Newton's Method and Symmetry for Semilinear Elliptic PDE on the Cube

Neuberger, John M.; Sieben, Nándor; Swift, James W.

doi:10.1137/120899054

Cited by 5 publications

(18 citation statements)

References 22 publications

(71 reference statements)

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“…Our methodology is mainly motivated by our work in computational combinatorics [27] and partial differential equations (PDE) [22,23,24]. In combinatorial applications, we have the needs and methods to follow trees.…”

Section: Introductionmentioning

confidence: 99%

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An MPI Implementation of a Self-Submitting Parallel Job Queue

Neuberger

Sieben

Swift

2012

Int J Parallel Prog

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Abstract. We present a simple and easy to apply methodology for using high-level self-submitting parallel job queues in an MPI environment. Using C++, we implemented a library of functions, MPQueue, both for testing our concepts and for use in real applications. In particular, we have applied our ideas toward solving computational combinatorics problems and for finding bifurcation diagrams of solutions of partial differential equations (PDE). Our method is general and can be applied in many situations without a lot of programming effort. The key idea is that workers themselves can easily submit new jobs to the currently running job queue. Our applications involve complicated data structures, so we employ serialization to allow data to be effortlessly passed between nodes. Using our library, one can solve large problems in parallel without being an expert in MPI. We demonstrate our methodology and the features of the library with several example programs, and give some results from our current PDE research. We show that our techniques are efficient and effective via overhead and scaling experiments.

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Section: Introductionmentioning

confidence: 99%

“…A computationally intensive and mathematically complex example result for PDE can be found in Section 6. The full mathematical details of the particular PDE we are interested in can be found in [24], along with our state of the art numerical results which use MPQueue. In Section 7 we give an overview of the implementation of MPQueue.…”

Section: Introductionmentioning

confidence: 99%

An MPI Implementation of a Self-Submitting Parallel Job Queue

Neuberger

Sieben

Swift

2012

Int J Parallel Prog

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“…Our initial experience as a collaborative group concerned solutions and numerical approximations of solutions to semilinear elliptic PDE and PdE (partial difference equations), e.g., Example 2.3, [20,21]. In such works we not only observe invariant subspaces but use them to make our Newton's method-based algorithms more robust and efficient.…”

Section: Resultsmentioning

confidence: 99%

“…Also note that δ A 0 (i) = 0 for all i because A 0 = −A 0 . With these definitions, we can write Equation (18) (20) implies that (21) f…”

Section: Matched Partitions and Anti-synchronymentioning

confidence: 99%

Synchrony and Antisynchrony for Difference-Coupled Vector Fields on Graph Network Systems

Neuberger¹,

Sieben²,

Swift³

2019

SIAM J. Appl. Dyn. Syst.

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We define a graph network to be a coupled cell network where there are only one type of cell and one type of symmetric coupling between the cells. For a difference-coupled vector field on a graph network system, all the cells have the same internal dynamics, and the coupling between cells is identical, symmetric, and depends only on the difference of the states of the interacting cells. We define four nested sets of difference-coupled vector fields by adding further restrictions on the internal dynamics and the coupling functions. These restrictions require that these functions preserve zero or are odd or linear. We characterize the synchrony and anti-synchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and anti-synchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and anti-synchrony subspaces for some examples of graph networks. We also apply our theory to systems of coupled van der Pol oscillators.2000 Mathematics Subject Classification. 34C15, 34C23, 34A34, 37C80.

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“…Current sampling methods are either not suitable in this context or too computationally expensive (when they are based on an optimization process). Symmetries play an important role in many bifurcation problems [16] and may lead to very complex bifurcation diagrams [36], whose efficient computation requires that the computational means (namely, the ROM in the context of this paper) preserve the symmetries. In addition, when many symmetries are present, reduced-order modeling could be combined with other means to efficiently treat symmetries, such as some recent ideas on automated symmetry analysis; see [36] and references therein.…”

Section: Discussionmentioning

confidence: 99%

Construction of Bifurcation Diagrams Using POD on the Fly

Terragni¹,

Vega²

2014

SIAM J. Appl. Dyn. Syst.

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An adaptive method is presented for the construction of bifurcation diagrams providing the large time dynamics of dissipative systems as a bifurcation parameter is varied. The method combines a standard numerical solver and a Galerkin system resulting from Galerkin projecting the governing equations onto a set of proper orthogonal decomposition (POD) modes that are computed on the fly as the bifurcation parameter is varied. The numerical solver provides some snapshots that are used to either completely calculate (at the outset of the process) or update (later on) the POD modes. This solver is run over short time intervals only, at specific values of the bifurcation parameter, meaning that the (much faster) Galerkin system is used for the majority of the calculation. The key ingredients are (i) detecting on the fly when updating is needed and (ii) reconstructing the POD modes. The necessity of updating the POD basis is detected monitoring (a) the amplitudes of the high-order modes (which estimate the accuracy of the approximation) and (b) consistency with the solution provided by a second instrumental Galerkin system that retains a larger number of POD modes (which detects truncation instabilities of the first Galerkin system). Updating the POD modes is performed applying POD to a mix of appropriately weighted old and new modes. The method is tested on the complex Ginzburg-Landau equation, considering bifurcation diagrams that exhibit a variety of bifurcations involving periodic, quasi-periodic, and chaotic attractors, and turns out to be flexible, robust, and computationally fast.

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Newton's Method and Symmetry for Semilinear Elliptic PDE on the Cube

Cited by 5 publications

References 22 publications

An MPI Implementation of a Self-Submitting Parallel Job Queue

An MPI Implementation of a Self-Submitting Parallel Job Queue

Synchrony and Antisynchrony for Difference-Coupled Vector Fields on Graph Network Systems

Construction of Bifurcation Diagrams Using POD on the Fly

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