MATHICSE Technical Report : Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method

The efficient solution of coupled PDEs/ODEs problems arising in cardiac electrophysiology is of key importance whenever interested to study the electrical behavior of the tissue for several instances of relevant physical and/or geometrical parameters. This poses significant challenges to reduced order modeling (ROM) techniques – such as the reduced basis method – traditionally employed when dealing with the repeated solution of parameter dependent differential equations. Indeed, the nonlinear nature of the problem, the presence of moving fronts in the solution, and the high sensitivity of this latter to parameter variations, make the application of standard ROM techniques very problematic. In this paper we propose a local ROM built through a k-means clustering in the state space of the snapshots for both the solution and the nonlinear term. Several comparisons among alternative local ROMs on a benchmark test case show the effectivity of the proposed approach. Finally, the application to a parametrized problem set on an idealized leftventricle geometry shows the capability of the proposed ROM to face complex problems.

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.

EPFL2016

Modélisation mathématique et simulation numérique des phénomènes dynamiques et thermiques apparaissant dans un glacier

Pierre Kirner

In this work, two problems linked to glacier modeling are investigated. We propose an optimisation method for studying the flow of the ice and we present a numerical study about glacier thermal phenomena. In the first chapter of this thesis, we expose the models of these two problems. On one hand, we note that the boundary conditions on the bedrock are misunderstood, which explains it is difficult to obtain an accurate simulation of the motion of the ice. Also we establish a mathematical model where the bedrock boundary conditions depend on a control parameter. The aim of this study is to minimize a cost functional describing the difference between the computed velocity at the surface and the measure done. We study the cost functional with respect to the control parameter and we detail an optimisation method to solve the optimal control problem. On the other hand, we introduce two thermodynamical model governing the temperature and the water content field. The models correspond to a Stefan problem for the temperature and a convection-diffusion equation for the water content. The second chapter deals with the numerical resolution of the optimisation problem. First, a Finite Element Method (FEM) is described to solve the partial differential equations. Then, the algorithms used for the optimal control problem are detailed. Finally, this techniques are applied on two glaciers : Griesgletcher for 2D and Storglaciaren for 3D. The third chapter deals with the numerical resolution of the temperature and the water content models. A FEM is used for each problem. Concerning the temperature problem, the Stefan problem is numerically solved and the results allow to detect a free surface between the temperated ice and the cold ice. The water content field is also simulated. Numerical results are discussed on the Storglaciaren.

EPFL2007

Structure-preserving approaches and data-driven closure modeling for model order reduction

Nicolò Ripamonti

In this thesis, we propose model order reduction techniques for high-dimensional PDEs that preserve structures of the original problems and develop a closure modeling framework leveraging the Mori-Zwanzig formalism and recurrent neural networks. Since high-fidelity approximations of PDEs often result in a large number of degrees of freedom, the need for iterative evaluations for numerical optimizations and rapid feedback is computationally challenging.The first part of this thesis is devoted to conserving the high-dimensional equation's invariants, symmetries, and structures during the reduction process. Traditional reduction techniques are not guaranteed to yield stable reduced systems, even if the target problem is stable. In the context of fluid flows, the skew-symmetric structure of the problem entails the preservation of the kinetic energy of the system. By preserving the same structure at the level of the reduced model, we obtain enhanced stability, and accuracy and the reduced model acquires physical significance by preserving a surrogate of the energy of the original problem. Next, we focus on Hamiltonian systems, which, being driven by symmetry, are a source of great interest in the reduction community. It is well known that the breaking of these symmetries in the reduced model is accompanied by a blowup of the system energy and flow volume. In this thesis, geometric reduced models for Hamiltonian systems are further developed and combined with the dynamically orthogonal methods, addressing the poor reducibility in time of advection-dominated problems. The reduced solution is expressed as a linear combination of a finite number of modes and coincides with the symplectic projection of the high-fidelity Hamiltonian problem onto the tangent space of the approximating manifold. An error surrogate is used to monitor the approximation ability of the reduced model and make a change in the rank of the approximating system if necessary. The method is further developed through a combination of DEIM and DMD to reduce non-polynomial nonlinearities while preserving the symplectic structure of the problem and applied to the Vlasov-Poisson system.In the second part of the thesis, we consider several data-driven methods to address the poor accuracy in the under-resolved regime for Galerkin reduced models via a closure term. The closure term is developed systematically from the Mori-Zwanzig formalism by introducing projection operators on the spaces of resolved and unresolved scales, thus resulting in an additional memory integral term. The interaction between different scales turns out to be nonlocal in time and dominated by a high-dimensional orthogonal dynamics equation, which cannot be solved precisely and efficiently. Several classical methods in the field of statistical mechanics are used to approximate the memory term, exploiting the finiteness of the memory kernel support. We conclude this thesis by showing through numerical experiments how long short-term memory networks, i.e., machine learning structures characterized by feedback connections, represent a valid tool for approximating the additional memory term.