2024

M. Berzins.
**“COMPUTATIONAL ERROR ESTIMATION FOR THE MATERIAL POINT METHOD IN 1D AND 2D,”** In *VIII International Conference on Particle-Based Methods, PARTICLES 2023*, 2024.

The Material Point Method (MPM) is widely used for challenging applications in engineering, and animation but lags behind some other methods in terms of error analysis and computable error estimates. The complexity and nonlinearity of the equations solved by the method and its reliance both on a mesh and on moving particles makes error estimation challenging. Some preliminary error analysis of a simple MPM method has shown the global error to be first order in space and time for a widely-used variant of the Material Point Method. The overall time dependent nature of MPM also complicates matters as both space and time errors and their evolution must be considered thus leading to the use of explicit error transport equations. The preliminary use of an error estimator based on this transport approach has yielded promising results in the 1D case. One other source of error in MPM is the grid-crossing error that can be problematic for large deformations leading to large errors that are identified by the error estimator used. The extension of the error estimation approach to two space higher dimensions is considered and together with additional algorithmic and theoretical results, shown to give promising results in preliminary computational experiments.

J. K. Holmen , M. Garcıa, A. Bagusetty, V. Madananth, A. Sanderson,, M. Berzins.
**“Making Uintah Performance Portable for Department of Energy Exascale Testbeds,”** In *Euro-Par 2023: Parallel Processing*, pp. 1--12. 2024.

To help ease ports to forthcoming Department of Energy (DOE) exascale systems, testbeds have been made available to select users. These testbeds are helpful for preparing codes to run on the same hardware and similar software as in their respective exascale systems. This paper describes how the Uintah Computational Framework, an open-source asynchronous many-task (AMT) runtime system, has been modified to be performance portable across the DOE Crusher, DOE Polaris, and DOE Sunspot testbeds in preparation for portable simulations across the exascale DOE Frontier and DOE Aurora systems. The Crusher, Polaris, and Sunspot testbeds feature the AMD MI250X, NVIDIA A100, and Intel PVC GPUs, respectively. This performance portability has been made possible by extending Uintah’s intermediate portability layer [18] to additionally support the Kokkos::HIP, Kokkos::OpenMPTarget, and Kokkos::SYCL back-ends. This paper also describes notable updates to Uintah’s support for Kokkos, which were required to make this extension possible. Results are shown for a challenging radiative heat transfer calculation, central to the University of Utah’s predictive boiler simulations. These results demonstrate single-source portability across AMD-, NVIDIA-, and Intel-based GPUs using various Kokkos back-ends.

N. Shingde, T. Blattner, A. Bardakoff, W. Keyrouz, M. Berzins.
**“An illustration of extending Hedgehog to multi-node GPU architectures using GEMM,”** In *Springer Nature (to appear)*, 2024.

Asynchronous task-based systems offer the possibility of making it easier to take advantage of scalable heterogeneous architectures. This paper extends the previous work, demonstrating how Hedgehog, a dataflow graph-based model developed at the National Institute of Standards and Technology, can be used to obtain high performance for numerical linear algebraic operations as a starting point for complex algorithms. While the results were promising, it was unclear how to scale them to larger matrices and compute node counts. The aim here is to show how the new, improved algorithm inspired by DPLASMA performs equally well using Hedgehog. The results are compared against the leading library DPLASMA to illustrate the performance of different asynchronous dataflow models. The work demonstrates that using general-purpose, high-level abstractions, such as Hedgehog’s dataflow graphs, makes it possible to achieve similar performance to the specialized linear algebra codes such as DPLASMA.

2023

M. Berzins.
**“Error Estimation for the Material Point and Particle in Cell Methods,”** In *admos2023*, 2023.

The Material Point Method (MPM) is widely used for challenging applications in engineering, and animation. The complexity of the method makes error estimation challenging. Error analysis of a simple MPM method is undertaken and the global error is shown to be first order in space and time for a widely-used variant of the method. Computational experiments illustrate the estimated accuracy.

