Related papers: Sparse grids vs. random points for high-dimensiona…
In this work, we study the Hermite interpolation on $n$-dimensional non-equally spaced, rectilinear grids over a field $\Bbbk $ of characteristic zero, given the values of the function at each point of the grid and the partial derivatives…
The Active Subspace (AS) method is a widely used technique for identifying the most influential directions in high-dimensional input spaces that affect the output of a computational model. The standard AS algorithm requires a sufficient…
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…
In this paper we study the problem of maximizing the distance to a given point over an intersection of balls. It was already known that this problem can be solved in polynomial time and space if the given point is not in the convex hull of…
We investigate the solution of low-rank matrix approximation problems using the truncated SVD. For this purpose, we develop and optimize GPU implementations for the randomized SVD and a blocked variant of the Lanczos approach. Our work…
This work investigates the use of sparse polynomial interpolation as a model order reduction method for the incompressible Navier-Stokes equations. Numerical results are presented underscoring the validity of sparse polynomial…
We consider approximation of diameter of a set $S$ of $n$ points in dimension $m$. E$\tilde{g}$ecio$\tilde{g}$lu and Kalantari \cite{kal} have shown that given any $p \in S$, by computing its farthest in $S$, say $q$, and in turn the…
In this paper, we introduce a new approach to proving the convergence of the Stochastic Approximation (SA) and the Stochastic Gradient Descent (SGD) algorithms. The new approach is based on a concept called GSLLN (Generalized Strong Law of…
In this paper, we study the \emph{sparse integer least squares problem} (SILS), an NP-hard variant of least squares with sparse $\{0, \pm 1\}$-vectors. We propose an $\ell_1$-based SDP relaxation, and a randomized algorithm for SILS, which…
We propose an adaptive sparse grid stochastic collocation approach based upon Leja interpolation sequences for approximation of parameterized functions with high-dimensional parameters. Leja sequences are arbitrarily granular (any number of…
A polynomial matrix inequality is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of…
Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a…
By combining a certain approximation property in the spatial domain, and weighted $\ell_2$-summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in [M. Bachmayr, A. Cohen, R. DeVore and G.…
We analyze a sublinear RAlSFA (Randomized Algorithm for Sparse Fourier Analysis) that finds a near-optimal B-term Sparse Representation R for a given discrete signal S of length N, in time and space poly(B,log(N)), following the approach…
We introduce sparse random projection, an important dimension-reduction tool from machine learning, for the estimation of discrete-choice models with high-dimensional choice sets. Initially, high-dimensional data are compressed into a…
Finding suitable points for multivariate polynomial interpolation and approximation is a challenging task. Yet, despite this challenge, there has been tremendous research dedicated to this singular cause. In this paper, we begin by…
Stochastic approximation (SA) is a powerful and scalable computational method for iteratively estimating the solution of optimization problems in the presence of randomness, particularly well-suited for large-scale and streaming data…
Generalized linear mixed models (GLMMs) are a widely used tool in statistical analysis. The main bottleneck of many computational approaches lies in the inversion of the high dimensional precision matrices associated with the random…
Many high dimensional integrals can be reduced to the problem of finding the relative measures of two sets. Often one set will be exponentially larger than the other, making it difficult to compare the sizes. A standard method of dealing…
In this paper, we show that under over-parametrization several standard stochastic optimization algorithms escape saddle-points and converge to local-minimizers much faster. One of the fundamental aspects of over-parametrized models is that…