Schur complement method

Abstract: Schur process is a time-dependent analog of the Schur measure on partitions studied in math.RT/9907127. Our first result is that the correlation functions of the Schur process are determinants with a kernel that has a nice contour integral representation in terms of the parameters of the process. This general result is then applied to a particular specialization of the Schur process, namely to random 3-dimensional Young diagrams. The local geometry of a large random 3-dimensional diagram is described in terms of a determinantal point process on a 2-dimensional lattice with the incomplete beta function kernel (which generalizes the discrete sine kernel). A brief discussion of the universality of this answer concludes the paper.

Abstract: Invited Talks.- Non-matching Grids and Lagrange Multipliers.- A FETI Method for a Class of Indefinite or Complex Second- or Fourth-Order Problems.- Hybrid Schwarz-Multigrid Methods for the Spectral Element Method: Extensions to Navier-Stokes.- Numerical Approximation of Dirichlet-to-Neumann Mapping and its Application to Voice Generation Problem.- Selecting Constraints in Dual-Primal FETI Methods for Elasticity in Three Dimensions.- Coupled Boundary and Finite Element Tearing and Interconnecting Methods.- Parallel Simulation of Multiphase/Multicomponent Flow Models.- Uncoupling-Coupling Techniques for Metastable Dynamical Systems.- Minisymposium: Domain Decomposition Methods for Wave Propagation in Unbounded Media.- On the Construction of Approximate Boundary Conditions for Solving the Interior Problem of the Acoustic Scattering Transmission Problem.- Approximation and Fast Calculation of Non-local Boundary Conditions for the Time-dependent Schrodinger Equation.- Domain Decomposition and Additive Schwarz Techniques in the Solution of a TE Model of the Scattering by an Electrically Deep Cavity.- Minisymposium: Parallel Finite Element Software.- A Model for Parallel Adaptive Finite Element Software.- Towards a Unified Framework for Scientific Computing.- Distributed Point Objects. A New Concept for Parallel Finite Elements.- Minisymposium: Collaborating Subdomains for Multi-Scale Multi-Physics Modelling.- Local Defect Correction Techniques Applied to a Combustion Problem.- Electronic Packaging and Reduction in Modelling Time Using Domain Decomposition.- Improving Robustness and Parallel Scalability of Newton Method Through Nonlinear Preconditioning.- Iterative Substructuring Methods for Indoor Air Flow Simulation.- Fluid-Structure Interaction Using Nonconforming Finite Element Methods.- Interaction Laws in Viscous-Inviscid Coupling.- Minisymposium: Recent Developments for Schwarz Methods.- Comparison of the Dirichlet-Neumann and Optimal Schwarz Method on the Sphere.- Finite Volume Methods on Non-Matching Grids with Arbitrary Interface Conditions and Highly Heterogeneous Media.- Nonlinear Advection Problems and Overlapping Schwarz Waveform Relaxation.- A New Cement to Glue Nonconforming Grids with Robin Interface Conditions: The Finite Element Case.- Acceleration of a Domain Decomposition Method for Advection-Diffusion Problems.- A Stabilized Three-Field Formulation and its Decoupling for Advection-Diffusion Problems.- Approximation of Optimal Interface Boundary Conditions for Two-Lagrange Multiplier FETI Method.- Optimized Overlapping Schwarz Methods for Parabolic PDEs with Time-Delay.- Minisymposium: Trefftz-Methods.- A More General Version of the Hybrid-Trefftz Finite Element Model by Application of TH-Domain Decomposition.- Minisymposium: Domain Decomposition on Nonmatching Grids.- Mixed Finite Element Methods for Diffusion Equations on Nonmatching Grids.- Mortar Finite Elements with Dual Lagrange Multipliers: Some Applications.- Non-Conforming Finite Element Methods for Nonmatching Grids in Three Dimensions.- On an Additive Schwarz Preconditioner for the Crouzeix-Raviart Mortar Finite Element.- Minisymposium: FETI and Neumann-Neumann Domain Decomposition Methods.- A FETI-DP Method for the Mortar Discretization of Elliptic Problems with Discontinuous Coefficients.- A FETI-DP Formulation for Two-dimensional Stokes Problem on Nonmatching Grids.- Some Computational Results for Dual-Primal FETI Methods for Elliptic Problems in 3D.- The FETI Based Domain Decomposition Method for Solving 3D-Multibody Contact Problems with Coulomb Friction.- Choosing Nonmortars: Does it Influence the Performance of FETI-DP Algorithms?.