ABSTRACT. In this talk, we consider several ways to precondition high-order finite element matrices whose entries are not readily available. A standard way to increase the accuracy of a finite element discretization is to increase the polynomial order of the method. This can be particularly advantageous on emerging computer architectures that exploit a high degree of parallelism, because the high-order discretization leads to dense submatrices whose memory access patterns can be known in advance, and in some finite element settings high-speed tensor contractions can be exploited. The use of high-order discretizations also makes it relatively more expensive to explicitly assemble finite element matrices, and correspondingly more attractive to apply the matrix without storing its entries. Though this "matrix-free" setting is attractive from one perspective, it makes preconditioning and especially algebraic preconditioning a challenge. Here we consider ways to assemble sparser versions of such an operator that maintain its essential features. Once such a sparsification is assembled, an algebraic multigrid preconditioner can be developed. We find that this AMG preconditioner can be effective as a preconditioner for the original matrix-free operator.

ABSTRACT. Modern hardware architectures provide a formidable challenge to the design of algorithms with portable performance across different flavors of multicore CPUs, manycore accelerators, and graphics processors. I will present a case study of an algebraic multigrid method for uncertainty quantification to show the applicability of explicit vectorization techniques as a general design tool for massively parallel software.

ABSTRACT. To solve large Electromagnetism problems, we use an Integral Equation formulation that requires to solve dense complex problems with millions of unknowns, using a direct Cholesky solver on the TERA supercomputers located at CEA/DAM.

In this talk, we will describe how our task-based programming model made it possible to scale up to 60000 cores.

We will also explain how we enhanced our solver with hierarchical compression techniques (H-matrix) thanks to recursion, and how task preemption allows us to efficiently integrate MPI within this task-based model.

Finally we will illustrate how this flexible approach makes it possible to take advantage of the Intel KNL architecture by the means of multithreaded tasks.

ABSTRACT. SNAP is a proxy application which simulates the computational motion of a neutral particle transport code. In this work, we have adapted parts of SNAP separately; we have re-implemented the iterative shell of SNAP in the task-model runtime Legion, showing an improvement to the original schedule, and we have created multiple Kokkos implementations of the computational kernel of SNAP, displaying similar performance to the native Fortran.

ABSTRACT. FleCSI is a compile-time configurable framework designed to support multi-physics application development. As such, FleCSI attempts to provide a very general set of infrastructure design patterns that can be specialized and extended to suit the needs of a broad variety of solver and data requirements. Current support includes multi-dimensional mesh topology, mesh geometry, mesh adjacency information, n-dimensional hashed-tree data structures, graph partitioning interfaces, and dependency closures. FleCSI also introduces a functional programming model with control, execution, and data abstractions that are consistent with both MPI and state-of-the-art task-based runtimes such as Legion and Charm++. The FleCSI abstraction layer provides the developer with insulation from the underlying runtime, while allowing support for multiple runtime systems, including conventional models like asynchronous MPI. The intent is to give developers a concrete set of user-friendly programming tools that can be used now, while allowing flexibility in choosing runtime implementations and optimizations that can be applied to architectures and runtimes that arise in the future. The control and execution models in FleCSI also provide formal nomenclature for describing poorly understood concepts like kernels and tasks. To provide a low barrier to entry, the FleCSI data model does not lock developers into particular layouts or data structure representations, thus providing a low-buy-in approach that makes FleCSI an attractive option for many application projects.

In this presentation, we will cover some of the core concepts and features of FleCSI, and discuss how it is being used in the Ristra project, LANL’s Advanced Technology Development & Mitigation (ATDM) project.

ABSTRACT. Multigrid methods that use Braess-Sarazin-type relaxation schemes have been successfully applied to several saddle-point problems, including those that arise from the discretization of the Stokes equations. In this paper, we present a local Fourier analysis (LFA) of Braess-Sarazin-type relaxation for the staggered finite-difference discretization of the Stokes equations to analyze the convergence behavior. Exact and inexact (weighted Jacobi) Braess-Sarazin-type relaxation schemes are included. With this analysis, optimal parameters are proposed. With these parameters, our LFA predicts that multigrid methods using the inexact Braess-Sarazin relaxation should obtain the same convergence rate as those using the exact relaxation. Finally, some numerical experiments are presented to validate the two-grid and multigrid convergence factors.

