Speaker: Susanne Brenner, Louisiana State University
Title: Additive Schwarz Theory
The theory for additive Schwarz methods, introduced by Dryja and Widlund circa 1987, has been a major driving force behind the development of domain decomposition methods in the last thirty years. In this talk I will give an account of this theory from a personal perspective.

Speaker: Xiao-Chuan Cai, University of Colorado at Boulder
Title: Domain Decomposition: From Poisson to Coupled Multi-physics Problems
In 1987, Dryja and Widlund introduced a domain decomposition method "Additive Schwarz Method" (ASM) and showed that it works well for the Poisson equation. Over the past 30 years, ASM has evolved into a major algorithm for solving many types of partial differential equations on large scale parallel computers with millions of cores. In this talk, we discuss some recent development of ASM for solving multi-physics problems involving fluid-structure interaction with applications in biomechanics.

Speaker: Juan Calvo, University of Costa Rica
Title: A virtual coarse space for two-level overlapping Schwarz methods
We present a coarse space for irregular subdomains based on virtual elements for nodal problems in 2D. Its dimension equals the number of subdomain vertices and results are insensitive to large jumps in the coefficients of the elliptic problem across the interface. The condition number of the preconditioned system grows linearly as a function of the relative overlap. Some numerical experiments that verify the result are shown.

Speaker: Raymond Chan, Chinese University of Hong Kong
Title: Point-spread function reconstruction in ground-based astronomy
Because of atmospheric turbulence, images of objects in outer space acquired via ground-based telescopes are usually blurry. One way to estimate the blurring kernel or point spread function (PSF) is to make use of the aberration of wavefront received at the telescope, i.e., the phase. However only the low-resolution wavefront gradients can be collected by wavefront sensors. In this talk, I will discuss how to use regularization methods to reconstruct high-resolution phase gradients and then use them to recover the phase and the PSF in high accuracy.

Speaker: Clark Dohrmann, Sandia National Laboratories
Title: On the Design and Analysis of Coarse Spaces for Domain Decomposition Methods
Coarse spaces are at the heart of many domain decomposition methods. In this talk, we present an overview of the many fruitful and pleasant collaborations the speaker has had with Olof Widlund in this area during the past 10+ years. Building on the foundation laid by Dryja, Smith, and Widlund in their 1994 SINUM paper, we describe a family of coarse spaces built on energy minimization concepts. These spaces have been applied to a variety of problems including almost incompressible elasticity, irregular shaped subdomains, and problems in H(curl) and H(div). A hybrid algorithm which combines both overlapping Schwarz and iterative substructuring concepts will also be discussed. Finally, we present very recent joint work on the design of small coarse spaces and some new results for overlapping Schwarz algorithms.

Speaker: Hyea Hyun Kim, Kyung Hee University, Korea
Title: Adaptive BDDC and FETI-DP algorithms
Adaptive BDDC and FETI-DP algorithms are developed for elliptic problems in both two and three dimensions to deal with highly oscillatory and random coefficients. For problems with such bad coefficients, the standard choice of coarse problem often fails to give good performance of both algorithms for such bad coefficients. The coarse problem is enriched by solving appropriate generalized eigenvalue problems on each equivalence classes which are edges in two dimensions, and faces and edges in three dimensions. The condition numbers of both algorithms with the enriched coarse problem are controlled by the user-defined tolerance value, which is used to select adaptive coarse components from the generalized eigenvalue problems. Numerical results are presented to show the performance of the proposed methods in both two and three dimensions.

Speaker: Axel Klawonn, University of Cologne
Title: FETI-DP and BDDC for Composite Materials - Adaptive Coarse Spaces in 3D
Authors: Axel Klawonn, Martin Kuehn, Oliver Rheinbach
Composite materials are a challenge for domain decomposition methods since arbitrary material coefficient jumps can lead to a severe deterioration of their convergence rate. A remedy can be the construction of coarse spaces which are automatically adapted to those coefficient jumps. In this talk, such coarse spaces are considered for FETI-DP and BDDC domain decomposition methods which are based on the adaptive computation of small local eigenvalue problems. A special emphasis is put on the three dimensional case and on materials with heterogeneous material parameters where coefficient jumps are not aligned with the interface of the domain decomposition. Our new approach is based on solving local eigenvalue problems on faces, enriched by a selected, small number of additional local eigenvalue problems on edges. The additional edge eigenvalue problems make the method provably robust with a condition number bound which depends only on the tolerance of the local eigenvalue problems and some properties of the domain decomposition. The introduction of relevant edge eigenvalue problems for FETI-DP and BDDC methods yields a condition number estimate which is independent of discontinuities of the material parameters. Deflation and transformation of basis techniques are considered for the implementation of the adaptively chosen coarse components. Numerical results are presented for linear elasticity and composite materials supporting our theoretical findings. The problems considered include those with randomly distributed coefficients and almost incompressible material components.

