Related papers: Sparse Representations of Solutions to a class of …

In this article, we establish a probabilistic representation for the second-order moment of the solution of stochastic heat equation in $[0,1] \times \bR^d$, with multiplicative noise, which is fractional in time and colored in space. This…

The Initial-Boundary Value Problem for the heat equation is solved by using a new algorithm based on a random walk on heat balls. Even if it represents a sophisticated generalization of the Walk on Spheres (WOS) algorithm introduced to…

Identifying damage of structural systems is typically characterized as an inverse problem which might be ill-conditioned due to aleatory and epistemic uncertainties induced by measurement noise and modeling error. Sparse representation can…

In this work we study convergence properties of sparse polynomial approximations for a class of affine parametric saddle point problems. Such problems can be found in many computational science and engineering fields, including the Stokes…

Increasingly larger data sets of processes in space and time ask for statistical models and methods that can cope with such data. We show that the solution of a stochastic advection-diffusion partial differential equation provides a…

We consider a focusing Davey-Stewartson system and construct the solution of the Cauchy problem in the possible presence of exceptional points (and/or curves).

Employing a limiting case of a conjecture for constructing piecewise separable-variables functions, the elements of the Pseudoanalytic Function Theory are used for numerically approaching solutions of the forward Dirichlet boundary value…

In this work, we consider the numerical solution of an initial boundary value problem for the distributed order time fractional diffusion equation. The model arises in the mathematical modeling of ultra-slow diffusion processes observed in…

Sparse autoencoders (SAEs) are a promising approach to extracting features from neural networks, enabling model interpretability as well as causal interventions on model internals. SAEs generate sparse feature representations using a…

We present a Bayesian non-parametric way of inferring stochastic differential equations for both regression tasks and continuous-time dynamical modelling. The work has high emphasis on the stochastic part of the differential equation, also…

This book is an introduction to the theory of stochastic partial differential equations (SPDEs), using the random field approach pioneered by J.B. Walsh (1986). It consists of two blocks: the core matter (Chapters 1 to 6) and the appendices…

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel…

We formulate an initial- and Dirichlet boundary- value problem for a linear stochastic heat equation, in one space dimension, forced by an additive space-time white noise. First, we approximate the mild solution to the problem by the…

Spectral methods for solving partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) often use Fourier or polynomial spectral expansions on either uniform and non-uniform grids. However, while very widely…

We study a semidiscrete analogue of the Unified Transform Method introduced by A. S. Fokas, to solve initial-boundary-value problems for linear evolution partial differential equations with constant coefficients on the finite interval $x…

Sparse coding has been popularly used as an effective data representation method in various applications, such as computer vision, medical imaging and bioinformatics, etc. However, the conventional sparse coding algorithms and its manifold…

In this paper we introduce a general stochastic representation for an important class of processes with resetting. It allows to describe any stochastic process intermittently terminated and restarted from a predefined random or non-random…

In the second part of the series papers, we set out to study the algorithmic efficiency of sparse sensing. Stemmed from co-prime sensing, we propose a generalized framework, termed Diophantine sensing, which utilizes generic Diophantine…

In this paper, we discuss application of iterative Stochastic Optimization routines to the problem of sparse signal recovery from noisy observation. Using Stochastic Mirror Descent algorithm as a building block, we develop a multistage…

Given a sample covariance matrix, we examine the problem of maximizing the variance explained by a linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This is known as sparse…