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.

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.

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 degrees of 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 efficient preconditioners 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.

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.

We present an implementation of a fully stage-parallel preconditioner for Radau IIA type fully implicit Runge--Kutta methods, which approximates the inverse of $A_Q$ 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 either use 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 when the block solvers operate at lower parallel efficiencies, which occurs near the scaling limit. Achievable speedup increases linearly with number of stages and are bounded by the number of stages. Furthermore, we show that the presented stage-parallel concepts are also applicable to the case that $A_Q$ is directly diagonalized, which requires complex arithmetic or the solution of two-by-two blocks and sequentializes parts of the algorithm. Alternatively to distributing stages and assigning them to distinct processes, we discuss the possibility of batching operations from different stages together.

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.

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.

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.

The radial point interpolation meshfree discretization is a very efficient numerical framework for the analysis of piezoelectricity, in which the fundamental electrostatic equations governing piezoelectric media are solved without mesh generation. Due to the mechanical‐electrical coupling property and the piezoelectric constant, the discrete linear system is sparse, of generalized saddle point form and often very ill conditioned. In this work, we propose a technique for constructing a family of cell‐by‐cell approximate Schur complement matrices, to be used in preconditioning to accelerate the convergence of Krylov subspace iteration methods for such problems. The approximate Schur complement matrices are simply and cheaply constructed in the process of the meshfree discretization and have a sparse structure. It is proved that the so‐constructed approximate Schur complement matrices are spectrally equivalent to the exact Schur complement matrix, which leads to very fast convergence when used in preconditioning. In addition, nondimensionalization of the piezoelectric equations is considered to make the computations more stable. The robustness and the efficiency of the proposed preconditioners is illustrated numerically on two test problems, arising from a piezoelectric strip shear deformation problem and a piezoelectric strip bending problem. Numerical results show that the number of iterations to achieve a given tolerance is independent of the number of degrees of freedom as well as of the various problem parameters.

In this study we consider an efficient implementation of Implicit Runge-Kutta methods for solving large systems of ordinary differential equations that originate from finite element discretization of the heat and similar equations, to be solved on large time intervals. The main contribution of this work is to show how to implement a fully stage-parallel version of the method, utilizing the dominance of the block lower triangular part of the quadrature matrix, and to illustrate it numerically. Its usage for the solution of algebraic-differential equations is also touched.

In this paper we briefly account for the structure of the matrices, arising in various optimal control problems, constrained by PDEs, and how it can be utilized when constructing preconditioners for the arising linear systems to be solved in the optimization framework.

We present a method for solving optimal control problems constrained by a partial differential equation, where we simultaneously impose sparsity-promoting L 1-regularization on the control as well as box constraints on both the control and the state. We focus on numerical implementation aspects and on preconditioners used when solving the arising linear systems.

PreviousNext No AccessInternational Workshop on Gravity, Electrical & Magnetic Methods and Their Applications, Xi'an, China, 19–22 May 2019A Spectral-Element Approach to 3D Controlled-Source Electromagnetic forward modellingAuthors: Michael Weiss*Thomas KalscheuerMaya NeytchevaZhengyong RenMichael Weiss*Dept. of Earth Sciences, Uppsala University, SwedenSearch for more papers by this author, Thomas KalscheuerDept. of Earth Sciences, Uppsala University, SwedenSearch for more papers by this author, Maya NeytchevaDept. of Information Technology, Uppsala University, SwedenSearch for more papers by this author, and Zhengyong RenSchool of Geosciences and Info-Physics, Central South University, ChinaSearch for more papers by this authorhttps://doi.org/10.1190/GEM2019-073.1 SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InRedditEmail Abstract The spectral-element method based on high-order polynomials combines features of the spectral method and the finite element method. Similar to the finite element method, the spectral element method is based on Galerkin’s weighted residual technique. However, the basis functions for the spectral element method are non-linear polynomials, such as Gauss-Lobatto-Legendre or Gauss-Lobatto-Chebyshev polynomials, rather than linear functions used in the finite element method. These interpolation functions are characterised by spectral accuracy with the error decreasing exponentially with increasing polynomial order. Thus, similar accuracy to that of the finite element method can be obtained using fewer degrees of freedom respectively fewer elements. In this abstract, we outline the mathematical formulation of the spectral-element method applied to 3D controlled-source electromagnetic forward modelling. Keywords: 3D, modelling, electromagneticPermalink: https://doi.org/10.1190/GEM2019-073.1FiguresReferencesRelatedDetails International Workshop on Gravity, Electrical & Magnetic Methods and Their Applications, Xi'an, China, 19–22 May 2019ISSN (online):2159-6832Copyright: 2019 Pages: 471 publication data© 2019 Published in electronic format with permission by the Society of Exploration Geophysicists and the Chinese Geophysical SocietyPublisher:Society of Exploration Geophysicists HistoryPublished Online: 28 Sep 2019 CITATION INFORMATION Michael Weiss*, Thomas Kalscheuer, Maya Neytcheva, and Zhengyong Ren, (2019), "A Spectral-Element Approach to 3D Controlled-Source Electromagnetic forward modelling," SEG Global Meeting Abstracts : 288-291. https://doi.org/10.1190/GEM2019-073.1 Plain-Language Summary Keywords3DmodellingelectromagneticPDF DownloadLoading ...

