European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
08:30   MS8: Matrix Computations for PDEs (Part 1)
Chair: Valeria Simoncini
25 mins
Reduced Basis Iterative Solvers for Stochastic Galerkin Matrix Equations
Catherine Powell, Valeria Simoncini
Abstract: Stochastic Galerkin finite element methods (SGFEMs) provide an alternative to sampling methods for performing forward uncertainty quantification (UQ) in PDE models with random inputs. The coefficient matrices of such systems have a characteristic Kronecker product structure which can be exploited to design bespoke solvers and preconditioners. Due to the extremely large dimension of the underlying tensor product approximation space, one often runs out of memory when using standard preconditioned Krylov iterative methods. Reformulating the systems as multiterm matrix equations opens up the possibility to approximate the solution matrix using more memory efficient low-rank techniques. In previous work, a new method (known as multiRB) was introduced that determines a low-rank approximation for symmetric and positive definite problems by performing a projection onto a low-dimensional space which is contructed using ideas inspired by rational Krylov approximation. In this talk, we report on recent work to extend this approach to the discrete indefinite problems associated with stochastic Galerkin mixed finite element approximations of systems of PDEs with random inputs.
25 mins
An Improved Penalty Algorithm for mixed integer and PDE constrained optimization problems
Dominik Garmatter, Martin Stoll, Margherita Porcelli, Francesco Rinaldi
Abstract: In this talk we want to investigate optimal control problems with a partial differential equation (PDE) as constraint and additional integer constraints for the control, so-called mixed integer PDE constrained optimization (MIPDECO) problems. In order to satisfy the integrality constraints, mixed integer optimizations problems are usually solved by Branch-and-Bound (BnB) solvers. BnB algorithms find the global optimum by searching the combinatorial tree provided by the integer constraints in a clever and efficient way. Unfortunately, BnB methods in a PDE constrained optimization framework can become time-infeasible for two reasons: either the combinatorial complexity of the integer constraints is simply too overwhelming, or the solution time of a subproblem in the BnB is quite large due to the discretization of the PDE and the amount of subproblems to be solved then becomes problematic. As a remedy to this situation, we will investigate penalization strategies that replace the integrality constraint in the MIPDECO problem by a suitable penalty term in the objective function. We utilize the framework of an existing Exact Penalty Algorithm to correctly increase the penalization throughout our iteration, while, in each iteration, we try to find an iterate that decreases the current objective function value. Such a decrease can be achieved via a problem-specific perturbation stragety. The result of this combination then is our novel Improved Penalty Algorithm (IPA). Based on a numerical toy problem, we will compare the IPA to existing penalization strategies, simple rounding schemes, and the BnB-routine of Cplex.
25 mins
Matrix-oriented discretization methods for reaction-diffusion PDEs: comparisons and applications
Maria Chiara D'Autilia, Ivonne Sgura, Valeria Simoncini
Abstract: Systems of reaction-diffusion partial differential equations (RD-PDEs) are widely applied for modelling life science and physico-chemical phenomena. In particular, the coupling between diffusion and nonlinear kinetics can lead to the so-called Turing instability, giving rise to a variety of spatial patterns (like labyrinths, spots, stripes, etc.) attained as steady state solutions for large time intervals. To capture the morphological peculiarities of the pattern itself, a very fine space discretization may be required, limiting the use of standard (vector-based) ODE solvers in time because of excessive computational costs. We show that the structure of the diffusion matrix can be exploited so as to use matrix-based versions of time integrators, such as Implicit-Explicit (IMEX) and exponential schemes. This implementation entails the solution of a sequence of discrete matrix problems of significantly smaller dimensions than in the vector case, thus allowing for a much finer problem discretization. We illustrate our findings by numerically solving the Schnackenberg model, prototype of RD-PDE systems with Turing pattern solutions, and the DIB-morphochemical model describing metal growth during battery charging processes.
25 mins
Robust preconditioning of a class of PDE-constrained optimization problems including hyperbolic and parabolic control problems
Alexander Beigl, Otmar Scherzer, Jarle Sogn, Walter Zulehner
Abstract: In this talk we consider a class of optimization problems, where the state problem is of the form \[ \theta_i(y) = q_i \quad \text{for} \ i = 1,\ldots,m \] and the objective functional contains regularization terms of the form \[ \frac{\alpha_j}{2} \|\theta_j(y)\|_{H_j}^2 \quad \text{for} \ j = m+1,\ldots,n. \] with different regularization parameters $\alpha_j$. Here $(\theta_1,\ldots,\theta_n) \colon Y \longrightarrow H_1\times \cdots \times H_n$ is a bijective mapping between Hilbert spaces. We derive the corresponing KKT system and show that it is well-posed. Furthermore, we provide a preconditioner which is robust with respect to all the regularization parameters. We apply this method on two inverse problems: One with a parabolic PDE and one with a hyperbolic PDE. Finally we present some numerical results.