ABSTRACT. An emerging theme across many domains of science and engineering is modeling materials that can change shape. In this talk, we focus on modeling the evolution of liquid crystals with free boundaries, known as tactoids, that are in contact with an isotropic fluid. We present promising results from applying a class of classical nonlinear numerical methods to this model and compare them with previously used gradient-descent methods. Moreover, by wrapping the algorithms in a multilevel nested iteration approach, we see significant improvements with a variety of initial guesses. These methods are implemented in Morpho, an easy-to-use, open-source, and domain-specific programmable environment. We show results for the shape optimization problem modeling the free boundary followed by results for the coupled nematic liquid crystal application.

ABSTRACT. In molecular dynamics, one is interested in sampling the invariant distribution of the system. However, in many molecular systems, there is a natural time-scale separation between the fast (microscopic) variables, and the slow (macroscopic) modes that determine the global conformation of the molecule.

Markov chain Monte Carlo (MCMC) has been designed for sampling from probability distributions given by an energy function. However, if the system has a time-scale separation, most MCMC methods remain stuck in one local minimum of the energy function, prohibiting a swift exploration of the state space.

In my talk, I will present a new micro-macro MCMC method (mM-MCMC) that can overcome the time-scale barrier. The mM-MCMC method first samples from a macroscopic distribution, before reconstructing a new microscopic instance that is consistent with the macroscopic value. I will explain the method, show its efficiency on two molecular examples, and go deeper into the free energy problem.

ABSTRACT. We discuss a multigrid algorithm for generalized magnetohydrodynamics (GMHD). This system has two different PDE expressions that can generate a large near null space. One expression contains the curl operator while the other is a Hall term arising from a generalized Ohms law. It is possible for either one or both terms to dominate within different sub-regions. Further, the near null space character is different depending on which term is responsible. For the curl operator, the null space corresponds to the space of gradients while for the Hall term it is a restriction of the magnetic field locally. The thrust of the talk is an adaptation of the Arnold-Falk-Winther relaxation method to GMHD problems so that it smooths oscillatory errors associated with both null spaces. We apply the resulting preconditioner to two test problems (a shock tube and a thin current sheet) to illustrate the effectiveness of the approach.

ABSTRACT. A stabilized finite element (FE) discretization of the single fluid low Mach number compressible resistive magnetohydrodynamics (MHD) equations is presented. The resulting method is intended to model macroscopic plasma instabilities and disruptions in complex 3D Tokamak devices used for exploring magnetic confinement fusion (MCF). The stabilized FE description is developed from a variational multiscale stabilization (VMS) approach to handle different shortcomings of the standard Galerkin FE discretization. The resulting VMS FE operators are used for stabilization of strongly convective flow effects, and for stabilizing the saddle point structure of the magnetic field elliptic divergence cleaning term, and the low Mach number nearly incompressible flow limit. The multiphysics block structure of the Newton linearized discrete system will be described along with the challenges it induces on scalable iterative solvers. The results of several numerical tests in evaluating the effectiveness of the VMS formulation will be presented. These will include computations of a vertical displacement event (VDE) and a few plasma instabilities in realistic ITER geometries.

* This work was partially supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics Program. It has also been partially supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research and Office of Fusion Energy Sciences, Scientific Discovery through Advanced Computing (SciDAC) program.

ABSTRACT. A base-level mathematical basis for the continuum fluid modeling of dissipative plasma system is the resistive magnetohydrodynamic model. This model requires the solution of the governing partial differential equations (PDEs) describing conservation of mass, momentum, and thermal energy, along with various reduced forms of Maxwell’s equations for the electromagnetic fields. The resulting systems are characterized by strong nonlinear and nonsymmetric coupling of fluid and electromagnetic phenomena, as well as the significant range of time- and length-scales that these interactions produce. These characteristics make scalable and efficient iterative solution, of the resulting poorly-conditioned discrete systems, extremely difficult. In this talk we consider the use of block preconditioners for solving the coupled physics block systems.

The block preconditioner considered here is based on an approximate operator splitting approach which can isolate certain coupled systems, allowing them to be handled independently. Here we use the splitting to create two independent 2x2 block systems, a magnetics-flow system and a magnetics-constraint system.

We demonstrate the scalability of this approach for various resistive MHD problems that are relevant to magnetic confinement fusion applications.

