[00840] Efficient and scalable solvers and algorithms for multiscale phenomena
Session Date & Time :
00840 (1/3) : 1D (Aug.21, 15:30-17:10)
00840 (2/3) : 1E (Aug.21, 17:40-19:20)
00840 (3/3) : 2C (Aug.22, 13:20-15:00)
Type : Proposal of Minisymposium
Abstract : Many physical systems involve interactions between different scales in space and/or time, which usually stem from the coexistence of complex micro-structures and phenomena taking place at much larger scales.
Prominent examples of multiscale systems are biological tissues, composed of millions of cells but treated as a continuum, with scale separations in both space and time.
The focus of this minisymposium is on efficient and scalable numerical methods, solvers and high performance software, which can solve such complex systems on modern HPC computing architectures.
Organizer(s) : Nicolas A. Barnafi, Ngoc Mai Monica Huynh, Luca F. Pavarino
Tommaso Bevilacqua (Università degli Studi di Milano)
Xiao-Chuan Cai (University of Macao)
Fatemeh Chegini (Zuse Institute Berlin)
Durkbin Cho (Dongguk University)
Fritz Goebel (Karlsruhe Institute of Technology)
Matthias Gsell (Medical University of Graz)
Hardik Kothari (Università della Svizzera Italiana)
Massimiliano Leoni (RICAM)
Olof Widlund (Courant Institute)
Stefano Zampini (KAUST)
Talks in Minisymposium :
[02611] An efficient parallel interpolation algorithm with applications to multi-physics simulation of cardiac radiofrequency ablation
Author(s) :
Massimiliano Leoni (Johann Radon Institute for Computational and Applied Mathematics)
Argyrios Petras (RICAM-Johann Radon Institute for Computational and Applied Mathematics)
Luca Gerardo-Giorda (JKU and RICAM)
Abstract : In this talk we discuss modelling and simulation of Cardiac Radiofrequency Ablation, a clinical procedure used to treat some forms of cardiac arrhythmia by accessing the patient's heart with a catheter and burning it locally to make it electrically insulating.
By its nature, this problem requires a complex multi-physics approach, which in turn yields many computational challenges.
In particular, we will focus on parallel interpolation, a trivial-looking step that is crucial to a performant implementation
[02988] A multiscale preconditioner for simulating blood flows in artery with aneurysm
Author(s) :
Xiao-Chuan Cai (University of Macau)
Abstract : In this talk, we discuss some recent development of numerical methods for the simulation of blood flows in patient-specific arteries with aneurysm. Depending on the branching geometry and the patient parameters, the flow can be quite complicated with local vortex structures, but the principal component of the flow is always along the centerline of the artery. Based on this observation, we introduce a two-scale domain decomposition method for unsteady incompressible Navier-Stokes equations in three-dimensional complex patient-specific arteries, and the key component of the preconditioner is a parameterized one-dimensional unsteady Navier-Stokes or Stokes coarse problem defined along the centerline of the artery. The one-dimensional preconditioner and some overlapping three-dimensional subdomain preconditioners are combined additively to form the two-scale method via interpolations using radial basis functions. The most important feature of the method is that the cost of solving the coarse problem is nearly neglectable compared with the subdomain solver. Numerical experiments indicate that the proposed method is highly effective and robust for complex arteries with many branches and aneurysm. This is a joint work with Yingzhi Liu and Fenfen Qi.
[03487] Efficient solvers for models of personalized whole heart electromechanics
Author(s) :
Matthias Gsell (Medical University of Graz)
Christop Augustin (Medical University of Graz)
Karli Gillette (Medical University of Graz)
Alexander Jung (Medical University of Graz)
Gernot Plank (Medical University of Graz)
Abstract : Anatomically accurate computer models of four-chamber electromechanics, which are able to
replicate electromechanical function of an individual patient’s heart show high potential for both
clinical and industrial applications such as diagnostics, treatment optimization and device
development. Methodology used to obtain a first fully mechanistic whole-heart electromechanics
models with non-invasively personalized electrophysiology and calibrated mechanical and vascular
function will be presented. We demonstrate goodness of fit of the calibrated model, and validation
against common physiological principles.
[03593] Adaptive BDDC preconditioners for 3D divergence free virtual element discretizations of the Stokes equations.
Author(s) :
Tommaso Bevilacqua (University of Milan)
Franco Dassi (University of Milano-Bicocca)
Stefano Zampini (King Abdullah University of Science and Tecnology, )
Simone Scacchi (University of Milan)
Abstract : The balancing domain decomposition by constraints (BDDC) preconditioners are domain decomposition methods based on the subdivision of the computational domain of a partial differential equation (PDE) into non-overlapping subdomains.
