arXiv subjects

Giuseppe Carere

Publications and source records attributed to Giuseppe Carere.

5 recordsLinked to original sources

arXiv2026

Generalised Rank-Constrained Approximations of Hilbert-Schmidt Operators on Separable Hilbert Spaces and Applications

In this work we solve, for given bounded operators $B,C$ and Hilbert-Schmidt operator $M$ acting on potentially infinite-dimensional separable Hilbert spaces, the reduced rank approximation problem, $\min\{\lVert M-BXC\rVert_{L_2}:\ \text{dim ran}\ X\leq r\}.$ This extends the result of Sondermann (Statistische Hefte, 1986) and Friedland and Torokhti (SIAM J. Matrix Analysis and Applications, 2007), which studies this problem in the case of matrices $M$, $B$, $C$, $X$, and the analysis involves the Moore-Penrose inverse. In classical approximation problems that can be solved by the singular value decomposition or Moore-Penrose inverse, the solution satisfies a minimal norm property. Friedland and Torokhti state such a minimal norm property of the solution. We show that this minimal norm property does not hold in general and give a modified minimality property that does hold. We show that the solution may be discontinuous in infinite-dimensional settings. We give conditions for continuity of the solutions and construct continuous approximations when such conditions are not met. Finally, we study problems from signal processing, reduced rank regression and linear operator learning under a rank constraint. Our theoretical results enable us to explicitly find solutions to these problems and to characterise their existence, uniqueness and minimality property.

math.FA↗

arXiv2026

Optimal low-rank posterior mean and distribution approximation in linear Gaussian inverse problems on Hilbert spaces

We construct optimal low-rank approximations for the Gaussian posterior distribution in linear Gaussian inverse problems with possibly infinite-dimensional separable Hilbert parameter spaces and finite-dimensional data spaces. We first consider approximate posteriors in which the means vary and the posterior covariance is kept fixed, for all possible realisations of the data simultaneously. We give necessary and sufficient conditions for these approximating posteriors to be equivalent to the exact posterior. For such approximations, we measure the data-averaged approximation error with the Kullback-Leibler, Rényi and Amari $α$-divergences for $α\in(0,1)$, and the Hellinger distance. With the loss in Kullback-Leibler and Rényi divergences, we find the optimal approximations and formulate an equivalent condition for their uniqueness, extending the work in finite dimensions of Spantini et al. (SIAM J. Sci. Comput. 2015). We then consider joint low-rank approximation of the mean and covariance. For the reverse Kullback-Leibler divergence, the optimal approximations of the mean and of the covariance yield an optimal joint approximation of the mean and covariance. We interpret one such joint approximation in terms of an optimal projector in parameter space, and show that this approximation amounts to solving a Bayesian inverse problem with projected forward model. Extensive numerical examples demonstrate some of our theoretical findings.

math.ST↗

arXiv2025

Optimal low-rank posterior covariance approximation in linear Gaussian inverse problems on Hilbert spaces

For linear inverse problems with Gaussian priors and Gaussian observation noise, the posterior is Gaussian, with mean and covariance determined by the conditioning formula. The covariance is the central object for uncertainty quantification, as it encodes the variability of the posterior distribution and thus the uncertainty in the posterior mean estimate. Using the Feldman-Hajek theorem, we analyse the prior-to-posterior update and its low-rank approximation for infinite-dimensional Hilbert parameter spaces and finite-dimensional observations. We show that the posterior distribution differs from the prior on a finite-dimensional subspace, and construct low-rank approximations to the posterior covariance, while keeping the mean fixed. Since in infinite dimensions, not all low-rank covariance approximations yield approximate posterior distributions which are equivalent to the posterior and prior distribution, we characterise the low-rank covariance approximations which do yield this equivalence, and their respective inverses, or `precisions'. For such approximations, a family of measure approximation problems is solved by identifying the low-rank approximations which are optimal for various losses simultaneously. These loss functions include the family of Rényi divergences, the Amari $α$-divergences for $α\in(0,1)$, the Hellinger metric and the Kullback-Leibler divergence. Our results extend those of Spantini et al. (SIAM J. Sci. Comput. 2015) to Hilbertian parameter spaces, and provide theoretical underpinning for the construction of low-rank approximations of discretised versions of the infinite-dimensional inverse problem, by formulating discretisation independent results.

math.ST↗

arXiv2025

Optimal low-rank approximations for linear Gaussian inverse problems on Hilbert spaces, Part I: posterior covariance approximation

For linear inverse problems with Gaussian priors and Gaussian observation noise, the posterior is Gaussian, with mean and covariance determined by the conditioning formula. Using the Feldman-Hajek theorem, we analyse the prior-to-posterior update and its low-rank approximation for infinite-dimensional Hilbert parameter spaces and finite-dimensional observations. We show that the posterior distribution differs from the prior on a finite-dimensional subspace, and construct low-rank approximations to the posterior covariance, while keeping the mean fixed. Since in infinite dimensions, not all low-rank covariance approximations yield approximate posterior distributions which are equivalent to the posterior and prior distribution, we characterise the low-rank covariance approximations which do yield this equivalence, and their respective inverses, or `precisions'. For such approximations, a family of measure approximation problems is solved by identifying the low-rank approximations which are optimal for various losses simultaneously. These loss functions include the family of Rényi divergences, the Amari $α$-divergences for $α\in(0,1)$, the Hellinger metric and the Kullback-Leibler divergence. Our results extend those of Spantini et al. (SIAM J. Sci. Comput. 2015) to Hilbertian parameter spaces, and provide theoretical underpinning for the construction of low-rank approximations of discretised versions of the infinite-dimensional inverse problem, by formulating discretization independent results.

math.ST↗

arXiv2021

A weighted POD-reduction approach for parametrized PDE-constrained Optimal Control Problems with random inputs and applications to environmental sciences

Reduced basis approximations of Optimal Control Problems (OCPs) governed by steady partial differential equations (PDEs) with random parametric inputs are analyzed and constructed. Such approximations are based on a Reduced Order Model, which in this work is constructed using the method of weighted Proper Orthogonal Decomposition. This Reduced Order Model then is used to efficiently compute the reduced basis approximation for any outcome of the random parameter. We demonstrate that such OCPs are well-posed by applying the adjoint approach, which also works in the presence of admissibility constraints and in the case of non linear-quadratic OCPs, and thus is more general than the conventional Lagrangian approach. We also show that a step in the construction of these Reduced Order Models, known as the aggregation step, is not fundamental and can in principle be skipped for noncoercive problems, leading to a cheaper online phase. Numerical applications in three scenarios from environmental science are considered, in which the governing PDE is steady and the control is distributed. Various parameter distributions are taken, and several implementations of the weighted Proper Orthogonal Decomposition are compared by choosing different quadrature rules.

math.NA↗