Parallel Algorithms for Successive Convolution

The development of modern computing architectures with ever-increasing amounts of parallelism has allowed for the solution of previously intractable problems across a variety of scientific disciplines. Despite these advances, multiscale computing problems continue to pose an incredible challenge to...

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Published in:	Journal of scientific computing Vol. 86; no. 1; p. 1
Main Authors:	Christlieb, Andrew J., Guthrey, Pierson T., Sands, William A., Thavappiragasm, Mathialakan
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.01.2021 Springer Nature B.V
Subjects:	Algorithms Approximation Boundary conditions Computation Computational Mathematics and Numerical Analysis Decomposition Distributed memory Domain decomposition methods Integral equations Iterative methods Linear systems Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Methods Operators (mathematics) Optimization Partial differential equations Theoretical Integral solution Numerical analysis High-performance computing Domain decomposition Method-of-lines-transpose Fast algorithms
ISSN:	0885-7474, 1573-7691
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Abstract	The development of modern computing architectures with ever-increasing amounts of parallelism has allowed for the solution of previously intractable problems across a variety of scientific disciplines. Despite these advances, multiscale computing problems continue to pose an incredible challenge to modern architectures because they require resolving scales that often vary by orders of magnitude in both space and time. Such complications have led us to consider alternative discretizations for partial differential equations (PDEs) which use expansions involving integral operators to approximate spatial derivatives (Christlieb et al. in J Comput Phys 379:214–236, 2019; Christlieb et al. J Sci Comput 82:52(3):1–29, 2020; Christlieb et al. J Comput Phys 415:1–25, 2020). These constructions use explicit information within the integral terms, but treat boundary data implicitly, which contributes to the overall speed of the method. This approach is provably unconditionally stable for linear problems and stability has been demonstrated experimentally for nonlinear problems. Additionally, it is matrix-free in the sense that it is not necessary to invert linear systems and iteration is not required for nonlinear terms. Moreover, the scheme employs a fast summation algorithm that yields a method with a computational complexity of O ( N ) , where N is the number of mesh points along a coordinate direction. While much work has been done to explore the theory behind these methods, their practicality in large scale computing environments is a largely unexplored topic. In this work, we explore the performance of these methods by developing a domain decomposition algorithm suitable for distributed memory systems along with shared memory algorithms. As a first pass, we derive an artificial Courant–Friedrichs–Lewy condition that enforces a nearest-neighbor (N-N) communication pattern and briefly discuss possible generalizations. We also analyze several approaches for implementing the parallel algorithms by optimizing predominant loop structures and maximizing data reuse. Using a hybrid design that employs MPI and Kokkos (Edwards and Trott in J Parallel Distrib Comput 74:3202–3216, 2014) for the distributed and shared memory components of the algorithms, respectively, we show that our methods are efficient and can sustain an update rate > 1 × 10 8 DOF/node/s. We provide results that demonstrate the scalability and versatility of our algorithms using several different PDE test problems, including a nonlinear example, which employs an adaptive time-stepping rule.
AbstractList	The development of modern computing architectures with ever-increasing amounts of parallelism has allowed for the solution of previously intractable problems across a variety of scientific disciplines. Despite these advances, multiscale computing problems continue to pose an incredible challenge to modern architectures because they require resolving scales that often vary by orders of magnitude in both space and time. Such complications have led us to consider alternative discretizations for partial differential equations (PDEs) which use expansions involving integral operators to approximate spatial derivatives (Christlieb et al. in J Comput Phys 379:214–236, 2019; Christlieb et al. J Sci Comput 82:52(3):1–29, 2020; Christlieb et al. J Comput Phys 415:1–25, 2020). These constructions use explicit information within the integral terms, but treat boundary data implicitly, which contributes to the overall speed of the method. This approach is provably unconditionally stable for linear problems and stability has been demonstrated experimentally for nonlinear problems. Additionally, it is matrix-free in the sense that it is not necessary to invert linear systems and iteration is not required for nonlinear terms. Moreover, the scheme employs a fast summation algorithm that yields a method with a computational complexity of O(N), where N is the number of mesh points along a coordinate direction. While much work has been done to explore the theory behind these methods, their practicality in large scale computing environments is a largely unexplored topic. In this work, we explore the performance of these methods by developing a domain decomposition algorithm suitable for distributed memory systems along with shared memory algorithms. As a first pass, we derive an artificial Courant–Friedrichs–Lewy condition