Related papers: Sparse Representations of Solutions to a class of …
Expand-and-sparsify representations are a class of theoretical models that capture sparse representation phenomena observed in the sensory systems of many animals. At a high level, these representations map an input $x \in \mathbb{R}^d$ to…
Computing solutions to partial differential equations using the fast Fourier transform can lead to unwanted oscillatory behavior. Due to the periodic nature of the discrete Fourier transform, waves that leave the computational domain on one…
This paper deals with numerical solutions to a partial differential equation of fractional order. Generally this type of equation describes a transition from anomalous diffusion to transport processes. From a phenomenological point of view,…
In this paper, we are concerned with the asymptotic behavior of solutions to the Cauchy problem (or initial-boundary value problem) of one-dimensional Keller-Segel model. For the Cauchy problem, we prove that the solutions…
In this paper we use a Variational Quantum Algorithm to solve Initial Value Problems with the Implicit Crank-Nicolson and the Method of Lines (MoL) evolution schemes. The unknown functions use a spectral decomposition with the Fourier…
A solver for the Poisson equation for 1D, 2D and 3D regular grids is presented. The solver applies the convolution theorem in order to efficiently solve the Poisson equation in spectral space over a rectangular computational domain.…
This article gives a new insight of kernel-based (approximation) methods to solve the high-dimensional stochastic partial differential equations. We will combine the techniques of meshfree approximation and kriging interpolation to extend…
In an M-type 2 Banach space, firstly we explore some properties of the set-valued stochastic integral associated with the stationary Poisson point process. By using the Hahn decomposition theorem and bounded linear functional, we obtain the…
A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure…
The problem of estimating sparse eigenvectors of a symmetric matrix attracts a lot of attention in many applications, especially those with high dimensional data set. While classical eigenvectors can be obtained as the solution of a…
Pseudospectral numerical schemes for solving the Dirac equation in general static curved space are derived using a pseudodifferential representation of the Dirac equation along with a simple Fourier-basis technique. Owing to the presence of…
The success of the compressed sensing paradigm has shown that a substantial reduction in sampling and storage complexity can be achieved in certain linear and non-adaptive estimation problems. It is therefore an advisable strategy for…
Deep latent generative models have attracted increasing attention due to the capacity of combining the strengths of deep learning and probabilistic models in an elegant way. The data representations learned with the models are often…
We develop fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications…
We derive fundamental sample complexity bounds for recovering sparse and structured signals for linear and nonlinear observation models including sparse regression, group testing, multivariate regression and problems with missing features.…
We develop a framework for Gaussian processes regression constrained by boundary value problems. The framework may be applied to infer the solution of a well-posed boundary value problem with a known second-order differential operator and…
We present a nonvariational setting for the Neumann problem for the Poisson equation for solutions that are H\"{o}lder continuous and that may have infinite Dirichlet integral. We introduce a distributional normal derivative on the boundary…
This paper studies the sparse identification problem of unknown sparse parameter vectors in stochastic dynamic systems. Firstly, a novel sparse identification algorithm is proposed, which can generate sparse estimates based on least squares…
We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian…
We propose a predictor-corrector adaptive method for the simulation of hyperbolic partial differential equations (PDEs) on networks under general uncertainty in parameters, initial conditions, or boundary conditions. The approach is based…