Speaker: Duk-Soon Oh, Rutgers University
Title: Multigrid methods for H(div) with nonoverlapping domain decomposition smoothers
We design and analyze V-cycle multigrid methods for H(div) problems discretized by the lowest order Raviart-Thomas element. Unlike standard elliptic problems, multigrid methods for the H(div) problems with traditional smoothers do not work well. We introduce smoothers in the multigrid methods using nonoverlapping domain decomposition preconditioners that are based on substructuring. We prove uniform convergence of the V-cycle method.

Speaker: Luca Pavarino, University of Pavia
Title: Scalable Domain Decomposition Preconditioners for cardiac electro-mechanical simulations
Authors: P. Colli Franzone, L. F. Pavarino, S. Scacchi, S. Zampini
We introduce and study some scalable domain decomposition preconditioners for cardiac electro-mechanical 3D simulations. The cardiac electro-mechanical model considered consisting of four coupled components: a) the quasi-static transversely isotropic finite elasticity equations for the deformation of the cardiac tissue; b) the active tension model for the intracellular calcium dynamics and cross-bridge binding; c) the anisotropic Bidomain model for the electrical current flow through the deforming cardiac tissue; d) the membrane model of ventricular myocytes, including stretch-activated channels. The 3D numerical simulations are based on our finite element parallel solver, which employs Multilevel Additive Schwarz preconditioners for the solution of the discretized Bidomain equations and Newton-Krylov methods with AMG or BDDC preconditioners for the solution of the discretized nonlinear finite elasticity equations. The results of several parallel simulations show the scalability of both linear and nonlinear solvers and their application to the study of re-entrant waves in the presence of different mechano-electrical feedbacks.

Speaker: Oliver Rheinbach, TU Bergakademie Freiberg
Title: Domain Decomposition for Extreme Scale Computing
Different Versions of Nonlinear Domain Decomposition Methods (DDM) are discussed which scale to the largest supercomputers currently available. This includes Nonlinear BDDC and FETI-DP methods applying multigrid methods for the coarse problem or for, both, the subdomain problems as well as the coarse problem. In these method, the subdomain problems are nonlinear which reduces communication and can help to localize the computational work. Moreover, numerical results for a highly parallel numerical homogenization method well known in nonlinear structural mechanics (FE2) are presented applying FETI-DP domain decomposition methods on the Representative Volume Elements (RVEs).

Speaker: Marcus Sarkis, Worcester Polytechnic Institute
Title: Robust Discretizations based on Adaptive Selection of Primal Constraints
Major progress has been made recently to make FETI-DP and BDDC preconditioners robust with respect to any variation of coefficients inside and/or across the subdomains. A reason for this success is the adaptive selection of primal constraints technique based on localized generalized eigenvalue problems. In this talk we discuss how to transfer this technique to the field of discretizations. We design discretizations where the number of degrees of freedom is the number of primal constraints on the coarse triangulation. We establish a priori energy error estimates with hidden constants independently of the coefficients, and for the analysis, the regularity of the solution is not used, only that the right-hand side be in L2. We note that the resulting multiscale basis functions are nonlocal however decay exponentially. In this talk we concentrate on the Raviart-Thomas case. (Joint work with Alexandre Madureira, National Laboratory for Scientific Computing, Brazil)

Speaker: Simone Scacchi, University of Milan
Title: BDDC preconditioners for isogeometric analysis
Authors: L. Beirao da Veiga, D. Cho, L. F. Pavarino, S. Scacchi, S. Zampini, O. B. Widlund
Isogeometric Analysis (IGA) is an innovative numerical technology for the solution of Partial Differential Equations (PDEs), where the geometry description of the PDE domain is adopted from a Computer Aided Design (CAD) parametrization usually based on Non-Uniform Rational B-Splines (NURBS) and the same NURBS basis functions are also used as the PDEs discrete basis, following an isoparametric paradigm. The aim of this talk is to present Balancing Domain Decomposition by Constraints (BDDC) preconditioners for IGA discretizations of scalar elliptic problems and compressible linear elasticity, based on three types of interface averaging functions: standard ρ-scaling, stiffness scaling and the novel deluxe scaling, with either full or reduced set of primal constraints. The resulting algorithms are proved to be quasi-optimal, scalable and robust with respect to jumps of the PDEs coefficients. Extensive two- and three-dimensional numerical results validate the theoretical estimates.