J. K. Holmen, V. G. Vergara Larrea, E. W. Draeger, E. T. Phipps, P. J. Smith, M. Berzins, S. T. Smith, J. N. Thornock, S. Parete-Koon.
**“Strengthening the US Department of Energy's Recruitment Pipeline: The DOE/NNSA Predictive Science Academic Alliance Program (PSAAP) Experience,”** In *Practice and Experience in Advanced Research Computing*, ACM, pp. 137--144. 2023.

The US Department of Energy (DOE) oversees a system of 17 national laboratories responsible for developing unique scientific capabilities beyond the scope of academic and industrial institutions. These labs strive to keep America at the forefront of discovery and are home to some of the Nation’s best minds and the world’s best scientific and research facilities. Collaborations between national laboratories and academic institutions are critical to develop and recruit talent for the DOE workforce. Academia’s cooperative education model poses challenges for DOE recruitment pipelines centered around traditional internships. This paper discusses a promising DOE recruitment pipeline, the National Nuclear Security Administration’s (NNSA) Predictive Science Academic Alliance Program (PSAAP) initiative. As a part of this, experiences capturing the successes and challenges faced by the University of Utah’s Carbon Capture Multidisciplinary Simulation Center (CCMSC) through their participation in the PSAAP-II initiative are shared. These experiences demonstrate the success of Utah’s PSAAP center as a recruitment pipeline with approximately 43% of CCMSC students going to a national laboratory after graduation. Potential opportunities to strengthen the DOE’s recruitment pipeline are also discussed.

T. A. J. Ouermi, R. M Kirby, M. Berzins.
**“HiPPIS A High-Order Positivity-Preserving Mapping Software for Structured Meshes,”** In *ACM Trans. Math. Softw*, ACM, Nov, 2023.

ISSN: 0098-3500

DOI: 10.1145/3632291

Polynomial interpolation is an important component of many computational problems. In several of these computational problems, failure to preserve positivity when using polynomials to approximate or map data values between meshes can lead to negative unphysical quantities. Currently, most polynomial-based methods for enforcing positivity are based on splines and polynomial rescaling. The spline-based approaches build interpolants that are positive over the intervals in which they are defined and may require solving a minimization problem and/or system of equations. The linear polynomial rescaling methods allow for high-degree polynomials but enforce positivity only at limited locations (e.g., quadrature nodes). This work introduces open-source software (HiPPIS) for high-order data-bounded interpolation (DBI) and positivity-preserving interpolation (PPI) that addresses the limitations of both the spline and polynomial rescaling methods. HiPPIS is suitable for approximating and mapping physical quantities such as mass, density, and concentration between meshes while preserving positivity. This work provides Fortran and Matlab implementations of the DBI and PPI methods, presents an analysis of the mapping error in the context of PDEs, and uses several 1D and 2D numerical examples to demonstrate the benefits and limitations of HiPPIS.

N. Shingde, M. Berzins, T. Blattner, W. Keyrouz, A. Bardakoff.
**“Extending Hedgehog’s dataflow graphs to multi-node GPU architectures,”** In *Workshop on Asynchronous Many-Task Systems and Applications (WAMTA23)*, 2023.

Asynchronous task-based systems offer the possibility of making it easier to take advantage of scalable heterogeneous architectures.

This paper extends the National Institute of Standards and Technology’s Hedgehog dataflow graph models, which target a single high-end

compute node, to run on a cluster by borrowing aspects of Uintah’s cluster-scale task graphs and applying them to a sample implementation

of matrix multiplication. These results are compared to implementations using the leading libraries, SLATE and DPLASMA, for illustrative purposes only. The motivation behind this work is to demonstrate that using general purpose high-level abstractions, such as Hedgehog’s dataflow graphs, does not negatively impact performance.

2022

M. Berzins.
**“Energy conservation and accuracy of some MPM formulations,”** In *Computational Particle Mechanics*, 2022.

