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DOE OSTI2020

PETSc Users Manual (Rev. 3.13)

This manual describes the use of PETSc for the numerical solution of partial differential equations and related problems on high-performance computers. The Portable, Extensible Toolkit for Scientific Computation (PETSc) is a suite of data structures and routines that provide the building blocks for the implementation of large-scale application codes on parallel (and serial) computers. PETSc uses the MPI standard for all message-passing communication. PETSc includes an expanding suite of parallel linear solvers, nonlinear solvers, and time integrators that may be used in application codes written in Fortran, C, C++, and Python. PETSc provides many of the mechanisms needed within parallel application codes, such as parallel matrix and vector assembly routines. The library is organized hierarchically, enabling users to employ the level of abstraction that is most appropriate for a particular problem. By using techniques of object-oriented programming, PETSc provides enormous flexibility for users. PETSc is a sophisticated set of software tools; as such, for some users it initially has a much steeper learning curve than a simple subroutine library. In particular, for individuals without some computer science background, experience programming in C, C++, python, or Fortran and experience using a debugger such as gdb or dbx, it may require a significant amount of time to take full advantage of the features that enable efficient software use. However, the power of the PETSc design and the algorithms it incorporates may make the efficient implementation of many application codes simpler than \rolling them" yourself.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

PETSc TSAdjoint: A Discrete Adjoint ODE Solver for First-Order and Second-Order Sensitivity Analysis

Here, we present a new software system PETSc TSAdjoint for first-order and second order adjoint sensitivity analysis of time-dependent nonlinear differential equations. The derivative calculation in PETSc TSAdjoint is essentially a high-level algorithmic differentiation process. The adjoint models are derived by differentiating the timestepping algorithms and implementing them based on the parallel infrastructure in PETSc. Full differentiation of the library code, including MPI routines, is avoided, and users do not need to derive their own adjoint models for their specific applications. PETSc TSAdjoint can compute the first-order derivative, that is, the gradient of a scalar functional, and the Hessian-vector product, which carries second-order derivative information, while requiring minimal input (a few callbacks) from the users. The adjoint model employs optimal checkpointing schemes in a manner that is transparent to users. Finally, usability, efficiency, and scalability are demonstrated through examples from a variety of applications.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2020

Extending Petsc's Composable Hierarchically Nested Linear Solvers

The Contributions from the RELACS group at both Rice University and the University at Buffalo in this phase of the PETSc Composable Solvers effort have centered around four main areas: scalable mesh processing, mesh adaptivity, solvers for subsurface flow, and performance modeling. The prominence of mesh processing demonstrates the tight relationship between meshing and discretization on the one hand, and optimal solvers on the other. All optimal solvers that we consider depend on some notion of hierarchy, and we express this using the DMPlex abstraction in PETSc. This relationship demands tight integration between the DM and SNES/TS components in PETSc that is the foundation of much of this work. In addition, interpretation of performance results for scalable solvers necessitates that information from the discretization and solver enter the performance model. Without this, comparing different solvers can be a fruitless exercise. Some major accomplishment of the past three years in these areas include: scalable mesh loading in PETSc on more than 10K cores, integrated mesh adaptivity using both p4est and Pragmatic, scalable multigrid for DG discretizations of subsurface flow, and predictive performance modeling incorporating error estimates.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Toward performance-portable PETSc for GPU-based exascale systems

The Portable Extensible Toolkit for Scientific computation (PETSc) library delivers scalable solvers for nonlinear time-dependent differential and algebraic equations and for numerical optimization. The PETSc design for performance portability addresses fundamental GPU accelerator challenges and stresses flexibility and extensibility by separating the programming model used by the application from that used by the library, and it enables application developers to use their preferred programming model, such as Kokkos, RAJA, SYCL, HIP, CUDA, or OpenCL, on upcoming exascale systems. Furthermore, a blueprint for using GPUs from PETSc-based codes is provided, and case studies emphasize the flexibility and high performance achieved on current GPU-based systems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

