Giacomo Garegnani

Biographie

Giacomo Garegnani est un postdoc à l'EPFL dans la chaire d'Analyse Numérique et Mathématiques Computationnelles (ANMC).Il a obtenu un diplôme traditionnel en piano au Conservatorio Vivaldi en 2012, et son Master en Sciences Computationnelles et Ingénierie à l'EPFL en 2017, avec un Projet de Master sous la supervision du Prof. A. Abdulle.

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https://people.epfl.ch/giacomo.garegnani?lang=fr

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A probabilistic finite element method based on random meshes: A posteriori error estimators and Bayesian inverse problems

Assyr Abdulle, Giacomo Garegnani

We present a novel probabilistic finite element method (FEM) for the solution and uncertainty quantification of elliptic partial differential equations based on random meshes, which we call random mesh FEM (RM-FEM). Our methodology allows to introduce a probability measure on classical FEMs to quantify the uncertainty due to numerical errors either in the context of a-posteriori error quantification or for FE based Bayesian inverse problems. The new approach involves only a perturbation of the mesh and an interpolation that are very simple to implement We present a posteriori error estimators and a rigorous a posteriori error analysis based uniquely on probabilistic information for standard piecewise linear FEM. A series of numerical experiments illustrates the potential of the RM-FEM for error estimation and validates our analysis. We furthermore demonstrate how employing the RM-FEM enhances the quality of the solution of Bayesian inverse problems, thus allowing a better quantification of numerical errors in pipelines of computations. (C) 2021 Elsevier B.V. All rights reserved.

ELSEVIER SCIENCE SA2021

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Drift Estimation of Multiscale Diffusions Based on Filtered Data

Assyr Abdulle, Giacomo Garegnani, Andrea Zanoni

We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure.

2021

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Learning effective models from multiscale data: filtering and Bayesian inference

Assyr Abdulle, Giacomo Garegnani, Andrea Zanoni

2021

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