ABSTRACT. Biot's consolidation model is used in a variety of applications, ranging from geophysics to biomechanics, in addition to being used in multiphysics problems. In this work we study a weak formulation of Biot's consolidation model using stress as a primary unknown, and further impose the symmetry of this stress weakly. This leads to a saddle point system depending on several parameters. We show that the formulation is well posed in a set of weighted Hilbert spaces, and propose a block diagonal preconditioner that is robust in a wide set of parameter ranges and mesh refinement.

ABSTRACT. Linear solvers for the systems of partial differential equations are the central process to many numerical simulations in science and engineering. It is often the most time-consuming part of the computation. Because of the extremely large problem size, iterative methods are normally the only option available. In order to properly design the iterative methods for these systems, one has to rely on the specific properties of the underlying differential systems themselves. As a consequence, an efficient preconditioning strategy based on the PDE systems is required. Two-stage preconditioners are one of the most effective preconditioning techniques. The method splits the resulting linear system in two major components and uses a suitable preconditioner for each component. In our work, we present a general framework for constructing two-stage preconditioners based on multigrid ideas for solving linear systems from the fully implicit formulation. The framework is based on a multigrid technique known as multigrid reduction (MGR). MGR starts by decomposing the unknowns into sets called C-points and F-points. Several examples are covered such as saddle point problems and the problems from reservoir simulation. The MGR framework provides both theoretical and practical ways to further understand and improve the two-stage preconditioners for different PDE systems. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE- AC52-07NA27344.

ABSTRACT. Multigrid methods are efficient iterative solvers for the solution of partial differential equations (PDEs). The efficiency of multigrid methods is due to the combination of suitable smoothers with a coarse grid correction. As one of the two key ingredients smoothers have a significant impact on the performance of multigrid solvers. Most of the literature on smoothers in multigrid methods is concerned with scalar PDEs, only. Systems are considered less often. In this talk, we present a comparative study of several smoothers for multigrid methods for the solution of the Stokes equations. Besides the commonly used Vanka smoother and the Braess-Sarazin smoother, we also consider a non-overlapping variant of the Vanka smoother. While the latter is computationally cheaper, the convergence depends much more on the implementation than that of the overlapping method. We consider discretization using appropriate finite elements as well as by finite differences on staggered grids. A comparison including the computational cost and the convergence properties of the different methods will be presented.

ABSTRACT. The need for parallelism in the time dimension is being driven by changes in computer architectures, where performance increases are now provided through greater concurrency, not faster clock speeds. Multigrid Reduction in Time (MGRIT) is an iterative procedure that allows for temporal parallelism by utilizing multigrid reduction techniques and a multilevel hierarchy of coarse time grids. MGRIT has been shown to be effective for both linear and nonlinear problems, with speedups of up to 50$x$ seen in the linear case.

Spatial and temporal adaptivity are powerful techniques used in most state of the art sequential time stepping routines. This talk will outline how those techniques can be used inside the MGRIT framework. In particular, we present a fully parallel, FMG based algorithm for adapting in time and space using MGRIT, backward Euler time stepping, and a first order systems of least squares (FOSLS) spatial solver. Spatial adaptation is completed using the FOSLS error estimator and a threshold method. Temporal error estimators are calculated locally using Richardson extrapolation, but applied globally using a threshold method. Numerical results highlighting the usefulness of this approach will also be presented.

ABSTRACT. For the numerical solution of time-dependent partial differential equations, time-parallel methods have recently shown to provide a promising way to extend prevailing strong-scaling limits of numerical codes. One of the most complex methods in this field is the ``Parallel Full Approximation Scheme in Space and Time'' (PFASST). PFASST already shows promising results for many use cases and many more is work in progress. However, a solid and reliable mathematical foundation is still missing. We show that under certain assumptions the PFASST algorithm can be conveniently and rigorously described as a multigrid-in-time method. Following this equivalence, first steps towards a comprehensive analysis of PFASST using block-wise local Fourier analysis are taken. The theoretical results are applied to examples of diffusive and advective type.

