scholarly journals Preconditioned Dirichlet-Dirichlet Methods for Optimal Control of Elliptic PDE

2018 ◽  
Vol 26 (2) ◽  
pp. 175-192
Author(s):  
Daniel Loghin
Keyword(s):  
Optimal Control ◽  
Block Structure ◽  
Regularization Parameter ◽  
Domain Decomposition Method ◽  
Interface Problem ◽  
Krylov Methods ◽  
The Matrix ◽  
Boundary Operators ◽  
Elliptic Control Problems ◽  
Block Preconditioners

Abstract The discretization of optimal control of elliptic partial differential equations problems yields optimality conditions in the form of large sparse linear systems with block structure. Correspondingly, when the solution method is a Dirichlet-Dirichlet non-overlapping domain decomposition method, we need to solve interface problems which inherit the block structure. It is therefore natural to consider block preconditioners acting on the interface variables for the acceleration of Krylov methods with substructuring preconditioners. In this paper we describe a generic technique which employs a preconditioner block structure based on the fractional Sobolev norms corresponding to the domains of the boundary operators arising in the matrix interface problem, some of which may include a dependence on the control regularization parameter. We illustrate our approach on standard linear elliptic control problems. We present analysis which shows that the resulting iterative method converges independently of the size of the problem. We include numerical results which indicate that performance is also independent of the control regularization parameter and exhibits only a mild dependence on the number of the subdomains.

scholarly journals Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ulrich Langer ◽  
Olaf Steinbach ◽  
Huidong Yang
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Optimal Control Problems ◽  
Regularization Parameter ◽  
Target Function ◽  
Mesh Size ◽  
Turbulence Theory ◽  
Diffusion Reaction ◽  
Control Problems ◽  
Element Approximation

Abstract We consider elliptic distributed optimal control problems with energy regularization. Here the standard L 2 {L_{2}} -norm regularization is replaced by the H - 1 {H^{-1}} -norm leading to more focused controls. In this case, the optimality system can be reduced to a single singularly perturbed diffusion-reaction equation known as differential filter in turbulence theory. We investigate the error between the finite element approximation u ϱ ⁢ h {u_{\varrho h}} to the state u and the desired state u ¯ {\overline{u}} in terms of the mesh-size h and the regularization parameter ϱ. The choice ϱ = h 2 {\varrho=h^{2}} ensures optimal convergence the rate of which only depends on the regularity of the target function u ¯ {\overline{u}} . The resulting symmetric and positive definite system of finite element equations is solved by the conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) or balancing domain decomposition by constraints (BDDC). We numerically study robustness and efficiency of the AMG preconditioner with respect to h, ϱ, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.

The inverse problem of bimorph mirror tuning on a beamline

2011 ◽  
Vol 18 (6) ◽  
pp. 930-937 ◽  
Author(s):  
Rong Huang
Keyword(s):  
Inverse Problem ◽  
Tikhonov Regularization ◽  
Measurement Errors ◽  
Regularization Parameter ◽  
X Rays ◽  
The Matrix ◽  
Matrix Condition Number ◽  
Matrix Condition ◽  
Safety Limit

One of the challenges of tuning bimorph mirrors with many electrodes is that the calculated focusing voltages can be different by more than the safety limit (such as 500 V for the mirrors used at 17-ID at the Advanced Photon Source) between adjacent electrodes. A study of this problem at 17-ID revealed that the inverse problem of the tuningin situ, using X-rays, became ill-conditioned when the number of electrodes was large and the calculated focusing voltages were contaminated with measurement errors. Increasing the number of beamlets during the tuning could reduce the matrix condition number in the problem, but obtaining voltages with variation below the safety limit was still not always guaranteed and multiple iterations of tuning were often required. Applying Tikhonov regularization and using the L-curve criterion for the determination of the regularization parameter made it straightforward to obtain focusing voltages with well behaved variations. Some characteristics of the tuning results obtained using Tikhonov regularization are given in this paper.

scholarly journals On an Optimal -Control Problem in Coefficients for Linear Elliptic Variational Inequality

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Olha P. Kupenko ◽  
Rosanna Manzo
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Direct Method ◽  
Dirichlet Boundary Conditions ◽  
Elliptic Variational Inequalities ◽  
The Matrix ◽  
Degenerate Elliptic ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Linear Degenerate

We consider optimal control problems for linear degenerate elliptic variational inequalities with homogeneous Dirichlet boundary conditions. We take the matrix-valued coefficients in the main part of the elliptic operator as controls in . Since the eigenvalues of such matrices may vanish and be unbounded in , it leads to the “noncoercivity trouble.” Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of the optimal control problem in the class of the so-called -admissible solutions.

scholarly journals Coupling Parareal with Non-Overlapping Domain Decomposition Method

2016 ◽  
Vol Volume 23 - 2016 - Special... ◽  
Author(s):  
Rim GUETAT
Keyword(s):  
Domain Decomposition ◽  
Convergence Analysis ◽  
Numerical Experiments ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Time Dependent ◽  
Interface Problem ◽  
Time Interval ◽  
Overlapping Domain Decomposition ◽  
Time Dependent Problems

