This thesis manuscript deals with the study of certain kernel regression methods, which are specifically built to approximate functions which are related to partial differential equations (PDEs). In particular, the matter of the construction of covariance kernels which respect the properties of the PDE is studied. This question is mostly studied fr...
We consider an unknown multivariate function representing a system—such as a complex numerical simulator—taking both deterministic and uncertain inputs. Our objective is to estimate the set of deterministic inputs leading to outputs whose probability (with respect to the distribution of the uncertain inputs) of belonging to a given set is less than...
Published in Acta Universitatis Sapientiae, Mathematica
In the present work, we establish the strong approximations of the empirical k-spacings process {αn(x): 0 ≤ x
In this article, we present the exact expression of the L 2-norm of the forward stochastic integral driven by the multi-dimensional fractional Brownian motion with parameter 1 2
This paper revisits single-channel audio source separation based on a probabilistic generative model of a mixture signal defined in the continuous time domain. We assume that each source signal follows a non-stationary Gaussian process (GP), i.e., any finite set of sampled points follows a zero-mean multivariate Gaussian distribution whose covarian...
Published in Machine Learning: Science and Technology
We develop a probabilistic Stokes flow framework, using physics informed Gaussian processes, which can be used to solve both forward/inverse flow problems with missing and/or noisy data. The physics of the problem, specified by the Stokes and continuity equations, is exactly encoded into the inference framework. Crucially, this means that we do not...
In this work, we present a detailed analysis on the exact expression of the $L^2$-norm of the symmetric-Stratonovich stochastic integral driven by a multi-dimensional fractional Brownian motion $B$ with parameter $\frac{1}{4}
Published in Journal of Physics A: Mathematical and Theoretical
This paper presents and explores a diffusion model that generalizes Brownian motion (BM). On the one hand, as BM: the model’s mean square displacement grows linearly in time, and the model is Gaussian and selfsimilar (with Hurst exponent 12 ). On the other hand, in sharp contrast to BM: the model is not Markov, its increments are not stationary, an...
Many engineering problems are described by complex multidisciplinary systems, whose behavior is dictated by a non-linear system of equations called multidisciplinary analysis (MDA). When optimizing these systems, the resolution of the MDA at each evaluated design space point often represents a heavy computational burden, particularly when high-fide...
Published in Biometrics
We propose a model-based approach that combines Bayesian variable selection tools, a novel spatial kernel convolution structure, and autoregressive processes for detecting a subject's brain activation at the voxel level in complex-valued functional magnetic resonance imaging (CV-fMRI) data. A computationally efficient Markov chain Monte Carlo algor...