ABSTRACT. This work is based on the rational approximation of exponential integrators (REXI). In contrast to a standard time stepping method which suffers of sequentially executed time steps, REXI allows a decomposition into a sum of independent subproblems for arbitrarily long time steps which are not restricted by the CFL. An increase in time step size would not result in an increase in step-by-step time step computations, but in an increase of terms which can be computed independent to each other.

We will put our focus on ocean and atmospheric simulations on the sphere with the discretization based on latitude-longitude grids. These grids are well-known to suffer of pole singularities which leads to, amongst others, tiny cells close to the poles. For these small cells, the CFL condition also forces smaller time step sizes. In contrast to other solutions which are mainly based on changing the grid we make use of a new time stepping method which allows arbitrarily long time steps for oscillatory linear terms, hence efficiently overcoming CFL-induced restrictions.

Results will be presented with this time stepping method for simulations with significant importance for atmospheric simulations on the sphere. Here, two major challenges were tackled: The first one is an application specific efficient reformulation of each REXI subproblem. This results in a Helmholtz-like formulation and an efficient solver is required which leads to the second challenge: the development of an efficient solver for this Helmholtz-like problem. Here, we are exploiting properties with the discretization using spherical harmonics. This global spectral space has intrinsic multi-resolution properties and further advantageous mathematical properties. Instead of solving the solution on various resolutions in physical space as it is typical for geometric multi-grid methods, the exploitation of properties of spherical harmonics results in a highly efficient direct solver in spectral space.

We will discuss numerical properties with benchmarks based on shallow water equations. Such shallow water benchmarks are used for the development of new dynamical cores for ocean and atmospheric simulations. Regarding HPC we will discuss performance issues and the efficiency of the suggested massively parallel solver for linear oscillatory problems.

ABSTRACT. The idea of parallelizing in the time dimension and, thus, adding an additional layer of parallelism to a numerical algorithm has become an increasing popular way to solve evolutionary problems on massively parallel supercomputers. Starting with the seminal work of Nievergelt in 1964 [1], various approaches for parallel-in-time integration have been explored. One of these approaches is applying multigrid in the time dimension, resulting in the multigrid-reduction-in-time (MGRIT) algorithm by Falgout et al. [2] The corresponding open source code XBraid [3] is flexible, allowing for a variety of time stepping, relaxation, and temporal coarsening options.

A key advantage of MGRIT over many other time-parallel methods is its non-intrusiveness, i.e., the effort one has to put into implementing the method when aiming at adding time-parallelism to an existing time-stepping code is relatively low. However, comparing the performance of MGRIT with the more invasive space-time multigrid (STMG) algorithm by Horton and Vandewalle [4], the speedup in comparison to space-parallel methods with sequential time stepping is larger for the invasive STMG approach than for the non-invasive MGRIT approach.

In this talk, we discuss incorporating aspects of the more invasive STMG algorithm into XBraid, extending the flexibility of the software to allow implementing an STMG-like algorithm.

References [1] J. Nievergelt, Comm. ACM 7, pp. 731–733 (1964). [2] R. D. Falgout et al., SIAM J. Sci. Comput. 36, pp. C635—C661 (2014). [3] XBraid: Parallel multigrid in time. http://llnl.gov/casc/xbraid. [4] G. Horton and S. Vandewalle, SIAM J. Sci. Comput. 16, pp. 848—864 (1995).