We consider the iterative solution of optimal control problems constrained by the time-harmonic parabolic equations. Due to the time-harmonic property of the control equations, a suitable discretization of the corresponding optimality systems leads to a large complex linear system with special two-by-two block matrix of saddle point form. For this algebraic system, an efficient preconditioner is constructed, which results in a fast Krylov subspace solver, that is robust with respect to the mesh size, frequency, and regularization parameters. Furthermore, the implementation is straightforward and the computational complexity is of optimal order, linear in the number of degrees of freedom. We show that the eigenvalue distribution of the corresponding preconditioned matrix leads to a condition number bounded above by 2. Numerical experiments confirming the theoretical derivations are presented, including comparisons with some other existing preconditioners.

The availability of high performance computing resources enables us to perform very large numerical simulations and in this way to tackle challenging real life problems. At the same time, in order to efficiently utilize the computational power at our disposal, the ever growing complexity of the computer architecture poses high demands on the algorithms and their implementation. Performing large scale high performance simulations can be done by utilizing available general libraries, writing libraries that suit particular classes of problems or developing software from scratch. Clearly, the possibilities to enhance the efficiency of the software tools in the three cases is very different, ranging from nearly impossible to full capacity. In this work we exemplify the efficiency of the three approaches on benchmark problems, using monitoring tools that provide a very rich spectrum of data on the performance of the applied codes as well as on the utilization of the supercomputer itself.

Two-by-two block matrices with square blocks arise in the numerical treatment of numerous applications of practical significance, such as optimal control problems, constrained by a state equation in the form of partial differential equations, multiphase models, solving complex linear systems in real arithmetics, to name a few. Such problems lead to algebraic systems of equations with matrices of a certain two-by-two block form. For such matrices, a number of preconditioners has been proposed, some of them with tight eigenvalue bounds. In this paper it is shown that in particular one of them, referred to as PRESB, is very efficient, not only giving robust, favourable properties of the spectrum but also enabling an efficient implementation with low computational complexity. Various applications and generalizations of this preconditioning technique, such as in time-harmonic parabolic and Stokes equations, eddy current electromagnetic problems and problems with additional box-constraints, i.e. upper and/or lower bounds of the solution, are also discussed. The method is based on the use of coupled inner-outer iterations, where the inner iteration can be performed to various relative accuracies. This leads to variable preconditioners, thus, a flexible version of a Krylov subspace iteration method must be used. Alternatively, some version of a defect-correction iterative method can be applied.

The recent development of the high performance computer platforms shows a clear trend towards heterogeneity and hierarchy. In order to utilize the computational power, particular attention must be paid to finding new algorithms or adjust existing ones so that they better match the HPC computer architecture. In this work we consider an alternative to classical time-stepping methods based on use of time-harmonic properties and discuss solution approaches that allow efficient utilization of modern HPC resources. The method in focus is based on a truncated Fourier expansion of the solution of an evolutionary problem. The analysis is done for linear equations and it is remarked on the possibility to use two- or multilevel mesh methods for nonlinear problems, which can enable further, even higher degree of parallelization. The arising block matrix system to be solved admits a two-by-two block form with square blocks, for which a very efficient preconditioner exists. It leads to tight eigenvalue bounds for the preconditioned matrix and, hence, to a very fast convergence of a preconditioned Krylov subspace or iterative refinement method. The analytical background is shown as well as some illustrating numerical examples.

Abstract An efficient preconditioning technique used earlier for two-by-two block matrix systems with square matrix blocks is shown to be applicable also for a state variable box-constrained optimal control problem. The problem is penalized by a standard regularization term for the control variable and for the box-constraint, using a Moreau–Yosida penalization method. It is shown that there occur very few nonlinear iteration steps and also few iterations to solve the arising linearized equations on the fine mesh. This holds for a wide range of the penalization and discretization parameters. The arising nonlinearity can be handled with a hybrid nonlinear-linear procedure that raises the computational efficiency of the overall solution method.