A Fast Solver for the Fractional Helmholtz Equation
Abstract not provided.
Publications
A fast solver for the fractional helmholtz equation
SIAM Journal on Scientific Computing
The purpose of this paper is to study a Helmholtz problem with a spectral fractional Laplacian, instead of the standard Laplacian. Recently, it has been established that such a fractional Helmholtz problem better captures the underlying behavior in geophysical electromagnetics. We establish the well-posedness and regularity of this problem. We introduce a hybrid spectral-finite element approach to discretize it and show well-posedness of the discrete system. In addition, we derive a priori discretization error estimates. Finally, we introduce an efficient solver that scales as well as the best possible solver for the classical integer-order Helmholtz equation. We conclude with several illustrative examples that confirm our theoretical findings.
A FETI approach to domain decomposition for meshfree discretizations of nonlocal problems
Computer Methods in Applied Mechanics and Engineering
We propose a domain decomposition method for the efficient simulation of nonlocal problems. Our approach is based on a multi-domain formulation of a nonlocal diffusion problem where the subdomains share “nonlocal” interfaces of the size of the nonlocal horizon. This system of nonlocal equations is first rewritten in terms of minimization of a nonlocal energy, then discretized with a meshfree approximation and finally solved via a Lagrange multiplier approach in a way that resembles the finite element tearing and interconnect method. Specifically, we propose a distributed projected gradient algorithm for the solution of the Lagrange multiplier system, whose unknowns determine the nonlocal interface conditions between subdomains. Several two-dimensional numerical tests on problems as large as 191 million unknowns illustrate the strong and the weak scalability of our algorithm, which outperforms the standard approach to the distributed numerical solution of the problem. This work is the first rigorous numerical study in a two-dimensional multi-domain setting for nonlocal operators with finite horizon and, as such, it is a fundamental step towards increasing the use of nonlocal models in large scale simulations.
A Fractional Model for Anomalous Diffusion with Increased Variability
Abstract not provided.
A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems
Numerical Methods for Partial Differential Equations
Fractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. In this work we introduce a new fractional operator characterized by a doubly-variable fractional order and possibly truncated interactions. Under certain conditions on the model parameters and on the regularity of the fractional order we show that the corresponding Poisson problem is well-posed. We also introduce a finite element discretization and describe an efficient implementation of the finite-element matrix assembly in the case of piecewise constant fractional order. Through several numerical tests, we illustrate the improved descriptive power of this new operator across media interfaces. Furthermore, we present one-dimensional and two-dimensional h-convergence results that show that the variable-order model has the same convergence behavior as the constant-order model.
A Maxwell Solver for Next-Generation Platforms
Abstract not provided.
Adaptive FEM for the fractional Laplacian
Abstract not provided.
Adaptive FEM for the fractional Laplacian
Abstract not provided.
An optimization-based approach to parameter learning for fractional type nonlocal models
Computers and Mathematics with Applications
Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters. In this work we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method.
ASC ATDM Level 2 Milestone #6358: Assess Status of Next Generation Components and Physics Models in EMPIRE
This report documents the outcome from the ASC ATDM Level 2 Milestone 6358: Assess Status of Next Generation Components and Physics Models in EMPIRE. This Milestone is an assessment of the EMPIRE (ElectroMagnetic Plasma In Realistic Environments) application and three software components. The assessment focuses on the electromagnetic and electrostatic particle-in-cell solu- tions for EMPIRE and its associated solver, time integration, and checkpoint-restart components. This information provides a clear understanding of the current status of the EMPIRE application and will help to guide future work in FY19 in order to ready the application for the ASC ATDM L 1 Milestone in FY20. It is clear from this assessment that performance of the linear solver will have to be a focus in FY19.
Asynchronous Domain Decomposition Solvers
Abstract not provided.
Asynchronous Domain Decomposition Solvers
Abstract not provided.
Asynchronous Domain Decomposition Solvers
Abstract not provided.
Asynchronous Domain Decomposition Solvers
Abstract not provided.
Asynchronous Iterative Solvers for Extreme Scale
Abstract not provided.
Asynchronous Iterative Solvers: The ACHILES library and domain decomposition methods
Abstract not provided.