ABSTRACT. We present a new class of iterative schemes for solving initial value problems (IVP) based on discontinuous Galerkin (DG) methods. Starting from the weak DG formulation of an IVP, we derive an iterative method based on a preconditioned Picard iteration. Using this new approach, we can systematically construct explicit, implicit and semi-implicit schemes with arbitrary order of accuracy. We also show that the same schemes can be constructed by solving a series of correction equations based on the DG weak formulation. The accuracy of the schemes is proven to be min{2p+1, K} with p the degree of the DG polynomial basis and K the number of iterations. The stability is explored numerically; we show that the implicit schemes are A-stable at least for 0 ≤ p ≤ 9. Furthermore, we combine the methods with a multilevel strategy to accelerate their convergence speed. The new multilevel scheme is intended to provide a flexible framework for high order space-time discretizations and to be coupled with space-time multigrid techniques for solving partial differential equations (PDEs). We present numerical examples for ODEs and PDEs to analyze the performance of the new methods. Moreover, the newly proposed class of methods, due to its structure, is also a competitive and promising candidate for parallel in time algorithms such as Parareal, PFASST, multigrid in time, etc.

ABSTRACT. We present a scalable multigrid block-based preconditioner for the low-speed compressible Navier-Stokes equations with phase change. The equations are discretized in conservative form with a reconstructed Discontinuous Galerkin method and integrated with fully-implicit time discretization schemes. To enable robust convergence of stiff systems at low-Mach numbers, we use the Newton-Krylov framework for the primitive-variable formulation (pressure, velocity, and temperature), which is better conditioned than the conservative-variable form. In this paper we developed a block-partitioned preconditioner, which employs approximate block factorization techniques using the Schur complement matrix and reduces a 3x3 block system into a sequence of two 2x2 block sub-systems: velocity-pressure (vP) and velocity-temperature (vT). We compare the performance of this vP-vT Schur complement based preconditioner to a physics-block preconditioner, a one-level additive Schwarz preconditioner, and a fully-coupled algebraic multigrid preconditoner, on two representative problems: low-Mach lid-driven cavity flow and laser-induced melt convection. Numerical results indicate that vP-vT Schur complement preconditioner is robust at high CFL/Fourier numbers and exhibits good algorithmic and parallel scalability.

ABSTRACT. In recent years, high-order finite-element schemes have received a significant attention in the computational fluid dynamics (CFD) community due to the several advantages they can offer over the more established 2nd order finite-volume counterparts. These advantages include the use of a nearest neighbor stencil, straight forward extension to high-order discretizations, more capabilities for adaptive grids, and the generation of dense computations kernels that are more suitable for emerging high performance computing (HPC) architectures. However, the resulting discretized equations that must be solved for steady-state and time-implicit turbulent flow problems have proven to be much stiffer and more difficult to converge efficiently and robustly. The objective of the present study is to investigate and develop robust, efficient, and scalable multilevel solution strategies and preconditioning techniques for stabilized finite-element flow solvers. The proposed solution strategy is essentially a spectral multigrid approach in which the solution on the mesh with lowest polynomial degree (p=1) is solved using a preconditioned Newton-Krylov method. Seeking robustness, there has been a recent trend towards precondition the Newton-Krylov methods using incomplete factorization methods such as ILU(k). However, it is well known that these methods are not scalable. To overcome this problem, we have developed an implicit line preconditioner which can be properly distrusted among processing elements without affecting the convergence behavior of the linear system and non-linear path. The Implicit lines are extracted from a stiffness matrix based on strong connections. To improve the robustness of the implicit-line relaxation, a double CFL strategy, with a lower CFL number in the preconditioner matrix, has been developed. In this study, we also employ algebraic multigrid (AMG) preconditioners and augment them with the double CFL strategy. To reach high-order discretizations, a set of hierarchical basis functions are employed and a non-linear p-multigrid approach is developed. In order to test the performance of the newly developed algorithms, a new flow solver for two and three dimensional mixed-element unstructured meshes has been developed in which the Reynolds averaged Navier-Stokes (RANS) equations and negative Spalart-Almaras (SA) turbulence model are discretized, in a coupled form, using the Streamline Upwind Petrov-Galerkin (SUPG) scheme. Several two- and three-dimensional numerical examples including turbulent flows over a three element airfoil and a wing-body-tail configuration from the 4th AIAA Drag Prediction Workshop are presented in which the performance of implicit line relaxation and algebraic multigrid preconditioners are compared with the incomplete lower upper factorization (ILU(k)) method. Results show positive steps toward development of scalable solution techniques.

