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We evaluate the thermal corrections to the generalized Gaussian effective potential. We carry out the calculations of the lowest order corrections in the case of self-interacting scalar fields in one and two spatial dimensions, and study…

We present a new adaptive method for electronic structure calculations based on novel fast algorithms for reduction of multivariate mixtures. In our calculations, spatial orbitals are maintained as Gaussian mixtures whose terms are selected…

We present an analytical closed form expression, which gives a good approximate propagator for diffusion on the sphere. Our formula is the spherical counterpart of the Gaussian propagator for diffusion on the plane. While the analytical…

Phenomenologically interesting scalar potentials are highly atypical in generic random landscapes. We develop the mathematical techniques to generate constrained random potentials, i.e. Slepian models, which can globally represent…

The Gromov-Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures…

We present a simple, robust and black-box approach to the implementation and use of local, periodic, atom-centered Gaussian basis functions within a plane wave code, in a computationally efficient manner. The procedure outlined is based on…

Gaussian scale spaces are a cornerstone of signal representation and processing, with applications in filtering, multiscale analysis, anti-aliasing, and many more. However, obtaining such a scale space is costly and cumbersome, in…

The use of Gaussian processes (GPs) is supported by efficient sampling algorithms, a rich methodological literature, and strong theoretical grounding. However, due to their prohibitive computation and storage demands, the use of exact GPs…

In this work, we study probability functions associated with Gaussian mixture models. Our primary focus is on extending the use of spherical radial decomposition for multivariate Gaussian random vectors to the context of Gaussian mixture…

Collocation boundary element methods for integral equations are easier to implement than Galerkin methods because the elements of the discretization matrix are given by lower-dimensional integrals. For that same reason, the matrix assembly…

Bayesian learning using Gaussian processes provides a foundational framework for making decisions in a manner that balances what is known with what could be learned by gathering data. In this dissertation, we develop techniques for…

A fundamental drawback of kernel-based statistical models is their limited scalability to large data sets, which requires resorting to approximations. In this work, we focus on the popular Gaussian kernel and on techniques to linearize…

3D Gaussian Splatting (3DGS) has demonstrated impressive Novel View Synthesis (NVS) results in a real-time rendering manner. During training, it relies heavily on the average magnitude of view-space positional gradients to grow Gaussians to…

An optimized Rayleigh-Schr\"{o}dinger expansion scheme of solving the functional Schr\"odinger equation with an external source is proposed to calculate the effective potential beyond the Gaussian approximation. For a scalar field theory…

We consider linear dynamical systems of ordinary differential equations or differential algebraic equations. Physical parameters are substituted by random variables for an uncertainty quantification. We expand the state variables as well as…

We consider Gaussian distributions on certain Riemannian symmetric spaces. In contrast to the Euclidean case, it is challenging to compute the normalization factors of such distributions, which we refer to as partition functions. In some…

Over the last few years there have been dramatic advances in our understanding of mathematical and computational models of complex systems in the presence of uncertainty. This has led to a growth in the area of uncertainty quantification as…

We consider single particle Schrodinger operators with a gap in the en ergy spectrum. We construct a complete, orthonormal basis function set for the inv ariant space corresponding to the spectrum below the spectral gap, which are…

Gaussian processes (GPs) are typically criticised for their unfavourable scaling in both computational and memory requirements. For large datasets, sparse GPs reduce these demands by conditioning on a small set of inducing variables…

In this paper, we develop an adaptive Generalized Multiscale Discontinuous Galerkin Method (GMs-DGM) for a class of high-contrast flow problems, and derive a-priori and a-posteriori error estimates for the method. Based on the a-posteriori…