Scalable bifurcation analysis algorithms for large parallel applications

Salinger, Andrew G.; Pawlowski, Roger P.; Romero, Louis A.

doi:10.1016/b978-008043944-0/50990-2

Cited by 4 publications

(3 citation statements)

References 15 publications

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“…With our simplifications near homogeneity, we directly compose an emergent pattern and exploit the principles of the method of parallel computation to switch over to a new branch. We then use a pseudo arclength continuation approach for branch tracing as presented by Salinger et al (2002) which is also easily scalable to large systems (Salinger et al, 2001).…”

Section: Related Workmentioning

confidence: 99%

Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

2017

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In this paper, we present computational techniques to investigate the effect of surface geometry on biological pattern formation. In particular, we study two-component, nonlinear reaction-diffusion (RD) systems on arbitrary surfaces. We build on standard techniques for linear and nonlinear analysis of RD systems and extend them to operate on large-scale meshes for arbitrary surfaces. In particular, we use spectral techniques for a linear stability analysis to characterise and directly compose patterns emerging from homogeneities. We develop an implementation using surface finite element methods and a numerical eigenanalysis of the Laplace-Beltrami operator on surface meshes. In addition, we describe a technique to explore solutions of the nonlinear RD equations using numerical continuation. Here, we present a multiresolution approach that allows us to trace solution branches of the nonlinear equations efficiently even for large-scale meshes. Finally, we demonstrate the working of our framework for two RD systems with applications in biological pattern formation: a Brusselator model that has been used to model pattern development on growing plant tips, and a chemotactic model for the formation of skin pigmentation patterns. While these models have been used previously on simple geometries, our framework allows us to study the impact of arbitrary geometries on emerging patterns.

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Section: Related Workmentioning

confidence: 99%

Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

2017

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“…This capability has been verified and validated for numerous fluid flow applications and has demonstrated parallel scaling to millions of unknowns [12,13], and is briefly described in Section 2.3. The LOCA (Library of Continuation Algorithms) library [14] has also been interfaced with the MPSalsa code for directly calculating bifurcations [15]. A Newtonbased algorithm in LOCA is used to converge directly to the instability, converging the parameter value and solution simultaneously.…”

Section: 0604x10 5 =mentioning

confidence: 99%

Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Salinger

Lehoucq

Pawlowski

et al. 2002

Numerical Methods in Fluids

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Summary:Stability analysis algorithms coupled with a robust Newton-Krylov steady state iterative solver are used to understand the behavior of the 2D model problem of thermal convection in a 8:1 differentially heated cavity. Parameter continuation methods along with bifurcation and linear stability analysis are used to study transition from steady to transient flow as a function of Rayleigh number. To carry out this study the steady state form of the governing PDEs is discretized using a Galerkin/Least Squares Finite Element formulation, and solved on parallel computers using a fully coupled Newton method and preconditioned Krylov iterative linear solvers. Linear stability analysis employing a large scale eigenvalue capability is used to determine the stability of the steady solutions. The boundary between steady and time dependent flows is determined by a Hopf bifurcation tracking capability that is used to directly track the instability with respect to the aspect ratio of the system and with respect to mesh resolution. The effect of upwinding stabilization terms in the finite element formulation on the computed value of critical Rayleigh number is investigated. The Hopf bifurcation signaling the onset of flow is determined to occur at a critical Rayleigh number of .

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“…Because the bifurcation analysis is facilitated by time-invariant inputs, we bypassed the motion detection stage, directly providing a synthetic, yet plausible, directional response to the MT neurons. We obtained solutions for systematic variations of the parameters through numerical continuation, an efficient family of techniques to follow solutions across parameter changes and for which multiple techniques and software have been developed (Doedel, 1981; Salinger et al, 2001; Henderson, 2002; Dhooge et al, 2003). …”

Section: Modelmentioning

confidence: 99%

Construction and evaluation of an integrated dynamical model of visual motion perception

2015

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Although numerous models describe the individual neural mechanisms that may be involved in the perception of visual motion, few of them have been constructed to take arbitrary stimuli and map them to a motion percept. Here, we propose an integrated dynamical motion model (IDM), which is sufficiently general to handle diverse moving stimuli, yet sufficiently precise to account for a wide-ranging set of empirical observations made on a family of random dot kinematograms. In particular, we constructed models of the cortical areas involved in motion detection, motion integration and perceptual decision. We analyzed their parameters through dynamical simulations and numerical continuation to constrain their proper ranges. Then, empirical data from a family of random dot kinematograms experiments with systematically varying direction distribution, presentation duration and stimulus size, were used to evaluate our model and estimate corresponding model parameters. The resulting model provides an excellent account of a demanding set of parametrically varied behavioral effects on motion perception, providing both quantitative and qualitative elements of evaluation.

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Scalable bifurcation analysis algorithms for large parallel applications

Cited by 4 publications

References 15 publications

Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

Computational bifurcation and stability studies of the 8: 1 thermal cavity problem

Construction and evaluation of an integrated dynamical model of visual motion perception

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