Nonoverlapping Domain Decomposition Methods for Time Harmonic Wave Problems

Claeys, Xavier; Collino, Francis; Joly, Patrick; Parolin, Emile

doi:10.1007/978-3-030-95025-5_5

Cited by 3 publications

(2 citation statements)

References 16 publications

(41 reference statements)

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“…Parallelization of calculations and decomposition of the computational domain are closely related concepts used to efficiently perform computing tasks on multiprocessor systems [1][2][3]. Parallelization of calculations is a method in which calculations are broken down into smaller subtasks that can be performed independently of each other.…”

Section: Introductionmentioning

confidence: 99%

“…Parallelization of calculations is a method in which calculations are broken down into smaller subtasks that can be performed independently of each other. These subtasks are then distributed among the available computing resources (processors, cores, nodes) for acceleration general performance [2,4,5]. Parallelization can only be done at the level of data, tasks, or instructions.…”

Section: Introductionmentioning

confidence: 99%

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Acceleration of computations in modelling of processes in complex objects and systems

Khaidurov,

Tatenko,

Lytovchenko

et al. 2024

Sist. dosl. energ.

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The development of methods of parallelization of computing processes, which involve the decomposition of the computational domain, is an urgent task in the modeling of complex objects and systems. Complex objects and systems can contain a large number of elements and interactions. Decomposition allows you to break down a system into simpler subsystems, which simplifies the analysis and management of complexity. By dividing the calculation area of the part, it is possible to perform parallel calculations, which increases the efficiency of calculations and reduces simulation time. Domain decomposition makes it easy to scale the model to work with larger or more detailed systems. With the right choice of decomposition methods, the accuracy of the simulation can be improved, since different parts of the system may have different levels of detail and require appropriate methods of additional analysis. Decomposition allows the simulation to be distributed between different participants or devices, which is relevant for distributed systems or collaborative work on a project. In this work, mathematical models are built, which consist in the construction of iterative procedures for "stitching" several areas into a single whole. The models provide for different complexity of calculation domains, which makes it possible to perform different decomposition approaches, in particular, both overlapping and non-overlapping domain decomposition. The obtained mathematical models of subject domain decomposition can be applied to objects and systems that have different geometric complexity. Domain decomposition models that do not use overlap contain different iterative methods of "stitching" on a common boundary depending on the types of boundary conditions (a condition of the first kind is a Dirichlet condition, or a condition of the second year is a Neumann condition), and domain decomposition models with an overlap of two or more areas consist of the minimization problem for constructing the iterative condition of "stitching" areas. It should be noted that the obtained models will work effectively on all applied tasks that describe the dynamic behavior of objects and their systems, but the high degree of efficiency of one model may be lower than the corresponding the degree of effectiveness of another model, since each task is individual. Keywords: mathematical modelling, decomposition of the computational domain, parallelization, optimization, complex objects and systems.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Acceleration of computations in modelling of processes in complex objects and systems

Khaidurov,

Tatenko,

Lytovchenko

et al. 2024

Sist. dosl. energ.

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Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation

et al. 2022

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We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the exchange of information between subdomains is achieved using first-order absorbing (impedance) transmission conditions, together with a partition of unity. We provide a novel analysis of this method at the PDE level (without discretization). First, we formulate the method as a fixed point iteration, and show (in dimensions 1, 2, 3) that it is well-defined in a tensor product of appropriate local function spaces, each with $$L^2$$ L 2 impedance boundary data. We then obtain a bound on the norm of the fixed point operator in terms of the local norms of certain impedance-to-impedance maps arising from local interactions between subdomains. These bounds provide conditions under which (some power of) the fixed point operator is a contraction. In 2-d, for rectangular domains and strip-wise domain decompositions (with each subdomain only overlapping its immediate neighbours), we present two techniques for verifying the assumptions on the impedance-to-impedance maps that ensure power contractivity of the fixed point operator. The first is through semiclassical analysis, which gives rigorous estimates valid as the frequency tends to infinity. At least for a model case with two subdomains, these results verify the required assumptions for sufficiently large overlap. For more realistic domain decompositions, we directly compute the norms of the impedance-to-impedance maps by solving certain canonical (local) eigenvalue problems. We give numerical experiments that illustrate the theory. These also show that the iterative method remains convergent and/or provides a good preconditioner in cases not covered by the theory, including for general domain decompositions, such as those obtained via automatic graph-partitioning software.

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Sharp bounds on Helmholtz impedance-to-impedance maps and application to overlapping domain decomposition

Lafontaine,

Spence

2023

Pure Appl. Analysis

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Nonoverlapping Domain Decomposition Methods for Time Harmonic Wave Problems

Cited by 3 publications

References 16 publications

Acceleration of computations in modelling of processes in complex objects and systems

Acceleration of computations in modelling of processes in complex objects and systems

Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation

Sharp bounds on Helmholtz impedance-to-impedance maps and application to overlapping domain decomposition

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