1,720,967 research outputs found
A Multigrid Solver for PDE-Constrained Optimization with Uncertain Inputs
No full textIn this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number N of samples used to discretized the probability space. We show that this reduced system can be solved with optimal O(N) complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and L1-norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest
Multilevel Optimized Schwarz Methods
No full textWe define a new two-level optimized Schwarz method (OSM), and we provide a convergence analysis both for overlapping and nonoverlapping decompositions. The two-level analysis suggests how to choose the optimized parameters. We also discuss an optimization procedure which relies only on the already studied one-level min-max problems, and we show that these two approaches are asymptotically equivalent. The two-level OSM has mesh independent convergence and it is scalable. We then generalize the two-level method defining a multilevel domain decomposition method which uses the OSM as a smoother. The main advantage of the method consists of its robustness and generality with respect to the equations under study. Thanks to the smoothing properties of the OSM, both with and without overlap, we can define a unique algorithm which can be applied to several equations, both with homogeneous and heterogeneous coefficients. We present extensive numerical results to compare the multilevel OSM, the one-level OSM, and the multigrid scheme. The experiments show that the multilevel OSM inherits robustness from the one-level OSM for heterogeneous elliptic problems, wave problems, and heterogeneous couplings. Finally, we apply the method to design a two-level solver for the heterogeneous Stokes-Darcy system
Substructured Two-grid and Multi-grid Domain Decomposition Methods
Full text linkTwo-level domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume. In this paper, we use the excellent smoothing properties of domain decomposition methods to define a new class of substructured two-level methods, for which both domain decomposition smoothers and coarse correction steps are defined on the interfaces. This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, our approach does not require the explicit construction of coarse spaces, and it permits a multilevel extension, which is desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.CSQ
Spectral Coarse Spaces for the Substructured Parallel Schwarz Method
Full text linkThe parallel Schwarz method (PSM) is an overlapping domain decomposition (DD) method to solve partial differential equations (PDEs). Similarly to classical nonoverlapping DD methods, the PSM admits a substructured formulation, that is, it can be formulated as an iteration acting on variables defined exclusively on the interfaces of the overlapping decomposition. In this manuscript, spectral coarse spaces are considered to improve the convergence and robustness of the substructured PSM. In this framework, the coarse space functions are defined exclusively on the interfaces. This is in contrast to classical two-level volume methods, where the coarse functions are defined in volume, though with local support. The approach presented in this work has several advantages. First, it allows one to use some of the well-known efficient coarse spaces proposed in the literature, and facilitates the numerical construction of efficient coarse spaces. Second, the computational work is comparable or lower than standard volume two-level methods. Third, it opens new interesting perspectives as the analysis of the new two-level substructured method requires the development of a new convergence analysis of general two-level iterative methods. The new analysis casts light on the optimality of coarse spaces: given a fixed dimension m, the spectral coarse space made by the first m dominant eigenvectors is not necessarily the minimizer of the asymptotic convergence factor. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.CSQ
Heterogeneous Optimized Schwarz Methods for Second Order Elliptic PDEs
Full text linkDue to their property of convergence in the absence of overlap, optimized Schwarz methods are the natural domain decomposition framework for heterogeneous problems, where the spatial decomposition is provided by the multiphysics of the phenomena. We study here heterogeneous problems which arise from the coupling of second order elliptic PDEs. Theoretical results and asymptotic formulas are proposed solving the corresponding min-max problems both for single and double sided optimizations, while numerical results confirm the effectiveness of our approach even when analytical conclusions are not available. Our analysis shows that optimized Schwarz methods do not suffer the heterogeneity, it is the opposite, they are faster the stronger the heterogeneity is. It is even possible to have h independent convergence choosing two independent Robin parameters. This property was proved for a Laplace equation with discontinuous coefficients, but only conjectured for more general couplings in [M. J. Gander and O. Dubois, Numer. Algorithms, 69 (2015), pp. 109-144]. Our study is completed by an application to a contaminant transport problem
