximations - such as polynomial chaos or stochastic collocation methods - are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses…
Bernstein polynomial approximation to a continuous function has a slower rate of convergence as compared to other approximation methods. "The fact seems to have precluded any numerical application of Bernstein polynomials from having been…
Spectral polynomial approximation of smooth functions allows real-time manipulation of and computation with them, as in the Chebfun system. Extension of the technique to two-dimensional and three-dimensional functions on hyperrectangles has…