DOI: 10.1007/s40571-021-00457-3

The success of the Material Point Method (MPM) in solving many challenging problems nevertheless raises some open questions regarding the fundamental properties of the method such as time integration accuracy and energy conservation. The traditional MPM time integration methods are often based upon the symplectic Euler method or staggered central differences. This raises the question of how to best ensure energy conservation in explicit time integration for MPM. Two approaches are used here, one is to extend the Symplectic Euler method (Cromer Euler) to provide better energy conservation and the second is to use a potentially more accurate symplectic methods, namely the widely-used Stormer-Verlet Method. The Stormer-Verlet method is shown to have locally third order time accuracy of energy conservation in time, in contrast to the second order accuracy in energy conservation of the symplectic Euler methods that are used in many MPM calculations. It is shown that there is an extension to the Symplectic Euler stress-last method that provides better energy conservation that is comparable with the Stormer-Verlet method. This extension is referred to as TRGIMP and also has third order accuracy in energy conservation. When the interactions between space and time errors are studied it is seen that spatial errors may dominate in computed quantities such as displacement and velocity. This connection between the local errors in space and time is made explicit mathematically and explains the observed results that displacement and velocity errors are very similar for both methods. The observed and theoretically predicted third-order energy conservation accuracy and computational costs are demonstrated on a standard MPM test example.

M. Berzins.
**“Computational Error Estimation for The Material Point Method,”** In *Computational Particle Mechanics*, Springer, 2022.

DOI: https://doi.org/10.1007/s40571-022-00530-5

A common feature of many methods in computational mechanics is that there is often a way of estimating the error in the computed solution. The situation for computational mechanics codes based upon the Material Point Method is very different in that there has been comparatively little work on computable error estimates for these methods. This work is concerned with introducing such an approach for the Material Point Method. Although it has been observed that spatial errors may dominate temporal ones at stable time steps, recent work has made more precise the sources and forms of the different MPM errors. There is then a need to estimate these errors computationally through computable estimates of the different errors in the material point method. Estimates of the different spatial errors in the Material Point Method are constructed based upon nodal derivatives of the different physical variables in MPM. These derivatives are then estimated using standard difference approximations calculated on the background mesh. The use of these estimates of the spatial error makes it possible to measure the growth of errors over time. A number of computational experiments are used to illustrate the performance of the computed error estimates. As the key feature of the approach is the calculation of derivatives on the regularly spaced background mesh, the extension to calculating derivatives and hence to error estimates for higher dimensional problems is clearly possible.

J.K. Holmen, D. Sahasrabudhe, M. Berzins.
**“Porting Uintah to Heterogeneous Systems,”** In *Proceedings of the Platform for Advanced Scientific Computing Conference (PASC22) Best Paper Award*, ACM, 2022.

The Uintah Computational Framework is being prepared to make portable use of forthcoming exascale systems, initially the DOE Aurora system through the Aurora Early Science Program. This paper describes the evolution of Uintah to be ready for such architectures. A key part of this preparation has been the adoption of the Kokkos performance portability layer in Uintah. The sheer size of the Uintah codebase has made it imperative to have a representative benchmark. The design of this benchmark and the use of Kokkos within it is discussed. This paper complements recent work with additional details and new scaling studies run 24x further than earlier studies. Results are shown for two benchmarks executing workloads representative of typical Uintah applications. These results demonstrate single-source portability across the DOE Summit and NSF Frontera systems with good strong-scaling characteristics. The challenge of extending this approach to anticipated exascale systems is also considered.

T.A.J. Ouermi, R.M. Kirby, M. Berzins.
**“ENO-Based High-Order Data-Bounded and Constrained Positivity-Preserving Interpolation,”** Subtitled **“https://arxiv.org/abs/2204.06168,”** In *Numerical Algorithms*, 2022.

A number of key scientific computing applications that are based upon tensor-product grid constructions, such as numerical weather prediction (NWP) and combustion simulations, require property-preserving interpolation. Essentially Non-Oscillatory (ENO) interpolation is a classic example of such interpolation schemes. In the aforementioned application areas, property preservation often manifests itself as a requirement for either data boundedness or positivity preservation. For example, in NWP, one may have to interpolate between the grid on which the dynamics is calculated to a grid on which the physics is calculated (and back). Interpolating density or other key physical quantities without accounting for property preservation may lead to negative values that are nonphysical and result in inaccurate representations and/or interpretations of the physical data. Property-preserving interpolation is straightforward when used in the context of low-order numerical simulation methods. High-order property-preserving interpolation is, however, nontrivial, especially in the case where the interpolation points are not equispaced. In this paper, we demonstrate that it is possible to construct high-order interpolation methods that ensure either data boundedness or constrained positivity preservation. A novel feature of the algorithm is that the positivity-preserving interpolant is constrained; that is, the amount by which it exceeds the data values may be strictly controlled. The algorithm we have developed comes with theoretical estimates that provide sufficient conditions for data boundedness and constrained positivity preservation. We demonstrate the application of our algorithm on a collection of 1D and 2D numerical examples, and show that in all cases property preservation is respected.

