10:40
MS8: Matrix Computations for PDEs (Part 2)
Chair: Valeria Simoncini
10:40
25 mins
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Symbol-based preconditioners for PDEs using multigrid
Matthias Bolten
Abstract: In the analysis and design of preconditioners for matrices steming from the discretization of PDEs the use of the generating symbol has proven to be useful, e.g., when special discretization techniques are used. The multi-level nature of the arising structured matrices demands iterative solution, as they are usually sparse, the cost of one iteration is optimal. Nevertheless, due to their asymptotic ill-conditioning, the use of a preconditioner is mandatory and multigrid is usually used here. In this talk we will discuss the use of symbol-based multigrid methods, the associated challenges, e.g., matrix-valued symbols, and their implementation.
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11:05
25 mins
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Preconditioners for space-time isogeometric problems
Mattia Tani, Gabriele Loli, Monica Montardini, Giancarlo Sangalli
Abstract: Isogeometric analysis (IGA) is a method to numerically solve partial differential equations. It is based on the idea of using B-splines (and their generalizations) both for the parametrization of the domain, as it is typically done by computer aided design software, and for the representation of the unknown solution. IGA can also be seen as an extension of the classical finite element method, where the basis functions are allowed to have a higher regularity.
In this talk we discuss preconditioning strategies suited for IGA. It is known that many approaches that are popular in the context of $C^0$ finite elements, both direct and iterative, tend to perform poorly when applied to IGA linear systems. In particular, their effectiveness deteriorates as the spline degree $p$ is increased.
We address this issue starting from the Poisson problem, and preconditioner that represents the same problem discretized on the reference domain. This preconditioner is robust with respect to $h$ and $p$ and can be applied in a very efficient way thanks to the Fast Diagonalization method, that exploits the tensor structure of the basis functions. We then consider the heat equation, and consider a space-time discretization where smooth splines are used both in space and in time. We develop two numerical formulations for this problem, a symmetric high-order least squares formulation, and a nonsymmetric low-order Galerkin formulation. For both approaches, we develop robust preconditioners that can be applied efficiently thanks to a variant to the Fast Diagonalization method. We finally highlight advantages and drawbacks of the two formulations.
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11:30
25 mins
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Identification of Dominant Modes via the Shifted Proper Orthogonal Decomposition
Philipp Schulze
Abstract: Model Order Reduction aims at approximating a certain input-output behavior of a high-dimensional system, the so-called full-order model (FOM), by a system of much smaller dimension, the so-called reduced-order model (ROM). Especially if the original system is nonlinear, this usually requires the simulation of the FOM for a few input or parameter configurations in order to generate characteristic solution trajectories and store them in a so-called snapshot matrix. These data are then used to identify appropriate ansatz functions which can afterwards be used for projecting the FOM onto their span and, thus, obtain a ROM. One of the most popular methods for identifying these basis functions or modes is the proper orthogonal decomposition (POD) which is based on a singular value decomposition of the snapshot matrix. Unfortunately, when it comes to systems whose dynamics is dominated by the advective transport of high-gradient structures, as shocks, the singular values decay very slowly which leads to a poor performance of the classical POD.
In this talk, we present the shifted POD [1--3] which extends the classical POD via introducing coordinate transformations for describing the advective transport. Especially, we demonstrate how to determine shifted POD modes via minimizing the residual between the original snapshot data and their shifted POD approximation. Numerical experiments illustrate that the shifted POD is capable of describing even complex advection-dominated dynamics with multiple advection speeds and directions with just a few modes, whereas the classical POD needs a lot more modes to obtain the same accuracy.
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11:55
25 mins
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Prescribing convergence behavior of block GMRES
Marie Kubínová, Kirk Soodhalter
Abstract: Block GMRES is a generalization of the generalized minimum residual (GMRES) method. GMRES is an iterative method for solving large, sparse linear systems, and block GMRES is a generalization thereof for solving systems with multiple right-hand sides simultaneously. We present new convergence results for block GMRES based on a different mathematical interpretation, wherein we now interpret our matrix and multiple linear systems as being a single linear system over a *-ring of complex matrices. With this interpretation, one can carefully extend many existing GMRES results to the block GMRES setting. We give an overview of these new results.
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