Partial Jacobian computation in the domain-specific program transformation system ADiCape

Petera, Monika; Lülfesmann, Michael; Bücker, H.

doi:10.1109/imcsit.2009.5352778

Cited by 2 publications

(2 citation statements)

References 13 publications

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“…Partial Jacobian computation is briefly mentioned in [10] where preconditioning is sketched as an area in which it could potentially be applied. It is also used in [16] in the context of a sensitivity analysis, but without any connection to preconditioning. Preliminary ideas on the conceptual framework of partial coloring are described in [14,13] The contribution of this article to the field of combinatorial scientific computing is (a) the introduction of a novel combinatorial problem arising from preconditioning using automatic differentiation, (b) the design of a suitable graph model encoding this situation, (c) the transformation of the scientific computing problem into a graph coloring problem, (d) the design of a heuristic for the solution of that graph coloring problem, and (e) the application of the overall methodology to a case study arising from computational fluid dynamics.…”

Section: Discussionmentioning

confidence: 99%

Enabling Implicit Time Integration for Compressible Flows by Partial Coloring: A Case Study of a Semi-matrix-free Preconditioning Technique

Bücker

Lülfesmann

Rostami

2016

2016 Proceedings of the Seventh SIAM Workshop on Combinatorial Scientific Computing

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Numerical techniques involving linearizations of nonlinear functions require the repeated solution of systems of linear equations whose coefficient matrix is the Jacobian of that nonlinear function. If the Jacobian is large and sparse, iterative methods offer the advantage that they involve the Jacobian solely in the form of matrix-vector products. Techniques of automatic differentiation are capable of evaluating these Jacobian-vector products efficiently and accurately in a matrix-free fashion. So, the numerical technique does not need to store the Jacobian explicitly. When the solution of the linear system is preconditioned, however, there is currently a considerable gap between automatic differentiation and preconditioning because the latter typically requires to explicitly store the Jacobian in a sparse data format. In an attempt to bridge this gap, we introduce an approach based on block diagonal preconditioning that brings together known computational building blocks in a novel way. The crucial methodological ingredient to that approach is the formulation and solution of a partial coloring problem in which colors are assigned to only a subset of the vertices of the underlying graph. Numerical experiments are reported that demonstrate the feasibility of this approach. Implicit Time Integration for FlowThe last decades have witnessed a steady increase in airline passenger traffic. This enormous air traffic and the resulting airport congestion has stimulated growing interest in the design of high-capacity aircrafts in which the number of passengers in a transport aircraft is large. Thus, it is no surprise that, in science and industry, numerical simulations that accurately and reliably predict the flow field around a high-capacity aircraft continue to be important.At RWTH Aachen University, Germany, an interdisciplinary team of researchers from mathematics, computer science, and engineering disciplines is developing the software package QUADFLOW [3,4,6]. This * Chair for Advanced Computing, Institute for Computer Science, Friedrich Schiller University Jena, Germany † Michael Stifel Center Jena for Data-driven and Simulation Science, 07737 Jena, Germany ‡ Düsseldorf, Germany adaptive finite volume flow solver represents the underlying physics of stationary flow by the Euler and Navier-Stokes equations. It employs two-and threedimensional meshes in multiblocks; see Fig. 1 for an example of such a mesh in two space dimensions. Summarizing the discussion in [6], let V denote any control volume and let u represent the vector of unknown conserved quantities. Then, spatial discretization results in the system of differential equations V

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Section: Discussionmentioning

confidence: 99%

Enabling Implicit Time Integration for Compressible Flows by Partial Coloring: A Case Study of a Semi-matrix-free Preconditioning Technique

Bücker

Lülfesmann

Rostami

2016

2016 Proceedings of the Seventh SIAM Workshop on Combinatorial Scientific Computing

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“…Like the previous approach in [3], the new approach is based on computing only a proper subset of the nonzero elements of the Jacobian J, which is referred to as partial Jacobian computation [5,[7][8][9][10]. We summarize partial Jacobian computation by considering Fig.…”

Section: Preconditioning Via Two Block Schemesmentioning

confidence: 99%

Preconditioning Jacobian Systems by Superimposing Diagonal Blocks

Rostami

Bücker

2020

Lecture Notes in Computer Science

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Preconditioning constitutes an important building block for the solution of large sparse systems of linear equations. If the coefficient matrix is the Jacobian of some mathematical function given in the form of a computer program, automatic differentiation enables the efficient and accurate evaluation of Jacobian-vector products and transposed Jacobian-vector products in a matrix-free fashion. Standard preconditioning techniques, however, typically require access to individual nonzero elements of the coefficient matrix. These operations are computationally expensive in a matrix-free approach where the coefficient matrix is not explicitly assembled. We propose a novel preconditioning technique that is designed to be used in combination with automatic differentiation. A key element of this technique is the formulation and solution of a graph coloring problem that encodes the rules of partial Jacobian computation that determines only a proper subset of the nonzero elements of the Jacobian matrix. The feasibility of this semi-matrix-free approach is demonstrated on a set of numerical experiments using the automatic differentiation tool ADiMat.

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Partial Jacobian computation in the domain-specific program transformation system ADiCape

Cited by 2 publications

References 13 publications

Enabling Implicit Time Integration for Compressible Flows by Partial Coloring: A Case Study of a Semi-matrix-free Preconditioning Technique

Enabling Implicit Time Integration for Compressible Flows by Partial Coloring: A Case Study of a Semi-matrix-free Preconditioning Technique

Preconditioning Jacobian Systems by Superimposing Diagonal Blocks

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