More details about each session

Session Synopses

Introduction to Numerical Software

In the first hour of our day with you we will give a brief introduction to our agenda and logistics and an overview of the various numerical packages you will learn about. In addition, we will do a hands-on exercise that amounts to the Hello, World! of numerical software, the one-dimensional heat equation, and use it to help answer a very simple science question.

Structured Discretization (with AMReX)

Slides

Block-structured adaptive mesh refinement (AMR) provides a natural framework in which to focus computing power on the most critical parts of the problem in the most computationally efficient way possible. AMReX supports the development of block-structured AMR algorithms for solving systems of partial differential equations (PDE’s) and other algorithms that require structured mesh and/or particle discretizations. We will begin with an overview of block-structured AMR, including several different time-stepping strategies, and then discuss the features of AMReX we might want to use to solve a multiphysics problem on machines from laptops to supercomputers. Hands-on exercises will include passive scalar advection with time-dependent adaptivity, the use of native linear solvers to impose incompressibility on a flow around obstacles, and “AMReX-Pachinko”, which demonstrates the interaction of particles with objects.

Unstructured Discretization (with MFEM/PUMI)

Slides

Unstructured meshes can yield required levels of accuracy using fewer degrees of freedom at the cost of more complex parallel data structures and algorithms. To support the ability of application code developers to take advantage of unstructured meshes, FASTMath develops core tools to support the development of unstructured mesh simulation capabilities. This lecture will first introduce the highly extendible MFEM high order finite element solver library and then overview the PUMI unstructured mesh tools developed to support mesh adaptation, load balancing and PIC calculations.

Nonlinear Solvers (with PETSc)

Slides

We will begin with a quick overview of iterative solvers for nonlinear systems, and then take a deeper look into Newton-Krylov methods and how to use them via the PETSc Scalable Nonlinear Equation Solvers (SNES) component. We will do some hands-on exploration with a classic computational fluid dynamics benchmark, the lid-driven cavity problem. We will end by looking at how nonlinear composition and preconditioning can be used to construct a wide array of nonlinear solvers from the algorithmic building blocks in SNES, and demonstrate how these techniques can handle particularly difficult nonlinearities.

Time Integration (with SUNDIALS)

Slides

In this lecture we will discuss the role and impact of high order, adaptive, and flexible time integration libraries in solution accuracy and computational efficiency of large-scale simulations. Due to the wide variety of backgrounds among ATPESC participants, we will briefly discuss

the location of time integrators in the HPC landscape, and their reliance on scalable nonlinear and linear solver libraries,
the different categories of time integration methods (explicit/implicit/IMEX),
the basic theoretical properties of time integration methods (order of accuracy, linear stability),
the role of temporal adaptivity for improving accuracy and efficiency,
an overview of DOE time integration packages.

We will spend approximately half of the time period in lecture, followed by hands-on exercises that examine stability, accuracy, temporal adaptivity, and the role of problem-specific preconditioning. All of the hands-on exercises focus on time-dependent PDEs, and use the SUNDIALS’ ARKODE library for time integration, along with the AMReX library for spatial semi-discretization.

Iterative Solvers & Preconditioners (with MueLu)

Slides

In this session, attendees will learn about linear solvers and preconditioners available in the Trilinos project. We will focus on Krylov solvers such as conjugate gradients (CG) and generalized minimum residual (GMRES); simple preconditioners like Jacobi, Gauss-Seidel, and Chebyshev polynomials; and scalable aggregation-based algebraic multigrid preconditioning. The two hands-on lessons will provide an opportunity to run a variety of stand-alone examples that demonstrate some of the many Trilinos solver capabilities on a model linear problem.

Iterative Solvers & Algebraic Multigrid (with HYPRE)

Slides

This session will present the basic concepts of iterative linear solvers with focus on Krylov solvers, including the generalized minimum residual method (GMRES), preconditioning and algebraic multigrid (AMG) methods. We will provide a brief description of the high performance linear solvers library HYPRE, its interfaces, and its most used multigrid solvers, BoomerAMG and PFMG, including a brief discussion of the effect of their data structures on performance. The lesson includes hands-on examples with structured and unstructured solvers from the HYPRE library applied to several test problems.

Optimization (with TAO)

Slides

This lecture will provide an introduction to numerical optimization with a theoretical focus on simulation-based problems. We will introduce the user interfaces for the Toolkit for Advanced Optimization (TAO) package within the PETSc library and exercise several gradient-based algorithms on a scalable synthetic test problem. We will observe and discuss the relative convergence of different classes of algorithms and sensitivity analysis methods in a parallel environment. Finally, we will also introduce constraints and study their effect on the problem solution.

Direct Solvers (with SuperLU/Strumpack)

Slides

In this lecture we discuss several aspects of direct solvers for large sparse linear systems. Sparse direct solvers, based on Gaussian elimination, are popular for a variety of applications mainly because of their numerical robustness and ease of use. Two software packages are introduced: SuperLU(_Dist) and STRUMPACK. We give a brief introduction to the underlying algorithms used in these solvers and illustrate their use.

In the second part of the lecture we introduce the concept of data-sparse matrix approximation, where a matrix is hierarchically partitioned in sub-blocks, which are compressed using low-rank approximations. Such a hierarchical matrix approximation allows for linear algebra operation that are much more efficient than the traditional O(n^3) dense linear algebra operations, such as LU decomposition or matrix-matrix multiplication, while also reducing memory usage. We then show how STRUMPACK uses such hierarchical matrix approximation to construct efficient preconditioners based on approximate sparse LU factorization.

Session 1 (10:30am-11:30am) includes a hands-on demo session with SuperLU.
Session 2 (11:45am-12:45pm) has a hands-on demo session with STRUMPACK.
Session 4 (3:40pm-4:30pm) includes a Q&A session (no hands-on).

Working with Numerical Packages in Practice

Slides

Developing high-quality, large-scale scientific computing applications in science and engineering involves expertise in many areas. Typically, no one person or group has all of the essential expertise and skills. Numerical software libraries and packages are a key way we share capability and know-how. Learning to leverage numerical packages to address new scientific computing challenges is part of becoming a member of the scientific computing community. In this wrap-up session of the day, we breifly discuss key tradeoffs in using numerical packages in practice.

Panel - Contributing to the Numerical Package Community

This will be a 45 minute panel question and answer period for ATPESC learners to ask questions about working with numerical package and the community of numerical package developers. If you have question(s) you know you would like to ask, we encourage attendees to submit questions ahead of time via the submission form. However, we expect there will also be ample opportunty to indicate your desire to ask questions via the main room slack channel, and then you may be called upon to un-mute and ask your question.

SME Speed Dating OPTIONAL ACTIVITY

This is an opportunity for you to meet and talk (1-on-1 or sometimes 2-on-1) with subject matter experts (SMEs) about their work and numerical packages they support. Each speed date will be a 20 minute Zoom meeting where you will be un-muted and able to have a conversation with an SME. Attendees may select up to 3 SMEs they would like to meet using this form Requests will be accomodated on a first-come, first-served basis until all available slots are taken. So, be sure to make your selections early in the day and no later than the end of the afternoon break, 3:40 PM CDT.