Related papers: Infinite-Dimensional Quadrature and Quantization
Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The approximation of such…
We introduce and analyze a parallel sequential Monte Carlo methodology for the numerical solution of optimization problems that involve the minimization of a cost function that consists of the sum of many individual components. The proposed…
In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of ANOVA-type. The weights model the relative importance…
We provide full theoretical guarantees for the convergence behaviour of diffusion-based generative models under the assumption of strongly log-concave data distributions while our approximating class of functions used for score estimation…
Integration over non-negative integrands is a central problem in machine learning (e.g. for model averaging, (hyper-)parameter marginalisation, and computing posterior predictive distributions). Bayesian Quadrature is a probabilistic…
This work is motivated by the need to study the impact of data uncertainties and material imperfections on the solution to optimal control problems constrained by partial differential equations. We consider a pathwise optimal control…
Two generalizations of It\^o formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations, when they occur in examples and it is applied to the case of a…
We extend the branching process based numerical algorithm of Bouchard et al. [3], that is dedicated to semilinear PDEs (or BSDEs) with Lipschitz nonlinearity, to the case where the nonlinearity involves the gradient of the solution. As in…
Let $\mu$ be a Borel probability measure associated with an iterated function system consisting of a countably infinite number of contracting similarities and an infinite probability vector. In this paper, we study the quantization…
Introducing inequality constraints in Gaussian process (GP) models can lead to more realistic uncertainties in learning a great variety of real-world problems. We consider the finite-dimensional Gaussian approach from Maatouk and Bay (2017)…
This paper concerns parameterized convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional Banach (resp. finite-dimensional) spaces and that are indexed by an arbitrary fixed set T…
Bayesian inference is a widely used technique for real-time characterization of quantum systems. It excels in experimental characterization in the low data regime, and when the measurements have degrees of freedom. A decisive factor for its…
Certain previously known upper bounds on the moments of the norm of martingales in 2-smooth Banach spaces are improved. Some of these improvements hold even for sums of independent real-valued random variables. Applications to concentration…
We propose an optimization algorithm to compute the optimal sensor locations in experimental design in the formulation of Bayesian inverse problems, where the parameter-to-observable mapping is described through an integral equation and its…
A generalized Bloch sphere, in which the states of a quantum entity of arbitrary dimension are geometrically represented, is investigated and further extended, to also incorporate the measurements. This extended representation constitutes a…
The Lipschitz space of an infinite (locally-finite) graph is defined as the set of functions on the vertices of the graph such that the differences of the values between adjacent vertices remain bounded. In this paper we prove that this set…
Computation of the marginal likelihood from a simulated posterior distribution is central to Bayesian model selection but is computationally difficult. I argue that the marginal likelihood can be reliably computed from a posterior sample by…
We consider a second-order elliptic boundary value problem with strongly monotone and Lipschitz-continuous nonlinearity. We design and study its adaptive numerical approximation interconnecting a finite element discretization, the…
Integration is affected by the curse of dimensionality and quickly becomes intractable as the dimensionality of the problem grows. We propose a randomized algorithm that, with high probability, gives a constant-factor approximation of a…
These lecture notes contain an introduction to some of the fundamental ideas and results in analysis and probability on infinite-dimensional spaces, mainly Gaussian measures on Banach spaces. They originated as the notes for a topics course…