PETSc Users Manual (Revision 3.15)

This manual describes the use of the Portable, Extensible Toolkit for Scientific Computation (PETSc) for the numerical solution of partial differential equations and related problems on high-performance computers. PETSc is a suite of data structures and routines that provide the building blocks for the implementation of large-scale application codes on parallel (and serial) computers. PETSc uses the MPI standard for all distributed memory communication.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1997

A Comparison of PETSC Library and HPF Implementations of an Archetypal PDE Computation

Two paradigms for distributed-memory parallel computation that free the application programmer from the details of message passing are compared for an archetypal structured scientific computation a nonlinear, structured-grid partial differential equation boundary value problem using the same algorithm on the same hardware. Both paradigms, parallel libraries represented by Argonne's PETSC, and parallel languages represented by the Portland Group's HPF, are found to be easy to use for this problem class, and both are reasonably effective in exploiting concurrency after a short learning curve. The level of involvement required by the application programmer under either paradigm includes specification of the data partitioning (corresponding to a geometrically simple decomposition of the domain of the PDE). Programming in SPAM style for the PETSC library requires writing the routines that discretize the PDE and its Jacobian, managing subdomain-to-processor mappings (affine global- to-local index mappings), and interfacing to library solver routines. Programming for HPF requires a complete sequential implementation of the same algorithm, introducing concurrency through subdomain blocking (an effort similar to the index mapping), and modest experimentation with rewriting loops to elucidate to the compiler the latent concurrency. Correctness and scalability are cross-validated on up to 32 nodes of an IBM SP2.

Hayder, M. Ehtesham↗

DOE OSTI2020

Evaluation of PETSc on a Heterogeneous Architecture, the OLCF Summit System: Part 1: Vector Node Performance

Our goal is to report on the basic performance of PETSc vector operations on a single node of the Oak Ridge Leadership Facility System Summit. We describe the Summit system and present data collected from several vector operations. Limited analysis is also presented.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Evaluation of PETSc on a Heterogeneous Architecture, the OLCF Summit System: Part II - Basic Communication Performance

Nearest-neighbor communication is at the heart of many high-performance parallel computations. We report on the performance of such communication on the Oak Ridge Leadership Computing Facility system Summit in the context of the PETSc communication module. The analysis in this report includes basic Ping-Pong point-to-point communication and regular and irregular nearest-neighbor communication.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Extending PETSc's Composable Hierarchical Solvers (Final Technical Report)

This report documents research activities conducted at CU Boulder as part of Extending PETSc’s Composable Hierarchical Solvers, which has been part of a collaboration with Argonne National Laboratory (separate award). Our work has focused on performance-portable end-to-end GPU solvers demonstrated via exemplary applications in nonlinear fluid and structural mechanics. We describe advances in algorithmic composition and analysis in the context of these applications, but the implementations are fully documented and decoupled, and in use by other projects. We believe the vertical integration achieved through collaboration with ECP’s CEED and the PSAAP center at CU was necessary to take risks with data structures and algorithms.

42 ENGINEERING↗

DOE OSTI2022

Extending PETSc’s Composable, Hierarchical, Nested Solvers (Final Report)

For this project, I have focused mainly on developing discretization tech nology in PETSc in order to allow us to support optimal solvers for com plex, multiphysics problems, and also outer-loop problems, such as PDE constrained optimization. There have been improvements to the unstruc tured mesh support in DMPlex and particle discretizations in DMSwarm. In addition, we have produced a number of physical examples, tutorials, and tools for understanding performance.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

PETSc DMNetwork: A Library for Scalable Network PDE-Based Multiphysics Simulations