2021

M. Berzins.
**“Symplectic Time Integration Methods for the Material Point Method, Experiments, Analysis and Order Reduction,”** In *WCCM-ECCOMAS2020 virtual Conference*, Note: *Minor typographical correction in March 2024*, January, 2021.

The provision of appropriate time integration methods for the Material Point Method (MPM) involves considering stability, accuracy and energy conservation. A class of methods that addresses many of these issues are the widely-used symplectic time integration methods. Such methods have good conservation properties and have the potential to achieve high accuracy. In this work we build on the work in [5] and consider high order methods for the time integration of the Material Point Method. The results of practical experiments show that while high order methods in both space and time have good accuracy initially, unless the problem has relatively little particle movement then the accuracy of the methods for later time is closer to that of low order methods. A theoretical analysis explains these results as being similar to the stage error found in Runge Kutta methods, though in this case the stage error arises from the MPM differentiations and interpolations from particles to grid and back again, particularly in cases in which there are many grid crossings.

M. Berzins.
**“Time Stepping with Space and Time Errors and Stability of the Material Point Method,”** In *Proceedings of VII International Conference on Particle-Based Methods, PARTICLES 2021*, Edited by P. Wriggers, M. Bischoff, E. Onate, M. Bischoff, A. Duster & T. Zohdi, 2021.

The choice of the time step for the Material Point Method (MPM) is often addressed by using a simple stability criterion, such as the speed of sound or a CFL condition. Recently there have been several advances in understanding the stability of MPM. These range from non-linear stability analysis, through to Von Neumann type approaches. While in many instances this works well it is important to understand how this relates to the overall errors present in the method. Although it has been observed that spatial errors may dominate temporal ones at stable time steps, recent work has made more precise the sources and forms of the different MPM errors. This now makes it possible to understand how the different errors and the stability analysis are connected. At the same time this also requires simple computable estimates of the different errors in the material point method. The use of simple estimates of these errors makes it possible to connect some of the errors introduced with the stability criteria used. A number of simple computational experiments are used to illustrate the theoretical results.

A. Dubey, M. Berzins, C. Burstedde, M.l L. Norman, D. Unat, M. Wahib.
**“Structured Adaptive Mesh Refinement Adaptations to Retain Performance Portability With Increasing Heterogeneity,”** In *Computing in Science & Engineering*, Vol. 23, No. 5, pp. 62-66. 2021.

ISSN: 1521-9615

DOI: 10.1109/MCSE.2021.3099603

Adaptive mesh refinement (AMR) is an important method that enables many mesh-based applications to run at effectively higher resolution within limited computing resources by allowing high resolution only where really needed. This advantage comes at a cost, however: greater complexity in the mesh management machinery and challenges with load distribution. With the current trend of increasing heterogeneity in hardware architecture, AMR presents an orthogonal axis of complexity. The usual techniques, such as asynchronous communication and hierarchy management for parallelism and memory that are necessary to obtain reasonable performance are very challenging to reason about with AMR. Different groups working with AMR are bringing different approaches to this challenge. Here, we examine the design choices of several AMR codes and also the degree to which demands placed on them by their users influence these choices.

J. K. Holmen, D. Sahasrabudhe, M. Berzins.
**“A Heterogeneous MPI+PPL Task Scheduling Approach for Asynchronous Many-Task Runtime Systems,”** In *Proceedings of the Practice and Experience in Advanced Research Computing 2021 on Sustainability, Success and Impact (PEARC21)*, ACM, 2021.

