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Person# Fabio Nobile

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Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathema

Approximation

An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word approximation is derived from Latin approximatus, from prox

Monte Carlo method

Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use rando

People doing similar research (86)

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Courses taught by this person (5)

MATH-105(a): Advanced analysis II

Etudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables.

MATH-305: Introduction to partial differential equations

This is an introductory course on Elliptic Partial Differential Equations. The course will cover the theory of both classical and generalized (weak) solutions of elliptic PDEs.

MATH-414: Stochastic simulation

The student who follows this course will get acquainted with computational tools used to analyze systems with uncertainty arising in engineering, physics, chemistry, and economics. Focus will be on sampling methods as Monte Carlo, quasi Monte Carlo, Markov Chain Monte Carlo.

We present a multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty. The algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size depends on the number of samples used to discretize the probability space. We show that this reduced system can be solved with optimal complexity. We test the multigrid method on three problems: a linear-quadratic problem for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and L^1 norm penalization on the control, in which the multigrid scheme is used within a semismooth Newton iteration; a risk-adverse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits very good performances and robustness with respect to all parameters of interest.

2023Juan Pablo Madrigal Cianci, Fabio Nobile

In this work, we present, analyze, and implement a class of multilevel Markov chain Monte Carlo(ML-MCMC) algorithms based on independent MetropolisHastings proposals for Bayesian inverse problems. In this context, the likelihood function involves solving a complex differential model, which is then approximated on a sequence of increasingly accurate discretizations. The key point of this algorithm is to construct highly coupled Markov chains together with the standard multilevel Monte Carlo argument to obtain a better cost-tolerance complexity than a single-level MCMC algorithm. Our method extends the ideas of Dodwell et al., [SIAM/ASA J. Uncertain. Quantif.,3 (2015), pp. 10751108] to a wider range of proposal distributions. We present a thorough convergence analysis of the ML-MCMC method proposed, and show, in particular, that (i) under some mild conditions on the (independent) proposals and the family of posteriors, there exists a unique invariant probability measure for the coupled chains generated by our method, and (ii) that such coupled chains are uniformly ergodic. We also generalize the cost-tolerance theorem of Dodwell et al. to our wider class of ML-MCMC algorithms. Finally, we propose a self-tuning continuation-type ML-MCMC algorithm. The presented method is tested on an array of academic examples, where some of our theoretical results are numerically verified. These numerical experiments evidence how our extended ML-MCMC method is robust when targeting some pathological posteriors, for which some of the previously proposed ML-MCMC algorithms fail.

2023Yoshihito Kazashi, Fabio Nobile

A kernel method for estimating a probability density function from an independent and identically distributed sample drawn from such density is presented. Our estimator is a linear combination of kernel functions, the coefficients of which are determined by a linear equation. An error analysis for the mean integrated squared error is established in a general reproducing kernel Hilbert space setting. The theory developed is then applied to estimate probability density func-tions belonging to weighted Korobov spaces, for which a dimension-independent convergence rate is established. Under a suitable smoothness assumption, our method attains a rate arbitrarily close to the optimal rate. Numerical results support our theory.