MATHICSE Technical Report : Preconditioners for robust optimal control problems under uncertainty
No full textThe discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle-point systems, which are fully coupled across the random realizations. Despite its relevance for numerous engineering problems, the solution of such systems is notoriusly challenging. In this manuscript, we study efficient preconditioners for all-at-once approaches using both an algebraic and an operator preconditioning framework. We show in particular that for values of the regularization parameter not too small, the saddle-point system can be efficiently solved by preconditioning in parallel all the state and adjoint equations. For small values of the regularization parameter, robustness can be recovered by the additional solution of a small linear system, which however couples all realizations. A mean approximation and a Chebyshev semi-iterative method are investigated to solve this reduced system. Our analysis considers a random elliptic partial differential equation whose diffusion coefficient κ(x,ω) is modeled as an almost surely continuous and positive random field, though not necessarily uniformly bounded and coercive. We further provide estimates on the dependence of the preconditioned system on the variance of the random field. Such estimates involve either the first or second moment of the random variables 1/min_{x\in \bar{D}} κ(x,ω) and max_{x\in \bar{D}} κ(x,ω), where D is the spatial domain. The theoretical results are confirmed by numerical experiments, and implementation details are further addressed.CSQ
Robust optimization of control parameters for WEC arrays using stochastic methods
Full text linkThis work presents a new computational optimization framework for the robust control of parks of Wave Energy Converters (WEC) in irregular waves. The power of WEC parks is maximized with respect to the individual control damping and stiffness coefficients of each device. The results are robust with respect to the incident wave direction, which is treated as a random variable. Hydrodynamic properties are computed using the linear potential model, and the dynamics of the system is computed in the frequency domain. A slamming constraint is enforced to ensure that the results are physically realistic. We show that the stochastic optimization problem is well posed. Two optimization approaches for dealing with stochasticity are then considered: stochastic approximation and sample average approximation. The outcomes of the above mentioned methods in terms of accuracy and computational time are presented. The results of the optimization for complex and realistic array configurations of possible engineering interest are then discussed. Results of extensive numerical experiments demonstrate the efficiency of the proposed computational framework
On the Derivation of Optimized Transmission Conditions for the Stokes-Darcy Coupling
No full textRecently a lot of attention has been devoted to the Stokes-Darcy coupling which is a system of equations used to model the flow of fluids in porous media. In [2, 1] a non standard behaviour of the optimized Schwarz method (OSM) has been observed: the optimized parameters obtained solving the classical min-max problems do not lead to an optimized convergence
Optimized Schwarz Methods for Advection Diffusion Equations in Bounded Domains
No full textOptimized Schwarz methods use better transmission conditions than the classical Dirichlet conditions that were used by Schwarz. These transmission conditions are optimized for the physical problem that needs to be solved to lead to fast convergence. The opimization is typically performed in the geometrically simplified setting of two unbounded subdomains using Fourier transforms. Recent studies for both homogeneous and heterogeneous domain decomposition methods indicate that the geometry of the physical domain has actually an influence on this opimization process. We study here this influence for an advection diffusion equation in a bounded domain using separation of variables. We provide theoretical results for the min-max problems characterizing the optimized transmission conditions. Our numerical experiments show significant improvements of the new transmission conditions which take the geometry into account, especially for strong tangential advection. </p
Heterogeneous Optimized Schwarz Methods for Coupling Helmholtz and Laplace Equations
No full textOptimized Schwarz methods have increasingly drawn attention over the last decades because of their improvements in terms of robustness and computational cost compared to the classical Schwarz method. Optimized Schwarz methods are also a natural framework to study heterogeneous phenomena, where the spatial decomposition is provided by the multi-physics of the problem, because of their good convergence properties in the absence of overlap. We propose here zeroth order optimized transmission conditions for the coupling between the Helmholtz equation and the Laplace equation, giving asymptotically optimized choices for the parameters, and illustrating our analytical results with numerical experiments
- …