ABSTRACT. Reservoir models can now include upwards of billions of grid cells and are pushing the limits of computational resources. History matching is conducted to reduce the number of simulations runs and is one of the primary time-consuming tasks. As models get larger the number of parameters to match increases, and the number of objective functions increases, traditional methods start to reach their limitations. To solve this we propose the use of Bayesian optimization (BO). BO iteratively searches for an optimal solution in the simulations campaign through the refinement of a set of priors initialized with a set of simulation results. The current simulation platform implements grid management and a suite of linear solvers, including AMG which look to improve, to perform the simulation on large scale distributed-memory systems. Preliminary results are encouraging, and we propose integrating BO as a built-in module to efficiently iterate to find an optimal history match of production data in a single package platform.

ABSTRACT. A common challenge in regression is that for many problems, the degrees of freedom required for a high-quality solution also allows for overfitting. Regularization is a class of strategies that seek to restrict the range of possible solutions so as to discourage overfitting while still enabling good solutions, and different regularization strategies impose different types of restrictions. In this talk, we present a multilevel regularization strategy that constructs and trains a hierarchy of neural networks, each of which has layers that are wider versions of the previous network's layers. We draw intuition and techniques from the field of Algebraic Multigrid (AMG), traditionally used for solving linear and nonlinear systems of equations, and specifically adapt the Full Approximation Scheme (FAS) for nonlinear systems of equations to the problem of deep learning. The resulting approach is a highly flexible framework that can be applied to a variety of layer types, which we demonstrate with both fully-connected and convolutional layers. We experimentally show with PDE regression problems that our multilevel training approach is an effective regularizer, improving the generalization performance of the neural networks studied.

ABSTRACT. Convolutional neural networks (CNNs) have demonstrated excellent performance on semantic segmentation tasks where objects in images need to be classified pixel-wise. Typical architectures work in a multiresolution manner, similar to geometric multigrid. However, CNNs typically struggle to capture the exact shapes of thin or small objects. The reason for this is the inability to accurately represent such objects on aggressively coarse feature maps---the scales on which CNNs invest the most computational effort. A similar issue arises in the solution of PDEs with heterogenous coefficients with geometric multigrid methods. For this reason, algebraic multigrid methods were invented, which are the unstructured multigrid counterpart that can automatically capture fine geometries on coarse grids. Similarly, graph neural networks (GNNs) are the unstructured counterpart of CNNs. In this work, we propose to equip popular CNN architectures with unstructured graph-based convolution operators, which are equivalent to standard convolution operators under certain assumptions but operate on unstructured grids. In particular, by using algebraic-distance-based AMG coarsening inside the GNN we are able to better capture small objects on the coarse grids and eventually segment them better. We benchmark our approach on several semantic segmentation datasets and demonstrate improved performance over the CNN counterparts, especially for small objects.

ABSTRACT. We propose a Machine Learning approach to reduce the operator complexity of Algebraic Multigrid (AMG) methods for solving parametric partial differential equations (PDEs). Following the guidance of multigrid convergence theories, this method exploits the Attention mechanism to sparsify coarse-grid matrices without deteriorating the overall convergence. A key feature of the proposed method is the capability of generalizing to not only problems of larger sizes, but also of different parameters in the training set. We will provide numerical experiments on anisotropic rotated Laplacian and linear elasticity problems to illustrate the performance compared with traditional non-Galerkin methods.

ABSTRACT. Multigrid methods are among the most efficient techniques for solving sparse linear systems arising from Partial Differential Equations (PDEs). One of the key components of Multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. Finding optimal smoothing coefficients is problem-dependent and can impose challenges in many cases. In this work, we propose a machine learning approach based on local Fourier analysis (LFA) to construct optimal smoothing stencils for structured problems. We will show preliminary numerical results on PDE problems to demonstrate improved convergence rates compared with classical relaxation methods.

ABSTRACT. Our talk introduces a one-reduce GMRES algorithm based on Gauss-Seidel (MGS) and Jacobi (CGS) iterations, together with a new AMG smoother. The correction matrix $T = (I + L)^{-1}$ for the projector $P = I - QTQ^T$ is approximated using a rank-1 perturbation of the identity, resulting in a low backward error. These ideas are applied to the AMG preconditioner. Inspired by Eirola and Nevanlinna (1989), the V-cycle pre-smoother performs an L-1 Jacobi followed by a Gauss-Seidel sweep or the product $(I - \gamma D^-1uv^T)D^{-1}r_k$ where $u = L_{k,1:k-1}$ and $v = e_k$, with shifts. The post-smoother updates a vector $x_{k+1}$ with one Gauss-Seidel or Jacobi iteration. The proposed approach is an efficient algebraic multigrid smoother whose convergence can be analysed with Neumann proxies and Gershgorin circles. Results from ill-conditioned Navier-Stokes pressure solvers exhibit a 3X decrease in compute time on GPUs. This iterative refinement approach is most effective when the $\kappa(D+L)$ is large, and convergence is accelerated by shifts.

