Flexible multi-output multifidelity uncertainty quantification via MLBLUE

Tuesday, September 20, 2022 - 1:25pm - 2:25pm
Walter Library 402

Matteo Croci (The University of Texas at Austin)

You may attend the talk either in person in Walter 402 or register via Zoom. Registration is required to access the Zoom webinar.

A central task in forward uncertainty quantification (UQ) is estimating the expectation of one or more quantities of interest (QoIs). In computational engineering UQ problems often involve multiple QoIs, and extremely heterogeneous models, both in terms of how they are constructed (varying grids, equations, or dimensions, different physics, surrogate and reduced-order models...) and in terms of their input-output structure (different models might have different uncertain inputs and yield different QoIs). In this complex scenario it is crucial to design estimators that are as flexible and as efficient as possible.

Multilevel (or multifidelity) Monte Carlo (MLMC) methods are often the go-to methods for estimating expectations as they are able to exploit the correlations between models to significantly reduce the estimation cost. However, multi-output strategies in MLMC methods are either sub-optimal, or non-existent.

In this talk we focus on multilevel best linear unbiased estimators (MLBLUEs, Schaden and Ullmann, SIAM/ASA JUQ, 2021). MLBLUEs are extremely flexible and have the appealing property of being provably optimal among all multilevel linear unbiased estimators, making them, in our opinion, one of the most powerful MLMC methods available in the literature. Nevertheless, MLBLUEs have two limitations: 1) their setup requires solving their model selection and sample allocation problem (MOSAP), which is a non-trivial nonlinear optimization procedure, and 2) they can only work with one scalar QoI at a time.

In this talk we show how the true potential of MLBLUEs can be unlocked:

1) We present a new formulation of their MOSAP that can be solved almost as easily and efficiently as a linear program.

2) We extend MLBLUEs to the multi- and infinite-dimensional output case.

3) We provide multi-output MLBUE MOSAP formulations that can be solved efficiently and consistently with widely available optimization software.

We show that the new multi-output MLBLUEs can be setup very efficiently and that they significantly outperform existing MLMC methods in practical problems with heterogeneous model structure.

Matteo Croci is a postdoctoral researcher at the Oden Institute for Computational Engineering and Sciences at the University of Texas at Austin working with Karen E. Willcox and Robert D. (Bob) Moser. Before moving to Austin in late 2022, Matteo worked for two years as a postdoctoral researcher in the Mathematical Institute at the University of Oxford (UK) under the supervision of Michael B. (Mike) Giles. Matteo obtained his PhD from the University of Oxford (UK) in March 2020 under the supervision of Patrick E. Farrell, Michael B. (Mike) Giles, and in collaboration with Marie E. Rognes from Simula Research Laboratory (Oslo, Norway). Matteo has a MSc in Mathematical Modelling and Scientific Computing from the University of Oxford (UK), and a BSc in Mathematical Engineering from the Politecnico of Milan (Italy).

Matteo’s research has always been interdisciplinary, working at the interface between different fields in applied mathematics and computational engineering. During his PhD, he developed numerical methods for uncertainty quantification, including multilevel Monte Carlo methods, finite element methods for the solution of partial differential equations (PDEs) with random coefficients, and stochastic modelling techniques using Gaussian fields. He applied these techniques to design, validate, and solve different models for brain solute movement. He also developed an optimization method for finding multiple solutions of semismooth equations, variational inequalities and constrained optimization problems. In his years as a postdoc, Matteo has become an expert in reduced- and mixed-precision (RP and MP) computing, in particular in the development of RP/MP methods for the numerical solution of PDEs, including RP finite difference and finite element methods, and MP time stepping methods.

Matteo won the Charles Broyden Prize for the best paper published in Optimization Methods and Software in 2020.