Related papers: Approximation by convolutions with probability den…
We investigate the convergence rates of variational posterior distributions for statistical inverse problems involving nonlinear partial differential equations (PDEs). Departing from exact Bayesian inference, variational inference…
Solutions to inverse problems that are ill-conditioned or ill-posed may have significant intrinsic uncertainty. Unfortunately, analysing and quantifying this uncertainty is very challenging, particularly in high-dimensional problems. As a…
We consider parameterized variational inverse problems that are constrained by partial differential equations (PDEs). We seek to efficiently compute the solution of the inverse problem when auxiliary model parameters, which appear in the…
We consider the convolution operator for a measure supported on complex curves. The measure which we consider here is an analogue of the affine arclength measure for real curves. By modifying a combinatorial argument called the band…
Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…
For a broad class of the Levy processes the new form (convolution type) of the infinitesimal generators is introduced. It leads to the new notions: a truncated generator, a quasi-potential. The probability of the Levy process remaining…
Motivated by recent progress in the numerical inversion of the Laplace transform, we investigate applications of finite-volume smeared spectral densities. These include the tuning of operator smearing, and the study of the finite-volume…
A new method for the solution of initial-boundary value problems for evolution PDEs recently introduced by Fokas is generalised to multidimensions. Also the relation of this method with the method of images and with the classical integral…
The objective in this paper is to demonstrate that four of the most used techniques in applied mathematics, viz., Fourier series, Fourier transform, Laplace transform and the Fourier-Laplace transform can be introduced using eigenvalue…
We investigate the variable-exponent Abel integral equations and corresponding fractional Cauchy problems. The main contributions of the work are enumerated as follows: (i) We develop an approximate inversion technique for variable-exponent…
We study computing the convolution of a private input $x$ with a public input $h$, while satisfying the guarantees of $(\epsilon, \delta)$-differential privacy. Convolution is a fundamental operation, intimately related to Fourier…
We initiate the study of \emph{inverse} problems in approximate uniform generation, focusing on uniform generation of satisfying assignments of various types of Boolean functions. In such an inverse problem, the algorithm is given uniform…
The classical Density Functional Theory (DFT) is introduced as an application of entropic inference for inhomogeneous fluids at thermal equilibrium. It is shown that entropic inference reproduces the variational principle of DFT when…
We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting…
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem…
In this work, we introduce a new generalized integral transform involving many potentially known or new transforms as special cases. Basic properties of the new integral transform, that investigated in this work, include the existence…
The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the…
The convolution potential arises in a wide variety of application areas, and its efficient and accurate evaluation encounters three challenges: singularity, nonlocality and anisotropy. We introduce a fast algorithm based on a far-field…
Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic formulation to…
Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of…