that enforces a nearest-neighbor (N-N) communication pattern and briefly discuss possible generalizations. We also analyze several approaches for implementing the parallel algorithms by optimizing predominant loop structures and maximizing data reuse. Using a hybrid design that employs MPI and Kokkos (Edwards and Trott in J Parallel Distrib Comput 74:3202–3216, 2014) for the distributed and shared memory components of the algorithms, respectively, we show that our methods are efficient and can sustain an update rate >1×108 DOF/node/s. We provide results that demonstrate the scalability and versatility of our algorithms using several different PDE test problems, including a nonlinear example, which employs an adaptive time-stepping rule. The development of modern computing architectures with ever-increasing amounts of parallelism has allowed for the solution of previously intractable problems across a variety of scientific disciplines. Despite these advances, multiscale computing problems continue to pose an incredible challenge to modern architectures because they require resolving scales that often vary by orders of magnitude in both space and time. Such complications have led us to consider alternative discretizations for partial differential equations (PDEs) which use expansions involving integral operators to approximate spatial derivatives (Christlieb et al. in J Comput Phys 379:214–236, 2019; Christlieb et al. J Sci Comput 82:52(3):1–29, 2020; Christlieb et al. J Comput Phys 415:1–25, 2020). These constructions use explicit information within the integral terms, but treat boundary data implicitly, which contributes to the overall speed of the method. This approach is provably unconditionally stable for linear problems and stability has been demonstrated experimentally for nonlinear problems. Additionally, it is matrix-free in the sense that it is not necessary to invert linear systems and iteration is not required for nonlinear terms. Moreover, the scheme employs a fast summation algorithm that yields a method with a computational complexity of O ( N ) , where N is the number of mesh points along a coordinate direction. While much work has been done to explore the theory behind these methods, their practicality in large scale computing environments is a largely unexplored topic. In this work, we explore the performance of these methods by developing a domain decomposition algorithm suitable for distributed memory systems along with shared memory algorithms. As a first pass, we derive an artificial Courant–Friedrichs–Lewy condition that enforces a nearest-neighbor (N-N) communication pattern and briefly discuss possible generalizations. We also analyze several approaches for implementing the parallel algorithms by optimizing predominant loop structures and maximizing data reuse. Using a hybrid design that employs MPI and Kokkos (Edwards and Trott in J Parallel Distrib Comput 74:3202–3216, 2014) for the distributed and shared memory components of the algorithms, respectively, we show that our methods are efficient and can sustain an update rate > 1 × 10 8 DOF/node/s. We provide results that demonstrate the scalability and versatility of our algorithms using several different PDE test problems, including a nonlinear example, which employs an adaptive time-stepping rule.
ArticleNumber	1
Author	Sands, William A. Guthrey, Pierson T. Christlieb, Andrew J. Thavappiragasm, Mathialakan
Author_xml	– sequence: 1 givenname: Andrew J. surname: Christlieb fullname: Christlieb, Andrew J. organization: Department of Computational Mathematics, Science and Engineering, Michigan State University – sequence: 2 givenname: Pierson T. surname: Guthrey fullname: Guthrey, Pierson T. organization: Department of Computational Mathematics, Science and Engineering, Michigan State University, Lawrence Livermore National Laboratory – sequence: 3 givenname: William A. orcidid: 0000-0003-3643-4751 surname: Sands fullname: Sands, William A. email: sandswi3@msu.edu organization: Department of Computational Mathematics, Science and Engineering, Michigan State University – sequence: 4 givenname: Mathialakan surname: Thavappiragasm fullname: Thavappiragasm, Mathialakan organization: Department of Computational Mathematics, Science and Engineering, Michigan State University, Oak Ridge National Laboratory
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Cites_doi	10.1137/0103004 10.1017/jfm.2020.228 10.1016/j.jcp.2011.04.023 10.1016/j.jcp.2018.07.037 10.1016/j.jcp.2003.08.011 10.1137/15M1035094 10.1007/s007910050013 10.1137/0907058 10.1016/j.jcp.2009.11.020 10.1016/j.camwa.2011.02.024 10.1137/070679065 10.1137/0103003 10.1137/19M1251953 10.1002/(SICI)1098-2426(200001)16:1<30::AID-NUM3>3.0.CO;2-V 10.1016/j.jcp.2016.09.048 10.1016/j.jcp.2006.03.021 10.1016/j.jcp.2020.109543 10.1137/S003614450036757X 10.1137/130932685 10.1016/j.jcp.2018.11.037 10.1016/j.jpdc.2014.07.003 10.3847/0067-0049/225/2/22 10.1007/BF01386295 10.1137/16M1104123 10.1002/nme.1910 10.1090/S0025-5718-2014-02834-2 10.1007/s10915-016-0328-0 10.2172/1169830
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SubjectTerms	Algorithms Approximation Boundary conditions Computation Computational Mathematics and Numerical Analysis Decomposition Distributed memory Domain decomposition methods Integral equations Iterative methods Linear systems Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Methods Operators (mathematics) Optimization Partial differential equations Theoretical
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Title	Parallel Algorithms for Successive Convolution
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