ABSTRACT. Multiphase flow is a critical process in a wide range of applications, including oil and gas recovery, carbon sequestration, and contaminant remediation. Numerical simulation of multiphase flow requires solving of a large, sparse linear system resulting from the discretization of the PDEs modeling the flow. In the case of multiphase, multicomponent flow with miscible effect, this is a very challenging task. The problem becomes even more difficult if phase transitions are considered. A new approach to handle phase transitions is to formulate the system as a nonlinear complementarity problem. The disadvantage of this approach is that when one phase disappears, the resulting linear system has the structure of a saddle point problem and becomes indefinite; current AMG algorithms cannot be applied directly. In this study, we develop a new multilevel strategy based on multigrid reduction to deal with problems of this type. Since the multigrid reduction algorithm relies on the interplay between the F-relaxation and coarse grid correction, the idea is to devise an appropriate C-F splitting at each level of reduction. We demonstrate the effectiveness of our method with some numerical results for the case of two-phase, two-component flow with phase disappearance. We also show that the strategy extends naturally to handle phase transitions in the case of multiphase, multicomponent flow.

ABSTRACT. This talk describes a First-order System Least Squares (FOSLS) formulations of a Nonlinear Stokes flow model for glaciers and ice sheets. In Glen's law, the most commonly used constitutive equation for ice rheology, the ice viscosity becomes infinite as the velocity gradients (strain rates) approach zero, which typically occurs near the ice surface where deformation rates are low, or when the basal slip velocities are high. The computational difficulties associated with the infinite viscosity are often overcome by an arbitrary modification of Glen's law that bounds the maximum viscosity. The fluidity formulation exploits the fact that only the product of the viscosity and strain rate appears in the nonlinear Stokes problem, a quantity that, in fact, approaches zero as the strain rate goes to zero. A Nested Iteration (NI) Newton-FOSLS approach is used to solve the nonlinear Stokes problems, in which most of the iterations are performed on the coarsest grid. This method is compared with a standard Galerkin method using Taylor-Hood elements on a benchmark test problem involving basal sliding.

ABSTRACT. Numerical simulation has become one of the major topics in Computational Science. To promote modelling and simulation of complex problems new strategies are needed allowing for the solution of large, complex model systems. Crucial issues for such strategies are reliability, efficiency, robustness, usability, and versatility. After discussing the needs of large-scale simulation we point out basic simulation strategies such as adaptivity, parallelism and multigrid solvers. To allow adaptive, parallel computations the load balancing problem for dynamically changing grids has to be solved efficiently by fast heuristics. These strategies are combined in the simulation system UG (“Unstructured Grids”) being presented in the following. In the second part of the seminar we show the performance and efficiency of this strategy in various applications. In particular, large scale parallel computations of density-driven groundwater flow in heterogenous porous media are discussed in more detail. Load balancing and efficiency of parallel adaptive computations is discussed and the benefit of combining parallelism and adaptivity is shown.

ABSTRACT. We consider PDE constrained optimization problems where the partial differential equation has uncertain coefficients modeled by means of random variables or random fields. The focus is on tracking type objective functions. The goal of the optimization is to determine a robust optimum, i.e., an optimum that is satisfactory in a broad parameter range, and as insensitive as possible to parameter uncertainties. First, a general overview is given of different deterministic goal functions which achieve the above aim with a varying degree of robustness. The relevant goal functions are deterministic because of the expected value operators they contain. Since the stochastic space is often high dimensional, a multilevel (Quasi-) Monte Carlo method is presented to efficiently calculate the gradient and the Hessian. The convergence behavior for gradient and Hessian based optimization methods is illustrated for a model elliptic diffusion problem with lognormal diffusion coefficient. We demonstrate the efficiency of the algorithm, in particular for a large number of optimization variables and a large number of uncertainties.​

ABSTRACT. We consider the numerical simulation of physical phenomena governed by partial differential equations (PDEs) with uncertain input data in a multilevel Monte Carlo (MLMC) framework. Generating samples of random fields with prescribed statistical properties efficiently is an important component of MLMC methods. We present a highly scalable multilevel, hierarchical sampling technique that involves solving a mixed discretization of a stochastic partial differential equation. This allows us to leverage existing scalable methods for solving the resulting sparse linear systems arising from the mixed finite element discretization.