We present DMNetwork, a high-level package included in the PETSc library for the simulation of multiphysics phenomena over large-scale networked systems. The library aims at applications that have networked structures such as those in electrical, gas, and water distribution systems. DMNetwork provides data and topology management, parallelization for multiphysics systems over a network, and hierarchical and composable solvers to exploit the problem structure. DMNetwork eases the simulation development cycle by providing the necessary infrastructure through simple abstractions to define and query the network components. This article presents the design of DMNetwork, illustrates its user interface, and demonstrates its ability to solve multiphysics systems, such as an electric circuit, a network of power grid and water subnetworks, and transient hydraulic systems over large networks with more than 2 billion variables on extreme-scale computers using up to 30,000 processors.

extreme-scale computing↗

DOE OSTI2026

Fast solvers for tokamak fluid models with PETSc

Multigrid (MG) is widely recognized as a highly effective solver for the model problem, the Laplacian, but textbook MG fails on most problems of interest. MG methods have been applied to complex, real-world applications with careful consideration of the physical model and discretization. In this work we develop the first step in applying MG methods to science and engineering relevant magnetohydrodynamics (MHD) tokamak models in the M3D-C1 (https://m3dc1.pppl.gov) fusion energy science code. The semi-implicit time integrator in M3D-C1 is composed of many linear solves. The implicit advance of the momentum equation is the most challenging and is the focus of this work. The current production solver in M3D-C1 is a block Jacobi (BJ) preconditioner within a Krylov solver, where blocks group degrees of freedom on planes of constant toroidal coordinate. BJ convergence degrades as the number of planes increases due to the spectral properties of the matrix preconditioned with BJ. The partially magnetic field-aligned, regular toroidal grid structure in M3D-C1 is amenable to semi-coarsening geometric MG in the toroidal direction. This paper develops such a solver and demonstrates competitive performance on a runaway electron model of a SPARC (https://cfs.energy/technology/sparc) disruption, and superior robustness on a stellarator model on which the BJ solver fails to converge.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2022

The PETSc Community as Infrastructure

Not Available

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

PETSc/TAO Users Manual (Rev. 3.19)

This manual describes the use of the Portable, Extensible Toolkit for Scientific Computation (PETSc) and the Toolkit for Advanced Optimization (TAO) for the numerical solution of partial differential equations and related problems on high-performance computers. PETSc/TAO is a suite of data structures and routines that provide the building blocks for the implementation of large-scale application codes on parallel (and serial) computers. PETSc uses the MPI standard for all distributed memory communication. PETSc/TAO includes a large suite of parallel linear solvers, nonlinear solvers, time integrators, and opti mization that may be used in application codes written in Fortran, C, C++, and Python (via petsc4py; see Getting Started). PETSc provides many of the mechanisms needed within parallel application codes, such as parallel matrix and vector assembly routines. The library is organized hierarchically, enabling users to employ the level of abstraction that is most appropriate for a particular problem. By using techniques of object-oriented programming, PETSc provides enormous flexibility for users. PETSc is a sophisticated set of software tools; as such, for some users it initially has a much steeper learning curve than packages such as MATLAB or a simple subroutine library. In particular, for individuals without some computer science background, experience programming in C, C++, python, or Fortran and experience using a debugger such as gdb or lldb, it may require a significant amount of time to take full advantage of the features that enable efficient software use. However, the power of the PETSc design and the algorithms it incorporates may make the efficient implementation of many application codes simpler than “rolling them” yourself. For many tasks a package such as MATLAB is often the best tool; PETSc is not intended for the classes of problems for which effective MATLAB code can be written. There are several packages, built on PETSc, that may satisfy your needs without requiring directly using PETSc. We recommend reviewing these packages functionality before starting to code directly with PETSc. PETSc can be used to provide a “MPI parallel linear solver” in an otherwise sequential, or OpenMP parallel code. This approach cannot provide extremely large improvements in the application time by utilizing large numbers of MPI processes but can still improve the performance. Certainly all parts of a previously sequential code need not be parallelized but the matrix generation portion must be parallelized to expect true scalability to large numbers of MPI processes. See PCMPI for details on how to utilize the PETSc MPI linear solver server. Since PETSc is under continued development, small changes in usage and calling sequences of routines will occur. PETSc has been supported for twenty-five years; see mailing list information on our website for information on contacting support.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