Asynchronous many-task runtime systems and MPI+X hybrid parallelism approaches have shown promise for helping manage the increasing complexity of nodes in current and emerging high performance computing (HPC) systems, including those for exascale. The increasing architectural diversity, however, poses challenges for large legacy runtime systems emphasizing broad support for major HPC systems. Performance portability layers (PPL) have shown promise for helping manage this diversity. This paper describes a heterogeneous MPI+PPL task scheduling approach for combining these promising solutions with additional consideration for parallel third party libraries facing similar challenges to help prepare such a runtime for the diverse heterogeneous systems accompanying exascale computing. This approach is demonstrated using a heterogeneous MPI+Kokkos task scheduler and the accompanying portable abstractions [15] implemented in the Uintah Computational Framework, an asynchronous many-task runtime system, with additional consideration for hypre, a parallel third party library. Results are shown for two challenging problems executing workloads representative of typical Uintah applications. These results show performance improvements up to 4.4x when using this scheduler and the accompanying portable abstractions [15] to port a previously MPI-Only problem to Kokkos::OpenMP and Kokkos::CUDA to improve multi-socket, multi-device node use. Good strong-scaling to 1,024 NVIDIA V100 GPUs and 512 IBM POWER9 processor are also shown using MPI+Kokkos::OpenMP+Kokkos::CUDA at scale.

J. K. Holmen, D. Sahasrabudhe, M. Berzins, A. Bardakoff, T. J. Blattner, . Keyrouz.
**“Uintah+Hedgehog: Combining Parallelism Models for End-to-End Large-Scale Simulation Performance,”** *Scientific Computing and Imaging Institute*, 2021.

The complexity of heterogeneous nodes near and at exascale has increased the need for “heroic” programming efforts. To accommodate this complexity, significant investment is required for codes not yet optimizing for low-level architecture features (e.g., wide vector units) and/or running at large-scale. This paper describes ongoing efforts to combine two codes, Hedgehog and Uintah, lying at both extremes to ease programming efforts. The end goals of this effort are (1) to combine the two codes to make an asynchronous many-task runtime system specializing in both node-level and large-scale performance and (2) to further improve the accessibility of both with portable abstractions. A prototype adopting Hedgehog in Uintah and a prototype extending Hedgehog to support MPI+X hybrid parallelism are discussed. Results achieving ∼60% of NVIDIA V100 GPU peak performance for a distributed DGEMM problem are shown for a naive MPI+Hedgehog implementation before any attempt to optimize for performance.

Authors note: This is a refereed but unpublished report that was

submitted to, reviewed for and accepted in revised form for a presentation of the same material at the Hipar Workshop at Supercomputing 21

W. T. Sołowski, M. Berzins, W. Coombs, J. Guilkey, M. Möller, Q. A. Tran, T. Adibaskoro, S. Seyedan, R. Tielen, K. Soga.
**“Material point method: Overview and challenges ahead (with videos),”** In *Advances in Applied Mechanics*, 1, Vol. 14, Ch. 2, Elsevier, pp. 113-204. 2021.

ISBN: 978-0-323-88519-5

The paper gives an overview of Material Point Method and shows its evolution over the last 25 years. The Material Point Method developments followed a logical order. The article aims at identifying this order and show not only the current state of the art, but explain the drivers behind the developments and identify what is currently still missing.The paper explores modern implementations of both explicit and implicit Material Point Method. It concentrates mainly on uses of the method in engineering, but also gives a short overview of Material Point Method application in computer graphics and animation. Furthermore, the article gives overview of errors in the material point method algorithms, as well as identify gaps in knowledge, filling which would hopefully lead to a much more efficient and accurate Material Point Method. The paper also briefly discusses algorithms related to contact and boundaries, coupling the Material Point Method with other numerical methods and modeling of fractures. It also gives an overview of modeling of multi-phase continua with Material Point Method. The paper closes with numerical examples, aiming at showing the capabilities of Material Point Method in advanced simulations. Those include landslide modeling, multiphysics simulation of shaped charge explosion and simulations of granular material flow out of a silo undergoing changes from continuous to discontinuous and back to continuous behavior.The paper uniquely illustrates many of the developments not only with figures but also with videos, giving the whole extend of simulation instead of just a timestamped image

W. T. Sołowski, M. Berzins, W. Coombs, J. Guilkey, M. Möller, Q. A. Tran, T. Adibaskoro, S. Seyedan, R. Tielen, K. Soga.
**“Material point method: Overview and challenges ahead (without videos),”** In *Advances in Applied Mechanics*, 1, Vol. 14, Ch. 2, Elsevier, pp. 113-204. 2021.

