Estimating the Ice Thickness of Mountain Glaciers from Surface Topography and Mass-Balance Data

The question addressed is the determination of a glacier’s subglacial topography, given surface topography and mass-balance data. The input data can be obtained relatively easily for a large number of glaciers. Several methods essentially based on the shallow ice approximation are proposed, some of which are extended to Stokes ice flows. Two gradient-free, iterative methods are first introduced, namely the quasi-stationary inverse method, that relies on the apparent surface mass-balance description of glacier dynamics, and the transient inverse method, consisting in the iterative update of the bedrock topography proportionally to the surface topography misfit at the end of the glacier’s considered evolution. Then, an optimal control algorithm is suggested that calculates the bedrock topography and some model parameters from surface observations through the minimization of a regularized misfit functional by means of a Lagrangian method. Numerical validations, along with sensitivity analyses and applications to real-world data are presented for each method.

Multi space reduced basis preconditioners for parametrized partial differential equations

Niccolò Dal Santo

The multiquery solution of parametric partial differential equations (PDEs), that is, PDEs depending on a vector of parameters, is computationally challenging and appears in several engineering contexts, such as PDE-constrained optimization, uncertainty quantification or sensitivity analysis. When using the finite element (FE) method as approximation technique, an algebraic system must be solved for each instance of the parameter, leading to a critical bottleneck when we are in a multiquery context, a problem which is even more emphasized when dealing with nonlinear or time dependent PDEs. Several techniques have been proposed to deal with sequences of linear systems, such as truncated Krylov subspace recycling methods, deflated restarting techniques and approximate inverse preconditioners; however, these techniques do not satisfactorily exploit the parameter dependence. More recently, the reduced basis (RB) method, together with other reduced order modeling (ROM) techniques, emerged as an efficient tool to tackle parametrized PDEs. In this thesis, we investigate a novel preconditioning strategy for parametrized systems which arise from the FE discretization of parametrized PDEs. Our preconditioner combines multiplicatively a RB coarse component, which is built upon the RB method, and a nonsingular fine grid preconditioner. The proposed technique hinges upon the construction of a new Multi Space Reduced Basis (MSRB) method, where a RB solver is built at each step of the chosen iterative method and trained to accurately solve the error equation. The resulting preconditioner directly exploits the parameter dependence, since it is tailored to the class of problems at hand, and significantly speeds up the solution of the parametrized linear system. We analyze the proposed preconditioner from a theoretical standpoint, providing assumptions which lead to its well-posedness and efficiency. We apply our strategy to a broad range of problems described by parametrized PDEs: (i) elliptic problems such as advection-diffusion-reaction equations, (ii) evolution problems such as time-dependent advection-diffusion-reaction equations or linear elastodynamics equations (iii) saddle-point problems such as Stokes equations, and, finally, (iv) Navier-Stokes equations. Even though the structure of the preconditioner is similar for all these classes of problems, its fine and coarse components must be accurately chosen in order to provide the best possible results. Several comparisons are made with respect to the current state-of-the-art preconditioning and ROM techniques. Finally, we employ the proposed technique to speed up the solution of problems in the field of cardiovascular modeling.

EPFL2018

A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices

Denis Devaud, Andrea Manzoni, Gianluigi Rozza

We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB-ANOVA-RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB-ANOVA-RB procedure computationally more feasible. (C) 2013 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Elsevier2013

Multi space reduced basis preconditioners for parametrized partial differential equations

Niccolò Dal Santo

EPFL2018