Winter School: Bridging multiscale, limited information, and low regularity in computational mathematics

Monday

Robert Scheichl

Catherine Powell

Tuesday

Robert Scheichl

Catherine Powell

Wednesday

Fabio Nobile

Daniel Peterseim

Thursday

Fabio Nobile

Friday

Daniel Peterseim

Fabio Nobile (EPFL):
Multilevel Monte Carlo methods for random differential equations
Part I

In this series of lectures we focus on random differential problems which may arise for instance when studying physical systems governed by partial differential equations with uncertainty in the model parameters described by means of random variables, or when studying dynamical systems subject to random fluctuations, described by stochastic differential equations. We will first introduce the Monte Carlo method, combined with a suitable discretization of the underlying differential equation, to compute expectations of output quantities of interest (QoIs) or of the whole solution, and investigate the interplay between the discretization error and the Monte Carlo error. We then introduce the Multilevel Monte Carlo paradigm, analyze its properties and practical implementation aspects, extend its formulation for computations of moments or other statistics of the QoIs. We will also discuss the related Multifidelity Monte Carlo approach. In the second part of the course we present how Multilevel Monte Carlo techniques can be used in some applications, such as PDE constrained optimization under uncertainty or sequential data assimilation.

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Part II

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Part III

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Part IV

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Daniel Peterseim (University of Augsburg):
An Introduction to Quantum Algorithms for Multiscale PDEs
Part I

This series of lectures explores the development of quantum algorithms for solving partial differential equations (PDEs). After introducing the key aspects of quantum computation that are relevant to scientific computing, we examine the algorithmic structure of computational PDEs and consider how this can be reinterpreted in a quantum setting. From this perspective, the lectures present quantum PDE solvers that, perhaps surprisingly, still rely on classical techniques such as finite elements, multilevel preconditioning and fast sampling. They also demonstrate how these concepts can be applied to practical quantum implementations. The discussion is centred on model PDEs that, despite their simplicity, capture the multiscale and low-regularity challenges posed by strongly heterogeneous coefficients in deterministic and stochastic contexts alike.

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Part II

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Part III

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Part IV

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Part V

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Catherine Powell (University of Manchester):
Parametric PDEs: Numerical Methods for Forward UQ & Surrogate Modelling
Part I

In applied mathematics we frequently encounter physics-based models consisting of partial differential equations (PDEs) with inputs such as material coefficients, boundary conditions and/or source terms that are uncertain in real-world settings. In uncertainty quantification (UQ), we represent uncertain model inputs as functions of random variables. The resulting PDEs may then be reformulated as parametric ones on a possibly high-(or even infinite-)dimensional parameter domain. In forward UQ, one aims to understand how uncertainty in model inputs affects uncertainty in model solutions. Naive sampling methods require the repeated numerical solution of the original PDE for different samples of the random inputs. When the cost of solving the problem for just one sample is already expensive (e.g. using a high-fidelity finite element method), obtaining accurate uncertainty assessments is infeasible. Over the last three decades, several families of numerical schemes have been developed to tackle both forward and inverse problems involving PDEs with uncertain inputs. Some of these are known as surrogate modelling techniques because they produce approximations in a functional form that can be cheaply evaluated for new choices of input parameters of interest (without additional PDE solves).

In this series of lectures, we will first discuss appropriate ways to model spatially-varying uncertain PDE inputs, introduce the concept of high-dimensional parametric PDEs and review the basic Monte Carlo method. We will then give an overview of some basic surrogate modelling approaches for facilitating forward UQ in parametric PDEs, including stochastic collocation and reduced basis methods, as well as an ‘intrusive’ approach known as the stochastic Galerkin method. We will then discuss more advanced adaptive multilevel approaches which construct approximation spaces tailored to the regularity of the problem at hand, and the crucial role of a posteriori error estimation.

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Part II

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Part III

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Part IV

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Robert Scheichl (University of Heidelberg):
An Introduction to Multiscale Methods and Localised Model Reduction
Part I

This series of lectures explores the development of multiscale methods and localised model reduction for the solution of PDEs that involve coefficients that vary across multiple scales. The variation may be inherent in the problem, e.g., when modelling composite materials, or it may arise in the context of uncertainty quantification. In both cases, this leads to high-dimensional parametric PDEs with multiscale behaviour that does not separate and is not easily amenable to homogenisation techniques. Due to the lack of scale separation, it is crucial that the local multiscale behaviour is captured well enough in the computational approach, but explicit discretisation with classical (polynomial) Finite Element Methods (FEM) is prohibitive. Using the framework of Generalised Finite Element Methods (GFEM), I will show how local fine scale information can be incorporated in the approximation space and how this can then be embedded in an efficient localised model reduction framework to develop surrogates for high-dimensional parameter spaces. This will encompass methods, such as the Multiscale FEM, the Generalised Multiscale FEM, Localised Orthogonal Decomposition (LOD), and the Multiscale-Spectral Generalised FEM.

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Part II

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Part III

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Part IV

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