Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters. In this work we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method.
Fractional equations have become the model of choice in several applications where heterogeneities at the microstructure result in anomalous diffusive behavior at the macroscale. Here, we introduce a new fractional operator characterized by a doubly-variable fractional order and possibly truncated interactions. Under certain conditions on the model parameters and on the regularity of the fractional order we show that the corresponding Poisson problem is well-posed. Additionally, we introduce a finite element discretization and describe an efficient implementation of the finite-element matrix assembly in the case of piecewise constant fractional order. Through several numerical tests, we illustrate the improved descriptive power of this new operator across media interfaces. Furthermore, we present one-dimensional and two-dimensional h-convergence results that show that the variable-order model has the same convergence behavior as the constant-order model.
Anomalous behavior is ubiquitous in subsurface solute transport due to the presence of high degrees of heterogeneity at different scales in the media. Although fractional models have been extensively used to describe the anomalous transport in various subsurface applications, their application is hindered by computational challenges. Simpler nonlocal models characterized by integrable kernels and finite interaction length represent a computationally feasible alternative to fractional models; yet, the informed choice of their kernel functions still remains an open problem. We propose a general data-driven framework for the discovery of optimal kernels on the basis of very small and sparse data sets in the context of anomalous subsurface transport. Using spatially sparse breakthrough curves recovered from fine-scale particle-density simulations, we learn the best coarse-scale nonlocal model using a nonlocal operator regression technique. Predictions of the breakthrough curves obtained using the optimal nonlocal model show good agreement with fine-scale simulation results even at locations and time intervals different from the ones used to train the kernel, confirming the excellent generalization properties of the proposed algorithm. A comparison with trained classical models and with black-box deep neural networks confirms the superiority of the predictive capability of the proposed model.
Proceedings of PMBS 2022: Performance Modeling, Benchmarking and Simulation of High Performance Computer Systems, Held in conjunction with SC 2022: The International Conference for High Performance Computing, Networking, Storage and Analysis
We propose a new benchmark for high-performance (HP) computers. Similar to High Performance Conjugate Gradient (HPCG), the new benchmark is designed to rank computers based on how fast they can solve a sparse linear system of equations, exhibiting computational and communication requirements typical in many scientific applications. The main novelty of the new benchmark is that it is now based on Generalized Minimum Residual method (GMRES) (combined with Geometric Multi-Grid preconditioner and Gauss-Seidel smoother) and provides the flexibility to utilize lower precision arithmetic. This is motivated by new hardware architectures that deliver lower-precision arithmetic at higher performance. There are other machines that do not follow this trend. However, using a lower-precision arithmetic reduces the required amount of data transfer, which alone could improve solver performance. Considering these trends, an HP benchmark that allows the use of different precisions for solving important scientific problems will be valuable for many different disciplines, and we also hope to promote the design of future HP computers that can utilize mixed-precision arithmetic for achieving high application performance. We present our initial design of the new benchmark, its reference implementation, and the performance of the reference mixed (double and single) precision Geometric Multi-Grid solvers on current top-ranked architectures. We also discuss challenges of designing such a benchmark, along with our preliminary numerical results using 16-bit numerical values (half and bfloat precisions) for solving a sparse linear system of equations.
We propose a domain decomposition method for the efficient simulation of nonlocal problems. Our approach is based on a multi-domain formulation of a nonlocal diffusion problem where the subdomains share “nonlocal” interfaces of the size of the nonlocal horizon. This system of nonlocal equations is first rewritten in terms of minimization of a nonlocal energy, then discretized with a meshfree approximation and finally solved via a Lagrange multiplier approach in a way that resembles the finite element tearing and interconnect method. Specifically, we propose a distributed projected gradient algorithm for the solution of the Lagrange multiplier system, whose unknowns determine the nonlocal interface conditions between subdomains. Several two-dimensional numerical tests on problems as large as 191 million unknowns illustrate the strong and the weak scalability of our algorithm, which outperforms the standard approach to the distributed numerical solution of the problem. Finally, this work is the first rigorous numerical study in a two-dimensional multi-domain setting for nonlocal operators with finite horizon and, as such, it is a fundamental step towards increasing the use of nonlocal models in large scale simulations.
Graph partitioning has been an important tool to partition the work among several processors to minimize the communication cost and balance the workload. While accelerator-based supercomputers are emerging to be the standard, the use of graph partitioning becomes even more important as applications are rapidly moving to these architectures. However, there is no distributed-memory-parallel, multi-GPU graph partitioner available for applications. We developed a spectral graph partitioner, Sphynx, using the portable, accelerator-friendly stack of the Trilinos framework. In Sphynx, we allow using different preconditioners and exploit their unique advantages. We use Sphynx to systematically evaluate the various algorithmic choices in spectral partitioning with a focus on the GPU performance. We perform those evaluations on two distinct classes of graphs: regular (such as meshes, matrices from finite element methods) and irregular (such as social networks and web graphs), and show that different settings and preconditioners are needed for these graph classes. The experimental results on the Summit supercomputer show that Sphynx is the fastest alternative on irregular graphs in an application-friendly setting and obtains a partitioning quality close to ParMETIS on regular graphs. When compared to nvGRAPH on a single GPU, Sphynx is faster and obtains better balance and better quality partitions. Sphynx provides a good and robust partitioning method across a wide range of graphs for applications looking for a GPU-based partitioner.
Support for lower precision computation is becoming more common in accelerator hardware due to lower power usage, reduced data movement and increased computational performance. However, computational science and engineering (CSE) problems require double precision accuracy in several domains. This conflict between hardware trends and application needs has resulted in a need for multiprecision strategies at the linear algebra algorithms level if we want to exploit the hardware to its full potential while meeting the accuracy requirements. In this paper, we focus on preconditioned sparse iterative linear solvers, a key kernel in several CSE applications. We present a study of multiprecision strategies for accelerating this kernel on GPUs. We seek the best methods for incorporating multiple precisions into the GMRES linear solver; these include iterative refinement and parallelizable preconditioners. Our work presents strategies to determine when multiprecision GMRES will be effective and to choose parameters for a multiprecision iterative refinement solver to achieve better performance. We use an implementation that is based on the Trilinos library and employs Kokkos Kernels for performance portability of linear algebra kernels. Performance results demonstrate the promise of multiprecision approaches and demonstrate even further improvements are possible by optimizing low-level kernels.
Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters. In this study we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method.
The purpose of this paper is to study a Helmholtz problem with a spectral fractional Laplacian, instead of the standard Laplacian. Recently, it has been established that such a fractional Helmholtz problem better captures the underlying behavior in Geophysical Electromagnetics. In this work, we establish the well-posedness and regularity of this problem. We introduce a hybrid spectral-finite element approach to discretize it and show well-posedness of the discrete system. In addition, we derive a priori discretization error estimates. Finally, we introduce an efficient solver that scales aswell as the best possible solver for the classical integer-order Helmholtz equation. We conclude withseveral illustrative examples that confirm our theoretical findings.