EMPIRE-PIC: A performance portable unstructured particle-in-cell code
Communications in Computational Physics
In this paper we introduce EMPIRE-PIC, a finite element method particle-in-cell (FEM-PIC) application developed at Sandia National Laboratories. The code has been developed in C++ using the Trilinos library and the Kokkos Performance Portability Framework to enable running on multiple modern compute architectures while only requiring maintenance of a single codebase. EMPIRE-PIC is capable of solving both electrostatic and electromagnetic problems in two- and three-dimensions to second-order accuracy in space and time. In this paper we validate the code against three benchmark problems - a simple electron orbit, an electrostatic Langmuir wave, and a transverse electromagnetic wave propagating through a plasma. We demonstrate the performance of EMPIRE-PIC on four different architectures: Intel Haswell CPUs, Intel's Xeon Phi Knights Landing, ARM Thunder-X2 CPUs, and NVIDIA Tesla V100 GPUs attached to IBM POWER9 processors. This analysis demonstrates scalability of the code up to more than two thousand GPUs, and greater than one hundred thousand CPUs.
Error estimates for the optimal control of a parabolic fractional pde
SIAM Journal on Numerical Analysis
We consider the integral definition of the fractional Laplacian and analyze a linearquadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. To discretize the state equation we propose a fully discrete scheme that relies on an implicit finite difference discretization in time combined with a piecewise linear finite element discretization in space. We derive stability results and a novel L2(0, T;L2(Ω)) a priori error estimate. On the basis of the aforementioned solution technique, we propose a fully discrete scheme for our optimal control problem that discretizes the control variable with piecewise constant functions, and we derive a priori error estimates for it. We illustrate the theory with one- and two-dimensional numerical experiments.
Experimental Evaluation of Multiprecision Strategies for GMRES on GPUs
2021 IEEE International Parallel and Distributed Processing Symposium Workshops, IPDPSW 2021 - In conjunction with IEEE IPDPS 2021
Support for lower precision computation is becoming more common in accelerator hardware due to lower power usage, reduced data movement and increased computational performance. However, computational science and engineering (CSE) problems require double precision accuracy in several domains. This conflict between hardware trends and application needs has resulted in a need for multiprecision strategies at the linear algebra algorithms level if we want to exploit the hardware to its full potential while meeting the accuracy requirements. In this paper, we focus on preconditioned sparse iterative linear solvers, a key kernel in several CSE applications. We present a study of multiprecision strategies for accelerating this kernel on GPUs. We seek the best methods for incorporating multiple precisions into the GMRES linear solver; these include iterative refinement and parallelizable preconditioners. Our work presents strategies to determine when multiprecision GMRES will be effective and to choose parameters for a multiprecision iterative refinement solver to achieve better performance. We use an implementation that is based on the Trilinos library and employs Kokkos Kernels for performance portability of linear algebra kernels. Performance results demonstrate the promise of multiprecision approaches and demonstrate even further improvements are possible by optimizing low-level kernels.
Experimental Evaluation of Multiprecision Strategies for GMRES on GPUs
Abstract not provided.
Hybrid Finite Element - Spectral Method for the Spectral Fractional Laplacian
Abstract not provided.
Hybrid finite element–spectral method for the fractional Laplacian: Approximation theory and efficient solver
SIAM Journal on Scientific Computing
A numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions. The scheme is based on reformulating the original problem posed over Ω on the extruded domain C = Ω × [0, ∞) following [L. Caffarelli and L. Silvestre, Comm. Partial Differential Equations, 32 (2007), pp. 1245–1260]. The resulting degenerate elliptic integer order PDE is then approximated using a hybrid FEM-spectral scheme. Finite elements are used in the direction parallel to the problem domain Ω, and an appropriate spectral method is used in the extruded direction. The spectral part of the scheme requires that we approximate the true eigenvalues of the integer order Laplacian over Ω. We derive an a priori error estimate which takes account of the error arising from using an approximation in place of the true eigenvalues. We further present a strategy for choosing approximations of the eigenvalues based on Weyl’s law and finite element discretizations of the eigenvalue problem. The system of linear algebraic equations arising from the hybrid FEM-spectral scheme is decomposed into blocks which can be solved effectively using standard iterative solvers such as multigrid and conjugate gradient. Numerical examples in two and three dimensions suggest that the approach is quasi-optimal in terms of complexity.
Krylov Solvers and Algebraic Multigrid
Abstract not provided.
Large Scale Parallel Solution Methods for Electromagnetic Simulations
Abstract not provided.
Linear Solvers for Plasma Simulations on Advanced Architectures
Abstract not provided.
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