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We study an element agglomeration coarsening strategy that requires data redistribution at coarse levels when the number of coarse elements becomes smaller than the used computational units (cores). The overall procedure generates coarse elements (general unstructured unions of fine grid elements) within the framework of element-based algebraic multigrid methods (or AMGe) studied previously. The AMGe generated coarse spaces have the ability to exhibit approximation properties of the same order as the fine-level ones since by construction they contain the piecewise polynomials of the same order as the fine level ones. These approximation properties are key for the successful use of AMGe in multilevel solvers for nonlinear partial differential equations as well as for multilevel Monte Carlo (MLMC) simulations. The ability to coarsen without being constrained by the number of available cores, as described in the present paper, allows to improve the scalability of these solvers as well as in the overall MLMC method. The paper illustrates this latter fact with detailed scalability study of MLMC simulations applied to model Darcy equations with a stochastic log-normal permeability field.

Molecular dynamics (MD) simulations have transformed our understanding of the nanoscale, driving breakthroughs in materials science, computational chemistry, and several other fields, including biophysics and drug design. Even on exascale supercomputers, however, runtimes are excessive for systems and timescales of scientific interest. Here, we demonstrate strong scaling of MD simulations on the Cerebras Wafer-Scale Engine. By dedicating a processor core for each simulated atom, we demonstrate a 179-fold improvement in timesteps per second versus the Frontier GPU-based Exascale platform, along with a large improvement in timesteps per unit energy. Reducing every year of runtime to two days unlocks currently inaccessible timescales of slow microstructure transformation processes that are critical for understanding material behavior and function. Our dataflow algorithm runs Embedded Atom Method (EAM) simulations at rates over 270,000 timesteps per second for problems with up to 800k atoms. This demonstrated performance is unprecedented for general-purpose processing cores., Comment: 10 pages, 10 figures, 5 tables

This paper presents the use of element-based algebraic multigrid (AMGe) hierarchies, implemented in the ParELAG (Parallel Element Agglomeration Algebraic Multigrid Upscaling and Solvers) library, to produce multilevel preconditioners and solvers for H(curl) and H(div) formulations. ParELAG constructs hierarchies of compatible nested spaces, forming an exact de Rham sequence on each level. This allows the application of hybrid smoothers on all levels and AMS (Auxiliary-space Maxwell Solver) or ADS (Auxiliary-space Divergence Solver) on the coarsest levels, obtaining complete multigrid cycles. Numerical results are presented, showing the parallel performance of the proposed methods. As a part of the exposition, this paper demonstrates some of the capabilities of ParELAG and outlines some of the components and procedures within the library.

In this paper, a few dual least-squares finite element methods and their application to scalar linear hyperbolic problems are studied. The purpose is to obtain $L^2$-norm approximations on finite element spaces of the exact solutions to hyperbolic partial differential equations of interest. This is approached by approximating the generally infeasible quadratic minimization, that defines the $L^2$-orthogonal projection of the exact solution, by feasible least-squares principles using the ideas of the original $\mathcal{L}\mathcal{L}^*$ method proposed in the context of elliptic equations. All methods in this paper are founded upon and extend the $\mathcal{L}\mathcal{L}^*$ approach which is rather general and applicable beyond the setting of elliptic problems. Error bounds are shown that point to the factors affecting the convergence and provide conditions that guarantee optimal rates. Furthermore, the preconditioning of the resulting linear systems is discussed. Numerical results are provided to illustrate the behavior of the methods on common finite element spaces.

This paper presents and studies an approach for constructing auxiliary space preconditioners for finite element problems using a constrained nonconforming reformulation, that is based on a proposed modified version of the mortar method. The well-known mortar finite element discretization method is modified to admit a local structure, providing an element-by-element or subdomain-by-subdomain assembly property. This is achieved via the introduction of additional trace finite element spaces and degrees of freedom (unknowns) associated with the interfaces between adjacent elements or subdomains. The resulting nonconforming formulation and a reduced via static condensation Schur complement form on the interfaces are used in the construction of auxiliary space preconditioners for a given conforming finite element discretization problem. The properties of these preconditioners are studied and their performance is illustrated on model second order scalar elliptic problems utilizing high order elements., Comment: arXiv admin note: text overlap with arXiv:1910.06986

In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax-Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when $L^2$ convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an $H^{-1}$-type norm, but not $L^2$-coercive. Nevertheless, the $L^2$ convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the $L^2$ norm, or by properly choosing the finite element spaces, providing implicit control of the $L^2$ norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the $L^2$ convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss-Newton technique with nested iteration. We believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).

We modify the well-known interior penalty finite element discretization method so that it allows for element-by-element assembly. This is possible due to the introduction of additional unknowns associated with the interfaces between neighboring elements. The resulting bilinear form, and a Schur complement (reduced) version of it, are utilized in a number of auxiliary space preconditioners for the original conforming finite element discretization problem. These preconditioners are analyzed on the fine scale and their performance is illustrated on model second order scalar elliptic problems discretized with high order elements.