10:30
MS30: Numerical methods for PDE-constrained and controlled optimization problems with appplications (Part 1)
Chair: Maya Neytcheva
10:30
25 mins
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PDE Constrained Optimization Problems for the Waste Clearance of the Human Brain
Kent-Andre Mardal, Lars Magnus Valnes, Sebastian Mitusch, Geir Ringstad, Per Kristian Eide, Simon Funke
Abstract: The brain is our most energy expensive organ, but its metabolic cycle is not
understood. In particular, its waste disposal system is a mystery because it lacks the
lymphatic system that is present elsewhere in our body. Understand how waste is
cleared under healthy and diseased condition is important as accumulation of waste
is associated with dementia such as Alzheimer’s and Parkinson’s diseases.
Novel imaging protocols are under investigation for assessing brain clearance on the
long time scales hours. In this talk we will present the newly proposed protocols and
discuss the PDE constrained optimization problems that arise. Furthermore, a crucial
component in this type of investigations is that the data exist everywhere in space,
albeit at coarse resolution, but at certain points in time. Order optimal algorithms
are derived for some simplified problems.
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10:55
25 mins
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Applications of the PRESB preconditioning method for OPT-PDE problems
Owe Axelsson
Abstract: Optimal control problems constrained by partial differential equations arise in a multitude of important applications. They lead mostly to the solution of very large scale algebraic systems to be solved, which must be done by iterative methods. The problems should then be formulated so that they can be solved fast and robust, which requires the construction of an efficient preconditioner. After reduction of a variable, a two-by-two block matrix system with square blocks arises for which such a preconditioner, PRESB is presented, involving the solution of two algebraic systems which are a linear combination of the matrix blocks. These systems can be solved by inner iterations, involving some available classical solvers to some relative, not very demanding tolerance.
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11:20
25 mins
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On Preconditioners for PDE-Constrained Optimization Problems with Higher-Order Discretization in the Time Variable
Santolo Leveque, John Pearson
Abstract: Many existing preconditioned iterative solvers for time-dependent PDE-constrained optimization problems make use of a backward Euler discretization, as this leads to particularly convenient structures within the resulting matrix system, which give rise to suitable preconditioners. In this article, we propose a new preconditioner for a heat control problem, following discretization using a higher-order (Crank--Nicolson) method in time. We investigate in which circumstances the reduced discretization error makes it beneficial to apply this new preconditioner.
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11:45
25 mins
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Computing function of large matrices by a preconditioned rational Krylov method
Fabio Durastante, Daniele Bertaccini
Abstract: Rational Krylov methods are a powerful alternative for computing the product of a function of a large matrix times a given vector. However, the creation of the underlying rational subspaces requires solving sequences of large linear systems, a delicate task that can require intensive computational resources and should be monitored to avoid the creation of subspace different to those required. We propose the use of robust preconditioned iterative techniques to speedup the underlying process. We also discuss briefly how the inexact solution of these linear systems can affect the computed subspace. A preliminary test approximating a fractional power of the Laplacian matrix is included.
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