PETSc/TAO developments for GPU-based early exascale systems

The Portable Extensible Toolkit for Scientific Computation (PETSc) library provides scalable solvers for nonlinear time-dependent differential and algebraic equations and for numerical optimization via the Toolkit for Advanced Optimization (TAO). PETSc is used in dozens of scientific fields and is an important building block for many simulation codes. During the U.S. Department of Energy’s Exascale Computing Project, the PETSc team has made substantial efforts to enable efficient utilization of the massive fine-grain parallelism present within exascale compute nodes and to enable performance portability across exascale architectures. We recap some of the challenges that designers of numerical libraries face in such an endeavor, and then discuss the many developments we have made, which include the addition of new GPU backends, features supporting efficient on-device matrix assembly, better support for asynchronicity and GPU kernel concurrency, and new communication infrastructure. In conclusion, we evaluate the performance of these developments on some pre-exascale systems as well as the early exascale systems Frontier and Aurora, using compute kernel, communication layer, solver, and mini-application benchmark studies, and then close with a few observations drawn from our experiences on the tension between portable performance and other goals of numerical libraries.

Exascale Computing Project (ECP)↗

DOE OSTI2024

PETSc/TAO Users Manual V.3.21

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1998

Parallel Implicit Algorithms for CFD

The main goal of this project was efficient distributed parallel and workstation cluster implementations of Newton-Krylov-Schwarz (NKS) solvers for implicit Computational Fluid Dynamics (CFD.) "Newton" refers to a quadratically convergent nonlinear iteration using gradient information based on the true residual, "Krylov" to an inner linear iteration that accesses the Jacobian matrix only through highly parallelizable sparse matrix-vector products, and "Schwarz" to a domain decomposition form of preconditioning the inner Krylov iterations with primarily neighbor-only exchange of data between the processors. Prior experience has established that Newton-Krylov methods are competitive solvers in the CFD context and that Krylov-Schwarz methods port well to distributed memory computers. The combination of the techniques into Newton-Krylov-Schwarz was implemented on 2D and 3D unstructured Euler codes on the parallel testbeds that used to be at LaRC and on several other parallel computers operated by other agencies or made available by the vendors. Early implementations were made directly in Massively Parallel Integration (MPI) with parallel solvers we adapted from legacy NASA codes and enhanced for full NKS functionality. Later implementations were made in the framework of the PETSC library from Argonne National Laboratory, which now includes pseudo-transient continuation Newton-Krylov-Schwarz solver capability (as a result of demands we made upon PETSC during our early porting experiences). A secondary project pursued with funding from this contract was parallel implicit solvers in acoustics, specifically in the Helmholtz formulation. A 2D acoustic inverse problem has been solved in parallel within the PETSC framework.

Keyes, David E.↗

DOE OSTI2021

The PetscSF Scalable Communication Layer

PetscSF, the communication component of the Portable, Extensible Toolkit for Scientific Computation (PETSc), is designed to provide PETSc's communication infrastructure suitable for exascale computers that utilize GPUs and other accelerators. PetscSF provides a simple application programming interface (API) for managing common communication patterns in scientific computations by using a star-forest graph representation. PetscSF supports several implementations based on MPI and NVSHMEM, whose selection is based on the characteristics of the application or the target architecture. An efficient and portable model for network and intra-node communication is essential for implementing large-scale applications. The Message Passing Interface, which has been the de facto standard for distributed memory systems, has developed into a large complex API that does not yet provide high performance on the emerging heterogeneous CPU-GPU-based exascale systems. Here, we discuss the design of PetscSF, how it can overcome some difficulties of working directly with MPI on GPUs, and we demonstrate its performance, scalability, and novel features.

97 MATHEMATICS AND COMPUTING↗