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Time-harmonic control problems, constrained by a linear differential equation, can be solved efficiently by utilizing a Fourier time series expansion in the angular frequency variable. Then the optimal solution consists of a series of complex variable space discretization equations, which are uncoupled with respect to the different frequencies. Hence, it suffices to consider a single equation with the angular frequency as a parameter. We consider here methods to solve the so-arising linear system of equations and describe, analyze and test the performance of two novel approaches based on its exact and approximate Schur complement. The performance of the methods is tested and compared with another existing method.

This special issue of Numerical Linear Algebra with Applications is dedicated to celebrating the life and work of Professor Owe Axelsson, who passed away in 2022 at the age of 88. His passing is a significant loss for the numerical analysis community, touching everyone who knew him personally and those familiar with his impactful research.

When solving very large scale problems on parallel computer platforms, we consider the advantages of domain decomposition in strips or layers, compared to general domain decomposition splitting techniques. The layer sub-domains are grouped in pairs, ordered as odd-even respectively even-odd and solved by a Schwarz alternating iteration method, where the solution at the middle interfaces of the odd-even groups is used as Dirichlet boundary conditions for the even-odd ordered groups and vice versa. To stabilize the method the commonly used coarse mesh method can be replaced by a coarse-fine mesh method. A component analysis of the arising eigenvectors demonstrates that this solution framework leads to very few Schwarz iterations. The resulting coarse-fine mesh method entails a coarse mesh of a somewhat large size. In this study it is solved by two methods, a modified Cholesky factorization of the whole coarse mesh matrix and a block-diagonal preconditioner, based on the coarse mesh points and the inner node points. Extensive numerical tests show that the latter method, being also computationally cheaper, needs very few iterations, in particular when the domain has been divided in many layers and the coarse to fine mesh size ratio is not too large.

The use of high order fully implicit Runge--Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space resolution with many millions of spatial degreesof freedom and long time intervals. In this study we consider strongly A-stable implicit Runge--Kutta methods of arbitrary order of accuracy, based on Radau quadratures, for which efficien tpreconditioners have been introduced. A refined spectral analysis of the corresponding matrices and matrix sequences is presented, both in terms of localization and asymptotic global distribution of the eigenvalues. Specific expressions of the eigenvectors are also obtained. The given study fully agrees with the numerically observed spectral behavior and substantially improves the theoretical studies done in this direction so far. Concluding remarks and open problems end the current work, with specific attention to the potential generalizations of the hereby suggested general approach.

Fully implicit Runge–Kutta methods offer the possibility to use high order accurate time discretization to match space discretization accuracy, an issue of significant importance for many large scale problems of current interest, where we may have fine space resolution with many millions of spatial degrees of freedom and long time intervals. In this work, we consider strongly A-stable implicit Runge–Kutta methods of arbitrary order of accuracy, based on Radau quadratures. For the arising large algebraic systems we introduce efficient preconditioners, that (1) use only real arithmetic, (2) demonstrate robustness with respect to problem and discretization parameters, and (3) allow for fully stage-parallel solution. The preconditioners are based on the observation that the lower-triangular part of the coefficient matrices in the Butcher tableau has larger in magnitude values, compared to the corresponding strictly upper-triangular part. We analyze the spectrum of the corresponding preconditioned systems and illustrate their performance with numerical experiments. Even though the observation has been made some time ago, its impact on constructing stage-parallel preconditioners has not yet been done and its systematic study constitutes the novelty of this article.

We develop an efficient and robust iterative framework suitable for solving the linear system of equations resulting from the spectral element discretisation of the curl-curl equation of the total electric field encountered in geophysical controlled-source electromagnetic applications. We use the real-valued equivalent form of the original complex-valued system and solve this arising real-valued two-by-two block system (outer system) using the generalised conjugate residual method preconditioned with a highly efficient block-based PREconditioner for Square Blocks (PRESB). Applying this preconditioner equates to solving two smaller inner symmetric systems which are either solved using a direct solver or iterative methods, namely the generalised conjugate residual or the flexible generalised minimal residual methods preconditioned with the multigrid-based auxiliary-space preconditioner AMS. Our numerical experiments demonstrate the robustness of the outer solver with respect to spatially variable material parameters, for a wide frequency range of five orders of magnitude (0.1-10’000 Hz), with respect to the number of degrees of freedom, and for stretched structured and unstructured as well as locally refined meshes. For all the models considered, the outer solver reaches convergence in a small (typically < 20) number of iterations. Further, our numerical tests clearly show that solving the two inner systems iteratively using the indicated preconditioned iterative methods is computationally beneficial in terms of memory requirement and time spent as compared to a direct solver. On top of that, our iterative framework works for large-scale problems where direct solvers applied to the original complex-valued systems succumb due to their excessive memory consumption, thus making the iterative framework better suited for large-scale 3D problems. Comparison to a similar iterative framework based on a block-diagonal and the auxiliary-space preconditioners reveals that the PRESB preconditioner requires slightly fewer iterations to converge yielding a certain gain in time spent to obtain the solution of the two-by-two block system.

We consider the iterative solution of algebraic systems, arising in optimal control problems, constrained by a partial differential equation, with additional box constraints on the state and the control variables, and sparsity imposed on the control. A nonsymmetric two-by-two block preconditioner is analysed and tested for a wide range of problem, regularization and discretization parameters. The constraint equation characterizes convection-diffusion processes.

We consider the iterative solution of algebraic systems, arising in optimal control problems constrained by a partial differential equation with additional box constraints on the state and the control variables, and sparsity imposed on the control. A nonsymmetric two-by-two block preconditioner is analysed and tested for a wide range of problem, regularization and discretization parameters. The constraint equation characterizes convection-diffusion processes.

We present an implementation of a stage-parallel preconditioner for Radau IIA type fully implicit Runge–Kutta methods, which approximates the inverse of the Runge–Kutta matrix AQ from the Butcher tableau by the lower triangular matrix resulting from an LU decomposition and diagonalizes the system with as many blocks as stages. For the transformed system, we employ a block preconditioner where each block is distributed and solved by a subgroup of processes in parallel. For combination of partial results, we use either a communication pattern resembling Cannon’s algorithm or shared memory. A performance model and a large set of performance studies (including strong-scaling runs with up to 150k processes on 3k compute nodes) conducted for a time-dependent heat problem, using matrix-free finite element methods, indicate that the stage-parallel implementation can reach higher throughputs near the scaling limit. The achievable speedup increases linearly with the number of stages and is bounded by the number of stages. Furthermore, we show that the presented stage-parallel concepts are also applicable to the case that AQ is directly diagonalized, which requires either complex arithmetic or solutions of two-by-two blocks, both exposing about half the parallelism. Alternatively to distributing stages and assigning them to distinct processes, we discuss the possibility of batching operations from different stages together.

By use of Fourier time series expansions in an angular frequency variable, time-harmonic optimal control problems constrained by a linear differential equation decouples for the different frequencies. Hence, for the analysis of a solution method one can consider the frequency as a parameter. There are three variables to be determined, the state solution, the control variable, and the adjoint variable. The first order optimality conditions lead to a three-by-three block matrix system where the adjoint optimality variable can be eliminated. For the so arising two-by-two block system, in this paper we study a factorization method involving an exact Schur complement method and illustrate the performance of an inexact version of it.