ABSTRACT. This is a particularly disruptive time for the development of future computing systems, and the next 10 years will see some very fundamental shifts in how systems are architected and deployed. Future high performance computing systems will be heterogenous. With the emergence of Heterogeneous Computing including AI and Quantum, we are seeing an explosive growth in computational techniques be supported on future computing systems and in computing solutions to support those techniques. In this talk we will present those opportunities and challenges for these heterogenous systems, and we’ll offer thoughts on the implications for future systems designs both at the hardware and software levels.

ABSTRACT. In the parallel Krylov iterative solvers of scientific and engineering applications by FEM and FDM, overlapping of halo communication and computation (CC-Overlapping) is widely used in combination with the dynamic loop scheduling capability of OpenMP. This method has been mainly applied to SpMV. In the previous work by the authors, we proposed a reordering method for applying CC-Overlapping to processes including data dependencies such as the ICCG, and obtained high parallel performance with massively parallel supercomputers. However, CC-Overlapping was only applied to SpMV. In the present work, we proposed a method to apply CC-Overlapping to forward-backward substitution, and verified it by parallel ICCG. Furthermore, we applied the proposed method to the MGCG method with parallel multi-grid preconditioning, and obtained a performance improvement of 8% to 18% using up to 1,024 nodes of Wisteria/BDEC-01 (Odyssey) at the University of Tokyo with A64FX processors.

ABSTRACT. We present an algebraic framework for operator splitting preconditioners for general sparse matrices. The framework leads to four different approaches: two with alternating splittings and two with a multiplicative ansatz. The ansatz generalizes ADI and ILU methods to multiple factors and more general factor form. The factors may be computed directly from the matrix coefficients or adaptively by incomplete sparse inversions. The special case of tridiagonal splittings is examined in more detail. We decompose the adjacency graph of the sparse matrix into multiple (almost) disjoint linear forests and each linear forest (union of disjoint paths) leads to a tridiagonal splitting. We obtain specialized variants of the four general approaches. Parallel implementations for all steps are provided on a GPU. We demonstrate the effectiveness and efficiency of these preconditioners combined with GMRES on various matrices.

ABSTRACT. In the European MICROCARD project we are developing a new cell-by-cell model for simulating the electrophysiology of the human heart in order to study effects that scarred tissue from ageing or infarcts has on the heartbeat. Since the fine resolution of this new model results in linear systems multiple orders of magnitude larger than current state-of-the-art models, we require a preconditioner that scales well and is tailored to the geometry of the simulation domain. We present some initial results we obtain with a new implementation of the Balancing Domain Decomposition by Constraints (BDDC) preconditioner in Ginkgo leveraging rank-local GPU-resident sparse direct solvers and distributed Krylov methods.

ABSTRACT. Block floating point (BFP) arithmetic arithmetic is currently seeing a resurgence in interest because it requires less power, less chip area, and is less complicated to implement in hardware than standard floating point arithmetic. This paper explores the application of BFP to mixed- and progressive-precision multigrid methods, enabling the solution of linear elliptic partial differential equations (PDEs) in energy- and hardware-efficient integer arithmetic. While most existing applications of BFP tend to use small block sizes, the block size here is chosen to be maximal such that matrices and vectors share a single exponent for all entries. We provide algorithms for BLAS-like routines for BFP arithmetic that ensure exact vector-vector and matrix-vector computations up to a specified precision, and using these algorithms, we experimentally study the asymptotic precision requirements to achieve discretization-error-accuracy.

ABSTRACT. In order to utilize modern exascale computers, Multigrid Reduction in Time (MGRIT) introduces parallelism to the time dimension by solving the initial value problem using multigrid. Although it is well known that the convergence of MGRIT depends on the choice of coarse-grid time-stepping operator, the derivation, in general, of a "good" coarse-grid operator remains an open problem. To address this, we introduce a general framework, called the $\theta$ method, for deriving accurate coarse operators in the family of Runge-Kutta methods. We motivate the problem by examining MGRIT convergence in the naive case, where the coarse-grid is a simple re-discretization of the fine-grid. We then derive order conditions for the coarse operator to match the fine-grid to a given accuracy. We derive several methods to demonstrate the technique, and demonstrate enhanced theoretical MGRIT convergence. Finally, we confirm the convergence bounds numerically on the linear advection-diffusion equation.