In this paper, we present a new parallel algorithm for time dependent problems based on coupling parareal with non-overlapping domain decomposition method in order to increase parallelism in time and in space. For this we focus on the iterative methods of parallization in space to solve the interface problem like Neumann-Neumann method. In the new algorithm, the coarse temporel propagator is defined on the global domain and the Neumann-Neumann method is chosen as a fine propagator with a few iterations. We present the rigorous convergence analysis of the new coupled algorithm on bounded time interval. Numerical experiments illustrate the performance of this new algorithm and confirm our analysis. RÉSUMÉ. Dans ce papier, nous présentons un nouvel algorithme parallèle pour les problèmes dé-pendant du temps basé sur le couplage du pararéel avec les méthodes de décomposition de domaine sans recouvrement afin d'augmenter le parallélisme dans le temps et l'espace. Nous nous concen-trons sur les méthodes itératives de parallélisation en espace pour résoudre le problème d'interface par la méthode de Neumann-Neumann. Dans ce nouvel algorithme, le propagateur grossier est dé-finie sur le domaine global et la méthode de Neumann-Neumann est choisi pour le propagateur fin avec quelques itérations. Nous présentons l'analyse rigoureuse de convergence du nouvel algorithme couplé sur un intervalle de temps borné. Des expèriences numériques illustrent les performances de ce nouvel algorithme et confirment notre analyse. Dans ce papier, nous présentons un nouvel algorithme parallèle pour les problèmes dépendantdu temps basé sur le couplage du pararéel avec les méthodes de décomposition de domainesans recouvrement afin d’augmenter le parallélisme dans le temps et l’espace. Nous nous concentronssur les méthodes itératives de parallélisation en espace pour résoudre le problème d’interfacepar la méthode de Neumann-Neumann. Dans ce nouvel algorithme, le propagateur grossier est définiesur le domaine global et la méthode de Neumann-Neumann est choisi pour le propagateur finavec quelques itérations. Nous présentons l’analyse rigoureuse de convergence du nouvel algorithmecouplé sur un intervalle de temps borné. Des expèriences numériques illustrent les performances dece nouvel algorithme et confirment notre analyse.

scholarly journals Flexible Krylov Methods for Edge Enhancement in Imaging

2021 ◽  
Vol 7 (10) ◽  
pp. 216
Author(s):  
Silvia Gazzola ◽  
Sebastian James Scott ◽  
Alastair Spence
Keyword(s):  
Least Squares ◽  
Regularization Parameter ◽  
Image Inpainting ◽  
Regularization Methods ◽  
Krylov Methods ◽  
Repeated Application ◽  
Linear Inverse Problems ◽  
New Methods ◽  
Krylov Projection ◽  
Theoretical Results

Many successful variational regularization methods employed to solve linear inverse problems in imaging applications (such as image deblurring, image inpainting, and computed tomography) aim at enhancing edges in the solution, and often involve non-smooth regularization terms (e.g., total variation). Such regularization methods can be treated as iteratively reweighted least squares problems (IRLS), which are usually solved by the repeated application of a Krylov projection method. This approach gives rise to an inner–outer iterative scheme where the outer iterations update the weights and the inner iterations solve a least squares problem with fixed weights. Recently, flexible or generalized Krylov solvers, which avoid inner–outer iterations by incorporating iteration-dependent weights within a single approximation subspace for the solution, have been devised to efficiently handle IRLS problems. Indeed, substantial computational savings are generally possible by avoiding the repeated application of a traditional Krylov solver. This paper aims to extend the available flexible Krylov algorithms in order to handle a variety of edge-enhancing regularization terms, with computationally convenient adaptive regularization parameter choice. In order to tackle both square and rectangular linear systems, flexible Krylov methods based on the so-called flexible Golub–Kahan decomposition are considered. Some theoretical results are presented (including a convergence proof) and numerical comparisons with other edge-enhancing solvers show that the new methods compute solutions of similar or better quality, with increased speedup.

scholarly journals Treatment of Block-Based Sparse Matrices in Domain Decomposition Method

2017 ◽  
Vol 2 (1) ◽  
pp. 1
Author(s):  
Abul Mukid Mohammad Mukaddes ◽  
Masao Ogino ◽  
Ryuji Shioya ◽  
Hiroshi Kanayama
Keyword(s):  
Finite Element ◽  
Domain Decomposition ◽  
Thermal Convection ◽  
Decomposition Method ◽  
Sparse Matrix ◽  
Domain Decomposition Method ◽  
Parallel Computer ◽  
Element Discretization ◽  
Sparsity Pattern ◽  
The Matrix

Abstract— The domain decomposition method involves the finite element solution of problems in the parallel computer. The finite element discretization leads to the solution of large systems of linear equation whose matrix is naturally sparse. The use of proper storing techniques for sparse matrix is fundamental especially when dealing with large scale problems typical of industrial applications. The aim of this research is to review the sparsity pattern of the matrices originating from the discretization of the elasto-plastic and thermal-convection problems. Some practical strategies dealing with sparsity pattern in the finite element code of adventure system are recalled. Several efficient storage schemes to store the matrix originating from elasto-plastic and thermal-convection problems have been proposed. In the proposed technique, inherent block pattern of the matrix is exploited to locate the matrix element. The computation in the high performance computer shows better performance compared to the conventional skyline storage method used by the most of the researchers.

scholarly journals The null identities for boundary operators in the (2, 2p + 1) minimal gravity

2020 ◽  
Vol 2020 (2) ◽  
Author(s):  
Goro Ishiki ◽  
Hisayoshi Muraki ◽  
Chaiho Rim
Keyword(s):  
Matrix Model ◽  
Model Representation ◽  
Physical Implication ◽  
Boundary Interaction ◽  
Liouville Gravity ◽  
The Matrix ◽  
Boundary Operators

Abstract By using the matrix model representation, we show that correlation numbers of boundary-changing operators (BCOs) in $(2,2p+1)$ minimal Liouville gravity satisfy some identities, which we call the null identities. These identities enable us to express the correlation numbers of BCOs in terms of those of boundary-preserving operators. We also discuss a physical implication of the null identities as the manifestation of the boundary interaction.

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