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Register a new userAchieving algorithmic resilience for temporal integration through spectral deferred corrections
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Added by
Ray Grout
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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Spectral deferred corrections (SDC) is an iterative approach for constructing higher- order accurate numerical approximations of ordinary differential equations. SDC starts with an initial approximation of the solution defined at a set of Gaussian or spectral collocation nodes over a time interval and uses an iterative application of lower-order time discretizations applied to a correction equation to improve the solution at these nodes. Each deferred correction sweep increases the formal order of accuracy of the method up to the limit inherent in the accuracy defined by the collocation points. In this paper, we demonstrate that SDC is well suited to recovering from soft (transient) hardware faults in the data. A strategy where extra correction iterations are used to recover from soft errors and provide algorithmic resilience is proposed. Specifically, in this approach the iteration is continued until the residual (a measure of the error in the approximation) is small relative to the residual on the first correction iteration and changes slowly between successive iterations. We demonstrate the effectiveness of this strategy for both canonical test problems and a comprehen- sive situation involving a mature scientific application code that solves the reacting Navier-Stokes equations for combustion research.
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Spectral deferred correction (SDC) methods are an attractive approach to iteratively computing collocation solutions to an ODE by performing so-called sweeps with a low-order time stepping method. SDC allows to easily construct high order split methods where e.g. stiff terms of the ODE are treated implicitly. This requires the solution to full accuracy of multiple linear systems of equations during each sweep, e.g. with a multigrid method. In this paper, we present an inexact variant of SDC, where each solve of a linear system is replaced by a single multigrid V-cycle and thus significantly reduces the cost for each sweep. For the investigated examples, this strategy results only in a small increase of the number of required sweeps and we demonstrate that inexact spectral deferred corrections can provide a dramatic reduction of the overall number of multigrid V-cycles required to complete an SDC time step.
High-performance computing (HPC) is a major driver accelerating scientific research and discovery, from quantum simulations to medical therapeutics. The growing number of new HPC systems coming online are being furnished with various hardware components, engineered by competing industry entities, each having their own architectures and platforms to be supported. While the increasing availability of these resources is in many cases pivotal to successful science, even the largest collaborations lack the computational expertise required for maximal exploitation of current hardware capabilities. The need to maintain multiple platform-specific codebases further complicates matters, potentially adding a constraint on the number of machines that can be utilized. Fortunately, numerous programming models are under development that aim to facilitate software solutions for heterogeneous computing. In this paper, we leverage the SYCL programming model to demonstrate cross-platform performance portability across heterogeneous resources. We detail our NVIDIA and AMD random number generator extensions to the oneMKL open-source interfaces library. Performance portability is measured relative to platform-specific baseline applications executed on four major hardware platforms using two different compilers supporting SYCL. The utility of our extensions are exemplified in a real-world setting via a high-energy physics simulation application. We show the performance of implementations that capitalize on SYCL interoperability are at par with native implementations, attesting to the cross-platform performance portability of a SYCL-based approach to scientific codes.
Large, complex, multi-scale, multi-physics simulation codes, running on high performance com-puting (HPC) platforms, have become essential to advancing science and engineering. These codes simulate multi-scale, multi-physics phenomena with unprecedented fidelity on petascale platforms, and are used by large communities. Continued ability of these codes to run on future platforms is as crucial to their communities as continued improvements in instruments and facilities are to experimental scientists. However, the ability of code developers to do these things faces a serious challenge with the paradigm shift underway in platform architecture. The complexity and uncertainty of the future platforms makes it essential to approach this challenge cooperatively as a community. We need to develop common abstractions, frameworks, programming models and software development methodologies that can be applied across a broad range of complex simulation codes, and common software infrastructure to support them. In this position paper we express and discuss our belief that such an infrastructure is critical to the deployment of existing and new large, multi-scale, multi-physics codes on future HPC platforms.
This work presents an efficient numerical method based on spectral expansions for simulation of heat and moisture diffusive transfers through multilayered porous materials. Traditionally, by using the finite-difference approach, the problem is discretized in time and space domains (Method of lines) to obtain a large system of coupled Ordinary Differential Equations (ODEs), which is computationally expensive. To avoid such a cost, this paper proposes a reduced-order method that is faster and accurate, using a much smaller system of ODEs. To demonstrate the benefits of this approach, tree case studies are presented. The first one considers nonlinear heat and moisture transfer through one material layer. The second case - highly nonlinear - imposes a high moisture content gradient - simulating a rain like condition - over a two-layered domain, while the last one compares the numerical prediction against experimental data for validation purposes. Results show how the nonlinearities and the interface between materials are easily and naturally treated with the spectral reduced-order method. Concerning the reliability part, predictions show a good agreement with experimental results, which confirm robustness, calculation efficiency and high accuracy of the proposed approach for predicting the coupled heat and moisture transfer through porous materials.
The spectral deferred correction (SDC) method is an iterative scheme for computing a higher-order collocation solution to an ODE by performing a series of correction sweeps using a low-order timestepping method. This paper examines a variation of SDC for the temporal integration of PDEs called multi-level spectral deferred corrections (MLSDC), where sweeps are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. Three different strategies to reduce the computational cost of correction sweeps on the coarser levels are examined: reducing the degrees of freedom, reducing the order of the spatial discretization, and reducing the accuracy when solving linear systems arising in implicit temporal integration. Several numerical examples demonstrate the effect of multi-level coarsening on the convergence and cost of SDC integration. In particular, MLSDC can provide significant savings in compute time compared to SDC for a three-dimensional problem.
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