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We propose a new approach to construct the eigenvalue expansion in a weighted Hilbert space of the solution to the Cauchy problem associated to Gauss-Laguerre invariant Markov semigroups that we introduce. Their generators turn out to be…
Problems with localized nonhomogeneous material properties present well-known challenges for numerical simulations. In particular, such problems may feature large differences in length scales, causing difficulties with meshing and…
We consider the task of generating discrete-time realisations of a nonlinear multivariate diffusion process satisfying an It\^o stochastic differential equation conditional on an observation taken at a fixed future time-point. Such…
Observations which are realizations from some continuous process are frequent in sciences, engineering, economics, and other fields. We consider linear models, with possible random effects, where the responses are random functions in a…
In this paper, we present a numerical approach to solve the McKean-Vlasov equations, which are distribution-dependent stochastic differential equations, under some non-globally Lipschitz conditions for both the drift and diffusion…
The moving average of the complex modulus of the analytic wavelet transform provides a robust time-scale representation for signals to small time shifts and deformation. In this work, we derive the Wiener chaos expansion of this…
This work presents a multiscale framework to solve a class of stochastic optimal control problems in the context of robot motion planning and control in a complex environment. In order to handle complications resulting from a large decision…
The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the…
In this paper, we develop an efficient numerical solver for unsteady diffusion-type partial differential equations with random coefficients. A major computational challenge in such problems lies in repeatedly handling large-scale linear…
Gaussian process regression is widely used because of its ability to provide well-calibrated uncertainty estimates and handle small or sparse datasets. However, it struggles with high-dimensional data. One possible way to scale this…
We investigate numerical behaviour of a convection diffusion equation with random coefficients by approximating statistical moments of the solution. Stochastic Galerkin approach, turning the original stochastic problem to a system of…
We study the spectral properties of a stochastic process obtained by multiplicative inversion of a non-zero-mean Gaussian process. We show that its autocorrelation and power spectrum exist for most regular processes, and we find a…
We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise. Owing to the damping term, under appropriate conditions on the nonlinearity, the solution admits a unique invariant distribution. We apply…
Gaussian processes are used in many machine learning applications that rely on uncertainty quantification. Recently, computational tools for working with these models in geometric settings, such as when inputs lie on a Riemannian manifold,…
The Gauss-Seidel method has been used for more than 100 years as the standard method for the solution of linear systems of equations under certain restrictions. This method, as well as Cramer and Jacobi, is widely used in education and…
Normalizing flows, diffusion normalizing flows and variational autoencoders are powerful generative models. This chapter provides a unified framework to handle these approaches via Markov chains. We consider stochastic normalizing flows as…
We study a finite system of diffusions on the half-line, absorbed when they hit zero, with a correlation effect that is controlled by the proportion of the processes that have been absorbed. As the number of processes in the system becomes…
Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing…
In this paper we study the high-dimensional super-resolution imaging problem. Here we are given an image of a number of point sources of light whose locations and intensities are unknown. The image is pixelized and is blurred by a known…
In many areas of science one aims to estimate latent sub-population mean curves based only on observations of aggregated population curves. By aggregated curves we mean linear combination of functional data that cannot be observed…