NAApr 14, 2016
Memory-efficient recycling of large Krylov-subspaces for sequences of Hermitian linear systemsMartin Peter Neuenhofen, Sven Groß
We present a new short-recurrence reaidual-optimal Krylov subspace recycling method for sequences of Hermitian systems of linear equations with a fixed system matrix and changing right-hand sides. Such sequences of linear systems occur while solving, e.g., discretized time-dependent partial differential equations. With this new method it is possible to recycle large-dimensional Krylov-subspaces with smaller computational overhead and storage requirements compared to current Krylov subspace recycling methods as e.g. R-MINRES. In this paper we derive the method from the residual-optimal preconditioned conjugate residual method and duscuss implementation issues. Numerical experiments illustrate the efficiency of our method.
NADec 21, 2017
High-order convergent Finite-Elements Direct Transcription Method for Constrained Optimal Control ProblemsMartin Peter Neuenhofen
In this paper we present a finite element method for the direct transcription of constrained non-linear optimal control problems. We prove that our method converges of high order under mild assumptions. Our analysis uses a regularized penalty-barrier functional. The convergence result is obtained from local strict convexity and Lipschitz-continuity of this functional in the finite-element space. The method is very flexible. Each component of the numerical solution can be discretized with a different mesh. General differential-algebraic constraints of arbitrary index can be treated easily with this new method. From the discretization results an unconstrained non-linear programming problem (NLP) with penalty- and barrier-terms. The derivatives of the NLP functions have a sparsity pattern that can be analysed and tailored in terms of the chosen finite-element bases in an easy way. We discuss how to treat the resulting NLP in a practical way with general-purpose software for constrained non-linear programming.
NAApr 20, 2016
M(s)stab(L): A Generalization of IDR(s)stab(L) for Sequences of Linear SystemsMartin Peter Neuenhofen
We propose Mstab, a novel Krylov subspace recycling method for the iterative solution of sequences of linear systems with fixed system matrix and changing right-hand sides. This new method is a straight and simple generalization of IDRstab. IDRstab in turn is a very efficient method and generalization of BiCGStab. The theory of Mstab is based on a generalization of the IDR theorem and Sonneveld spaces. Numerical experiments indicate that Mstab can solve sequences of linear systems faster than its corresponding IDRstab variant. Instead, when solving a single system both methods are identical.