The proposed sampling technique is then used to generate different realizations of random fields to be used as input coefficient realizations within the MLMC method. An application to subsurface flow using the MLMC method with algebraically coarsened spaces with the proposed sampling technique will be presented to demonstrate the scalability of the method for large-scale simulations.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

ABSTRACT. In order to assess the predictive accuracy of numerically estimating the effective permeability of a subsurface flow, we consider the forward propagation of uncertain hydraulic conductivity, represented as a spatially correlated Gaussian random field, and the incorporation of observational data via Bayes' law. For large-scale problems of interest, the cost of implementing these two methods does not scale well, as sampling from the spatially correlated Gaussian random field requires an expensive eigenvalue solve, and estimating of the statistics of the effective permeability with respect to the posterior distribution may be intractable. In this work we consider scalable methods to approach these two issues. For the first, we use a hierarchical method to sample spatially correlated random Gaussian field for prior samples. For the second, we apply a ratio estimator, formed from Bayes' law and estimated via Monte Carlo and multilevel Monte Carlo, in order to estimate the effective permeability using samples from the prior distribution. In addition to cost and error analysis, we provide numerical results to illustrate the performance of this method.

ABSTRACT. The support vector machine is a flexible optimization-based technique widely used for classification problems. In practice, its training part becomes computationally expensive on large-scale data sets because of such reasons as the complexity and number of iterations in parameter fitting methods, underlying optimization solvers, and nonlinearity of kernels. We introduce a fast multilevel framework for solving support vector machine models that is inspired by the algebraic multigrid. Significant improvement in the running has been achieved without any loss in the quality. The proposed technique is highly beneficial on imbalanced sets. We demonstrate computational results on publicly available and industrial data sets.

ABSTRACT. Electromagnetic surveys are one of many techniques used for geophysical exploration for hydrocarbon reservoirs and ore deposits. While these techniques are commonplace in industry, current algorithms still fall short of taking full advantage of the data collected in real-world surveys. The focus of this talk is on improving algorithms for the forward simulation of electromagnetic data. This forward model is naturally expressed in terms of Maxwell's equations in the frequency domain, where the electrical field is then further decomposed into a vector potential and a solenoidal part. A suitable finite-element discretization uses Nedelec (edge) elements for the vector potential, and Lagrange (nodal) elements for the solenoidal part, leading to a complex-valued block-structured matrix system for the the two solution components. In this talk, we consider block-structured multigrid preconditioners for the equivalent real form of the discretization matrix. In particular, we propose a block-diagonal preconditioner, which treats the curl-curl parts of the system using the Auxiliary-space Maxwell Solver (Hiptmair-Xu) algorithm, and the solenoidal parts by applying algebraic multigrid to a suitable approximate Schur complement. We highlight the construction of the problem as well as the preconditioner and present results which demonstrate the efficacy of the proposed solution technique.

ABSTRACT. The shifted Laplacian multigrid is a well known approach for preconditioning the indefinite linear system arising from discretizing the acoustic Helmholtz equation. This equation is used to model wave propagation in the frequency domain. However, in some cases the acoustic equation is not sufficient for modeling the physics of the wave propagation, and one has to consider the elastic Helmholtz equation. Such a case arises in geophysical seismic imaging applications, where the earth subsurface is the elastic medium. In this talk we extend the shifted Laplacian approach to the elastic Helmholtz equation, by combining the complex shift idea with approaches for linear elasticity. We show numerical experiments for problems with heterogeneous media.

