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Register a new userOptimised finite difference computation from symbolic equations
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Added by
Michael Lange
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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Domain-specific high-productivity environments are playing an increasingly important role in scientific computing due to the levels of abstraction and automation they provide. In this paper we introduce Devito, an open-source domain-specific framework for solving partial differential equations from symbolic problem definitions by the finite difference method. We highlight the generation and automated execution of highly optimized stencil code from only a few lines of high-level symbolic Python for a set of scientific equations, before exploring the use of Devito operators in seismic inversion problems.
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Domain specific languages (DSL) have been used in a variety of fields to express complex scientific problems in a concise manner and provide automated performance optimization for a range of computational architectures. As such DSLs provide a powerful mechanism to speed up scientific Python computation that goes beyond traditional vectorization and pre-compilation approaches, while allowing domain scientists to build applications within the comforts of the Python software ecosystem. In this paper we present Devito, a new finite difference DSL that provides optimized stencil computation from high-level problem specifications based on symbolic Python expressions. We demonstrate Devitos symbolic API and performance advantages over traditional Python acceleration methods before highlighting its use in the scientific context of seismic inversion problems.
In addition to hardware wall-time restrictions commonly seen in high-performance computing systems, it is likely that future systems will also be constrained by energy budgets. In the present work, finite difference algorithms of varying computational and memory intensity are evaluated with respect to both energy efficiency and runtime on an Intel Ivy Bridge CPU node, an Intel Xeon Phi Knights Landing processor, and an NVIDIA Tesla K40c GPU. The conventional way of storing the discretised derivatives to global arrays for solution advancement is found to be inefficient in terms of energy consumption and runtime. In contrast, a class of algorithms in which the discretised derivatives are evaluated on-the-fly or stored as thread-/process-local variables (yielding high compute intensity) is optimal both with respect to energy consumption and runtime. On all three hardware architectures considered, a speed-up of ~2 and an energy saving of ~2 are observed for the high compute intensive algorithms compared to the memory intensive algorithm. The energy consumption is found to be proportional to runtime, irrespective of the power consumed and the GPU has an energy saving of ~5 compared to the same algorithm on a CPU node.
Exascale computing will feature novel and potentially disruptive hardware architectures. Exploiting these to their full potential is non-trivial. Numerical modelling frameworks involving finite difference methods are currently limited by the static nature of the hand-coded discretisation schemes and repeatedly may have to be re-written to run efficiently on new hardware. In contrast, OpenSBLI uses code generation to derive the models code from a high-level specification. Users focus on the equations to solve, whilst not concerning themselves with the detailed implementation. Source-to-source translation is used to tailor the code and enable its execution on a variety of hardware.
By analogy with the theory of Backward Stochastic Differential Equations, we define Backward Stochastic Difference Equations on spaces related to discrete time, finite state processes. This paper considers these processes as constructions in their own right, not as approximations to the continuous case. We establish the existence and uniqueness of solutions under weaker assumptions than are needed in the continuous time setting, and also establish a comparison theorem for these solutions. The conditions of this theorem are shown to approximate those required in the continuous time setting. We also explore the relationship between the driver $F$ and the set of solutions; in particular, we determine under what conditions the driver is uniquely determined by the solution. Applications to the theory of nonlinear expectations are explored, including a representation result.
Discovering the underlying behavior of complex systems is an important topic in many science and engineering disciplines. In this paper, we propose a novel neural network framework, finite difference neural networks (FDNet), to learn partial differential equations from data. Specifically, our proposed finite difference inspired network is designed to learn the underlying governing partial differential equations from trajectory data, and to iteratively estimate the future dynamical behavior using only a few trainable parameters. We illustrate the performance (predictive power) of our framework on the heat equation, with and without noise and/or forcing, and compare our results to the Forward Euler method. Moreover, we show the advantages of using a Hessian-Free Trust Region method to train the network.
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