Jonathan S. Graf

FEM Convergence Studies for 2-D and 3-D Elliptic PDEs with Smooth and Non-Smooth Source Terms in COMSOL 5.1

The hardware used in the computational studies is part of the UMBC High Performance Computing Facility (HPCF). The facility is supported by the U.S. National Science Foundation through the MRI program (grant nos. CNS{0821258 and CNS{1228778) and the SCREMS program (grant no. DMS{0821311), with additional substantial support from the University of Maryland, Baltimore County (UMBC). See hpcf.umbc.edu for more information on HPCF and the projects using its resources. Co-author Graf acknowledges financial support as HPCF RA.

Long-time simulations with complex code using multiple nodes of Intel Xeon Phi Knights Landing

Abstract Modern partial differential equation (PDE) models across scientific disciplines require sophisticated numerical methods resulting in complex codes as well as large numbers of simulations for analysis like parameter studies and uncertainty quantification. To evaluate the behavior of the model for sufficiently long times, for instance, to compare to laboratory time scales, often requires long-time simulations with small time steps and high mesh resolutions. This motivates the need for very efficient numerical methods and the use of parallel computing on the most recent modern architectures. We use complex code resulting from a PDE model of calcium dynamics in a heart cell to analyze the performance of the recently released Intel Xeon Phi Knights Landing (KNL). The KNL is a second-generation many-integrated-core (MIC) processor released in 2016 with a theoretical peak performance of over 3 TFLOP/s of double-precision floating-point operations for which complex codes can be easily ported because of the x86 compatibility of each KNL core. We demonstrate the benefit of hybrid MPI+OpenMP code when implemented effectively and run efficiently on the KNL including on multiple KNL nodes. For multi-KNL runs for our sample code, it is shown to be optimal to use all cores of each KNL, one MPI process on every other tile, and only two of the maximum of four threads per core.

Modeling the Links Between the Chemical, Electrical and Contractile Calcium Dynamics in a Heart Cell

These results were obtained as part of the REU Site: Interdisciplinary Program in High Performance Computing (hpcreu.umbc.edu) in the Department of Mathematics and Statistics at the University of Maryland, Baltimore County (UMBC), in Summer 2016. This program is funded by the National Science Foundation (NSF), the National Security Agency (NSA), and the Department of Defense (DOD), with additional support from UMBC, the Department of Mathematics and Statistics, the Center for Interdisciplinary Research and Consulting (CIRC), and the UMBC High Performance Computing Facility (HPCF). HPCF is supported by the U.S. National Science Foundation through the MRI program (grant nos. CNS{0821258 and CNS (1228778) and the SCREMS program (grant no. DMS{0821311), with additional substantial support from UMBC. Co-author Uchenna Osia was supported in part by the UMBC National Security Agency (NSA) Scholars Program through a contract with the NSA. Graduate assistant Jonathan Graf was supported by UMBC.

Applications of Tensor Decompositions

These results were obtained as part of the REU Site: Interdisciplinary Program in High Performance Computing (hpcreu.umbc.edu) in the Department of Mathematics and Statistics at the University of Maryland, Baltimore County (UMBC) in Summer 2016. This program is funded by the National Science Foundation (NSF), the National Security Agency (NSA), and the Department of Defense (DOD), with additional support from UMBC, the Department of Mathematics and Statistics, the Center for Interdisciplinary Research and Consulting (CIRC), and the UMBC High Performance Computing Facility (HPCF). HPCF is supported by the U.S. National Science Foundation through the MRI program (grant nos. CNS{0821258 and CNS{1228778) and the SCREMS program (grant no. DMS{0821311), with additional substantial support from UMBC. Co-author Darren Stevens II was supported, in part, by the UMBC National Security Agency (NSA) Scholars Program through a contract with the NSA. Graduate assistant Jonathan Graf was supported by UMBC.

Parallelization for Fast Image Reconstruction using the Stochastic Origin Ensemble Method for Proton Beam Therapy

Proton beam therapy is becoming increasingly common in the field of cancer treatment because of the advantages over other forms of radiation therapy. These advantages arise from the finite range of the proton beams, the relatively low dosage of radiation upon entering a patient, and the large spike in dose at the end of the beam range known as the Bragg peak. A new computer code has been developed that uses the stochastic origin ensemble method to reconstruct an image of the gamma radiation produced by the proton beam. The objective of this research is to significantly improve the run time of the given computer code. For the reconstruction algorithm to be useful in medicine, it must be fast and precise, since it is impractical to ask that a patient lie completely still for several minutes. The original C++ code using OpenMP multi-threading on CPUs was ported to hybrid CPU/GPU code using CUDA. It shows very good speedup on the GPU up to the maximum possible number of threads, achieving a 5x speedup over the serial CPU run.

Comparison of Performance Analysis Tools for Parallel Programs Applied to CombBLAS

Performance analysis tools are powerful tools for high performance computing. By breaking down a program into how long the CPUs are taking on each process (profiling) or showing when events take place on a timeline over the course of running a program (tracing), a performance analysis tool can tell the programmer exactly, where the computer is running slowly. With this information, the programmer can focus on these performance “hotspots,” and the code can be optimized to run faster. We compared the performance analysis tools TAU, ParaTools ThreadSpotter, Intel VTune, Scalasca, HPCToolkit, and Score-P to the example code CombBLAS (combinatorial BLAS) which is a C++ implemenation of the GraphBLAS, a set of graph algorithms using BLAS (Basic Linear Algebra Subroutines). Using these performance analysis tools on CombBLAS, we found three major “hotspots” and attempted to improve the code. We were unsuccessful in improving these “hotspots” due to a time limitation but still gave suggestions on improving the OpenMP calls in the CombBLAS code.

FEM Convergence for PDEs with Point Sources in 2-D and 3-D

Numerical theory provides the basis for quantification of the accuracy and reliability of a FEM solution by error estimates on the FEM error vs. the mesh spacing of the FEM mesh. This paper presents techniques needed in COMSOL 5.1 to perform computational studies for elliptic test problems in two and three space dimensions that demonstrate this theory by computing the convergence order of the FEM error. In particular, we show how to perform these techniques for a problem involving a point source modeled by a Dirac delta distribution as forcing term. This demonstrates that PDE problems with a non-smooth source term necessarily have degraded convergence order compared to problems with smooth righthand sides and thus can be most efficiently solved by low-order FEM such as linear Lagrange elements.