ABSTRACT. Partitioned algorithms enable effective simulation of physical processes comprising multiple mathematical models by using separate analysis codes for each constituent component. Such algorithms allow reuse of existing investments in simulation codes and can improve computational efficiency by running the computer programs for the different models in their ``sweet spot.'' In many cases, though, this entails solving the constituent models on non-overlapping domains that are meshed independently of each other. The discrete representations of these subdomains may have gaps and/or overlaps, which significantly complicates the formulation of partitioned methods. In this talk we present a partitioned explicit elastodynamics approach for spatially non-coincident interfaces, based on a generalized dual Schur complements. To obtain a Lagrange multiplier formulation that is fully compatible with explicit time integration we start with an alternative coupling condition, which enforces the continuity of the accelerations across the interface. The dual Schur complement of the resulting monolithic problem then yields an explicit system for the interface flux, which can be used to drive the independent solution of the subdomain problems for the next time increment. We then extend this approach to spatially non-coincident interfaces by introducing state extension operators that map interface accelerations to a virtual common refinement interface, and a virtual Lagrange multiplier that enforces continuity of the state extensions on that interface. Although the resulting generalized monolithic problem is not necessarily symmetric, it conserves global flux and remains compatible with explicit time integration. In particular, the corresponding generalized dual Schur complement yields an explicit equation for the virtual Lagrange multiplier, which provides the necessary boundary conditions for the independent solution of the subdomain problems. The talk will discuss effective multigrid preconditioning strategies for the generalized Schur complement problem, and present the results of numerical experiments demonstrating optimal convergence rates, global flux conservation, and the effectiveness of the proposed preconditioner.

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research and the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multi-mission laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

ABSTRACT. Multigrid methods are among the fastest methods for solving many numerical problems. However, their performance is problem-dependent, varying significantly for particular problems. In fact, multiphysics coupled problems are difficult to deal with fully implicit algorithms as multigrid methods. In particular, the main difficulty lies in the design of the smoother, which often represents the most important ingredient in a multigrid method.

Biot's model, which represents the interrelation between the deformation of a porous material and the fluid flow inside of it, is an example of a relevant multi-physics problem due to its wide range of applications. The monolithic solution of this problem by multigrid methods has been studied and different smoothers have been proposed, as coupled Vanka smoothers and decoupled distributive relaxations. Here we propose different segregated smoothers, including one relaxation procedure based on the physics of the problem, which give rise to very efficient multigrid algorithms.

ABSTRACT. For many inverse problems, regularization is a key step in ensuring fidelity of the recovered solution and overcoming noisy data or uncertain forward models. For imaging problems, in particular, classical regularization based on the L2 norm of the solution gradient is well-known to be a poor choice, as it fails to preserve natural edges in the recovered solution, and so minimization based on the L1 norm or total variation is generally preferred. In this talk, we consider solution of the sequence of linear systems that arise when such a regularized problem is solved using a reweighed least-squares approach to resolve the regularization term. Particular attention is paid to the selection of components of a multigrid preconditioner for different ranges of the regularization parameter value.

ABSTRACT. Expected duration 60-70 minutes.

XBraid is an open source, non-intrusive, and general purpose parallel- in-time code developed at Lawrence Livermore National Lab. The need for parallelism in time is being driven by changes in computer architectures, where future speedups will be available through greater concurrency, but not faster clock speeds, which are stagnant. This leads to a bottleneck for sequential time marching schemes, because they lack parallelism in the time dimension. To address this bottleneck, XBraid non-intrusively implements the multigrid reduction in time (MGRIT) algorithm, which allows for the addition of temporal parallelism to existing codes.

XBraid - Is a scalable, multilevel parallel-in-time method based on multigrid - Non-intrusively wraps existing sequential time integration codes - Allows for a variety of time stepping schemes - Supports a variety of space-time refinement approaches - Uses FAS multigrid to solve nonlinear problems - Is equivalent to parareal with certain two-level settings

The XBraid code - Is written in MPI/C with C++ and F90 interfaces - Is released under LGPL 2.1

The tutorial can simply be watched, or it can be interactive. If you want to run the examples, you'll need the following. Requirements: - XBraid 2.1 (or higher) For code and user's manual, see http://llnl.gov/casc/xbraid - GCC compiler Recommended: - MPI - Python 2.7 (or higher) with NumPy and Matplotlib