The paper gives an overview of Material Point Method and shows its evolution over the last 25 years. The Material Point Method developments followed a logical order. The article aims at identifying this order and show not only the current state of the art, but explain the drivers behind the developments and identify what is currently still missing.The paper explores modern implementations of both explicit and implicit Material Point Method. It concentrates mainly on uses of the method in engineering, but also gives a short overview of Material Point Method application in computer graphics and animation. Furthermore, the article gives overview of errors in the material point method algorithms, as well as identify gaps in knowledge, filling which would hopefully lead to a much more efficient and accurate Material Point Method. The paper also briefly discusses algorithms related to contact and boundaries, coupling the Material Point Method with other numerical methods and modeling of fractures. It also gives an overview of modeling of multi-phase continua with Material Point Method. The paper closes with numerical examples, aiming at showing the capabilities of Material Point Method in advanced simulations. Those include landslide modeling, multiphysics simulation of shaped charge explosion and simulations of granular material flow out of a silo undergoing changes from continuous to discontinuous and back to continuous behavior.The paper uniquely illustrates many of the developments not only with figures but also with videos, giving the whole extend of simulation instead of just a timestamped image

R. Zambre, D. Sahasrabudhe, H. Zhou, M. Berzins, A. Chandramowlishwaran, P. Balaji.
**“Logically Parallel Communication for Fast MPI+Threads Communication,”** In *Proceedings of the Transactions on Parallel and Distributed Computing*, IEEE, April, 2021.

Supercomputing applications are increasingly adopting the MPI+threads programming model over the traditional “MPI everywhere” approach to better handle the disproportionate increase in the number of cores compared with other on-node resources. In practice, however, most applications observe a slower performance with MPI+threads primarily because of poor communication performance. Recent research efforts on MPI libraries address this bottleneck by mapping logically parallel communication, that is, operations that are not subject to MPI’s ordering constraints to the underlying network parallelism. Domain scientists, however, typically do not expose such communication independence information because the existing MPI-3.1 standard’s semantics can be limiting. Researchers had initially proposed user-visible endpoints to combat this issue, but such a solution requires intrusive changes to the standard (new APIs). The upcoming MPI-4.0 standard, on the other hand, allows applications to relax unneeded semantics and provides them with many opportunities to express logical communication parallelism. In this paper, we show how MPI+threads applications can achieve high performance with logically parallel communication. Through application case studies, we compare the capabilities of the new MPI-4.0 standard with those of the existing one and user-visible endpoints (upper bound). Logical communication parallelism can boost the overall performance of an application by over 2x.

2020

T. A. J. Ouermi, R. M. Kirby, M. Berzins.
**“Numerical Testing of a New Positivity-Preserving Interpolation Algorithm,”** Subtitled **“arXiv,”** 2020.

An important component of a number of computational modeling algorithms is an interpolation method that preserves the positivity of the function being interpolated. This report describes the numerical testing of a new positivity-preserving algorithm that is designed to be used when interpolating from a solution defined on one grid to different spatial grid. The motivating application is a numerical weather prediction (NWP) code that uses spectral elements as the discretization choice for its dynamics core and Cartesian product meshes for the evaluation of its physics routines. This combination of spectral elements, which use nonuniformly spaced quadrature/collocation points, and uniformly-spaced Cartesian meshes combined with the desire to maintain positivity when moving between these necessitates our work. This new approach is evaluated against several typical algorithms in use on a range of test problems in one or more space dimensions. The results obtained show that the new method is competitive in terms of observed accuracy while at the same time preserving the underlying positivity of the functions being interpolated.