- Minisymposium: Heterogeneous Domain Decomposition with Applications in Multiphysics.- Domain Decomposition Methods in Electrothermomechanical Coupling Problems.- A Multiphysics Strategy for Free Surface Flows.- Minisymposium: Robust Decomposition Methods for Parameter Dependent Problems.- Weighted Norm-Equivalences for Preconditioning.- Preconditioning for Heterogeneous Problems.- Minisymposium: Recent Advances for the Parareal in Time Algorithm.- On the Convergence and the Stability of the Parareal Algorithm to Solve Partial Differential Equations.- A Parareal in Time Semi-implicit Approximation of the Navier-Stokes Equations.- The Parareal in Time Iterative Solver: a Further Direction to Parallel Implementation.- Stability of the Parareal Algorithm.- Minisymposium: Space Decomposition and Subspace Correction Methods for Linear and Nonlinear Problems.- Multilevel Homotopic Adaptive Finite Element Methods for Convection Dominated Problems.- A Convergent Algorithm for Time Parallelization Applied to Reservoir Simulation.- Nonlinear Positive Interpolation Operators for Analysis with Multilevel Grids.- Minisymposium: Discretization Techniques and Algorithms for Multibody Contact Problems.- On Scalable Algorithms for Numerical Solution of Variational Inequalities Based on FETI and Semi-monotonic Augmented Lagrangians.- Fast Solving of Contact Problems on Complicated Geometries.- Contributed Talks.- Generalized Aitken-like Acceleration of the Schwarz Method.- The Fat Boundary Method: Semi-Discrete Scheme and Some Numerical Experiments.- Modelling of an Underground Waste Disposal Site by Upscaling and Simulation with Domain Decomposition Method.- Non-Overlapping DDMs to Solve Flow in Heterogeneous Porous Media.- Domain Embedding/Controllability Methods for the Conjugate Gradient Solution of Wave Propagation Problems.- An Accelerated Block-Parallel Newton Method via Overlapped Partitioning.- Generation of Balanced Subdomain Clusters with Minimum Interface for Distributed Domain Decomposition Applications.- Iterative Methods for Stokes/Darcy Coupling.- Preconditioning Techniques for the Bidomain Equations.- Direct Schur Complement Method by Hierarchical Matrix Techniques.- Balancing Neumann-Neumann Methods for Elliptic Optimal Control Problems.- Domain Decomposition Preconditioners for Spectral Nedelec Elements in Two and Three Dimensions.- Parallel Distributed Object-Oriented Framework for Domain Decomposition.- A Domain Decomposition Based Two-Level Newton Scheme for Nonlinear Problems.- Domain Decomposition for Discontinuous Galerkin Method with Application to Stokes Flow.- Hierarchical Matrices for Convection-Dominated Problems.- Parallel Performance of Some Two-Level ASPIN Algorithms.- Algebraic Analysis of Schwarz Methods for Singular Systems.- Schwarz Waveform Relaxation Method for the Viscous Shallow Water Equations.- A Two-Grid Alternate Strip-Based Domain Decomposition Strategy in Two-Dimensions.- Parallel Solution of Cardiac Reaction-Diffusion Models.- Predictor-Corrector Methods for Solving Continuous Casting Problem.

Abstract: Saddle point systems arise widely in optimization problems with constraints. The utility of Schur complement approximation is now broadly appreciated in the context of solving such saddle point systems by iteration. In this short manuscript, we present a new Schur complement approximation for PDE constrained optimization, an important class of these problems. Block diagonal and block triangular preconditioners have previously been designed to be used to solve such problems along with MINRES and non-standard Conjugate Gradients respectively; with appropriate approximation blocks these can be optimal in the sense that the time required for solution scales linearly with the problem size, however small the mesh size we use. In this paper, we extend this work to designing such preconditioners for which this optimality property holds independently of both the mesh size and of the Tikhonov regularization parameter \beta that is used. This also leads to an effective symmetric indefinite preconditioner that exhibits mesh and \beta-independence. We motivate the choice of these preconditioners based on observations about approximating the Schur complement obtained from the matrix system, derive eigenvalue bounds which verify the effectiveness of the approximation, and present numerical results which show that these new preconditioners work well in practice.