Speaker: Barry Smith, Argonne National Laboratory
Title: Composability in algebraic solvers and ODE integrators (canceled)
It is by now relatively commonly known that most efficient linear preconditioners can be obtained by composing a small set of simple operations, what is less well understood is that this also holds true for nonlinear solvers as well as ODE integrators. I will discuss composability at all three levels and how it can simplify the development of convergence results and numerical solver libraries.

Speaker: Nicole Spillane, Ecole Polytechnique
Title: Adaptive Multipreconditioning for Symmetric and Non-Symmetric Problems
Multipreconditioning is a technique that allows to use multiple preconditioners within a Krylov subspace method. For domain decomposition, this means that each contribution to the preconditioner (generated by one subdomain) is used to enlarge the search space instead of the sum of all of these directions. Quite naturally, this significantly enlarged search space leads to robust solvers that converge in a small number of iterations. In order for the overall cost of computation to be competitive, adaptivity is needed: only some iterations of the Krylov subspace methods are multipreconditioned. In this talk I will discuss how to choose the adaptivity process and what results can be guaranteed depending on the Domain Decomposition method and the nature of the linear system (symmetric or nonsymmetric).

Speaker: Daniel Szyld, Temple University
Title: Asynchronous Optimized Schwarz, Theory and Experiments
Asynchronous methods refer to parallel iterative procedures where each process performs its task without waiting for other processes to be completed, i.e., with whatever information it has locally available and with no synchronizations with other processes. In this talk, an asynchronous version of the optimized Schwarz method is presented for the solution of differential equations on a parallel computational environment. In a one-way subdivision of the computational domain, with overlap, the method is shown to converge when the optimal artificial interface conditions are used. Convergence is also proved under very mild conditions on the size of the subdomains, when approximate (non-optimal) interface conditions are utilized. Numerical results are presented on large three-dimensional problems illustrating the efficiency of the proposed asynchronous parallel implementation of the method. The main application shown is the calculation of the gravitational potential in the area around the Chicxulub crater, in Yucatan, where an asteroid is believed to have landed 66 million years ago contributing to the extinction of the dinosaurs. (Joint work with Frédéric Magoulès and Cedric Venet, CentraleSupélec, Châtenay-Malabry, France)

Speaker: Xuemin Tu, University of Kansas
Title: BDDC domain decomposition for hybrid discontinuous Galerkin Methods
A Balancing domain decomposition by constraints (BDDC) algorithm is studied for solutions of large sparse linear algebraic systems arising from hybrid discontinuous Galerkin discretization. The condition number for the preconditioned system is estimated and numerical results are provided to confirm the results.

Speaker: Barbara Wohlmuth, TU Munich
Title: Domain decomposition meets multigrid
We discuss classical tearing and interconnecting strategies in combination with reduced order modeling, adaptive strategies and large scale multigrid solvers. The concepts of overlapping domain decomposition techniques are applied to eigenvalue problems in structural mechanics. A greedy algorithm is used to set up a reduced basis for rather large multi-storey timber construction. Material parameters enter by the wall type and elastomer. The abstract framework of non-overlapping Dirichlet-Neumann or Dirichlet-Dirichlet components is used in combination with over-balancing techniques to design fault robust multigrid solver. The reconnecting step is controlled by an hierarchical error estimator accounting for the algebraic error and a residual type one for the discretization error.

Speaker: Jinchao Xu, Pennsylvania State University
Title: Overlapping Schwarz Methods with Randomized Ordering
We consider the iterative method of subspace corrections, especially overlapping Schwarz methods, with random ordering. We prove identities for the expected convergence rate, which can provide sharp estimates for the error reduction per iteration. We also study the fault-tolerant feature of the randomized successive subspace correction method by simply rejecting all the corrections when error occurs and show that the results iterative method converges with probability 1. Moreover, we also provide sharp estimates on the expected convergence rate for the fault-tolerant, randomized, subspace correction method.

Speaker: Stefano Zampini, KAUST
Title: On the robustness and prospects of BDDC and FETI-DP methods for finite element discretizations of elliptic PDEs
In the last decade, Balancing Domain Decomposition by Constraints (BDDC) and Finite Element Tearing and Interconnecting Dual-Primal (FETI-DP) methods have proven to be powerful solvers for large and sparse linear systems arising from the finite element discretization of elliptic PDEs. Condition number bounds can be theoretically established, and these bounds are independent of the number of subdomains of the decomposition, and, more important, on the heterogeneity of the coefficients of the PDE. After a brief introduction to the methods, I will present the current state of their implementation in the PETSc library, together with numerical results for a variety of finite element discretizations and partial differential equations. Robustness, algorithmic aspects, and prospects of these methods will be also discussed.