We apply BDDC to solve PDEs discretized by Virtual Element Methods (VEM) proving scalability and quasi-optimality of the algorithm. Numerical results with adaptively generated coarse spaces confirm the method's robustness in the presence of large jumps in the viscosity and with high-order VEM discretizations.
[03711] Higher Order Time Integration for EMI Cardiac Electrophysiology Simulations with Nested Subset Selection and BDDC Preconditioning
Author(s) :
Fatemeh Chegini (Zuse Institute Berlin(ZIB))
Martin Weiser (Zuse Institute Berlin(ZIB))
Abstract : Cardiac electrophysiology simulations call for adaptive methods due to locality of solution features. Traditional mesh refinement and coarsening approaches incur significant overheads. We investigate a novel approach using nested subset selection for algebraic degrees of freedom in hierarchical spectral deferred correction methods. This enables multi-rate integration with minimal overhead, and reduces the computational cost significantly. We also propose a novel domain decomposition preconditioner of BDDC type for cell-by-cell electrophysiology models and show numerical results.
[03783] Monolithic solution strategies for large-scale computational problems from physiology and astrophysics
Author(s) :
pietro benedusi (Simula Research Laboratory )
Rolf Krause (Università della Svizzera italiana)
Patrick Zulian (Università della Svizzera italiana)
Abstract : Currently, many problems in applied mathematics result in large-scale computational challenges which require the use of massively parallel machines and scalable solution strategies to minimize the time to solution. In this talk, we present monolithic strategies to numerically solve partial differential equations, for various applications. These strategies consist in framing, whenever possible, a computational problem in a large and possibly sparse (non) linear system which can be solved, in parallel, combining efficient preconditioning techniques and Krylov methods. By contrast, many traditional staggered approaches are based on the solution of a sequence of smaller computational problems. An example of such a paradigm can be found when solving evolutionary problems, where a monolithic strategy (also known as all-at-once approach) can be used, resulting in the assembly of large a space-time system with a block Toeplitz structure, in contrast to standard sequential time-stepping techniques. In this context, we consider the space-time discretization of the anisotropic diffusion equation, using isogeometric analysis in space and a discontinuous Galerkin approximation in time. Drawing inspiration from a former spectral analysis of space-time operators, we propose a parallel multigrid preconditioned GMRES method. The application of this multilevel space-time strategy to non-linear reaction-diffusion problems from electrophysiology (i.e. the monodomain equation and the EMI model) will be also discussed, considering comparison with other recently developed methods. Moreover, we present a monolithic approach to simulate radiative transfer in stellar atmospheres; in this scenario, we present a matrix-free implementation of a multi-fidelity preconditioner. Through simulations on massively parallel systems, we show how monolithic strategies can improve software scalability and discuss the trade-offs of this approach.
[03815] Overlapping Schwarz methods for Isogeometric analysis based on generalized B-splines
Author(s) :
Durkbin Cho (Dongguk University)
Abstract : \ifx\justbeingincluded\undefined
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\myTalk{Overlapping Schwarz methods for Isogeometric analysis based on generalized B-splines}
\myAuthor{\underline{Durkbin Cho}}
{Department of Mathematics, Dongguk University,}
{Pil-dong 3-ga, Jung-gu, Seoul, 04620, South Korea.}
{}
{}
\myAbstract{\indent Isogeometric analysis (IGA) is an innovative numerical methodology for the solution of partial differential equations (PDEs), introduced by Hughes {\it et al.} in \cite{HuCoBa05}, that potentially allows for a direct connection with CAD, thus providing a much easier and exact representation of the computational domain in a wide range of applications \cite{CHB}. Generalized B-splines (GB-splines) are a special class of Tchebycheff B-splines that are smooth piecewise function with sections in more general spaces \cite{LMS,CLM,M1,M2}. GB-splines allow for an exact representation of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation. They have been proposed in \cite{MPS1} as an attractive tool in isogeometric analysis. Since then, isogeometric analysis based on GB-splines have been studied in e.g. \cite{MPS2,MPSp}. In this talk, we present overlapping Schwarz preconditioners for {\sf elliptic} and {\sf biharmonic} problems discretized with isogeometric analysis based on GB-splines \cite{Cho1,Cho2}.
An h-analysis of the proposed preconditioners shows an optimal convergence rate bound. Numerical results in two- and three-dimensional tests confirm our
theory and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters.
}
\begin{thebibliography}{00}
\bibitem{HuCoBa05} T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs.
Isogeometric analysis: {CAD}, finite elements, {NURBS}, exact geometry, and mesh refinement. \textit{Comp. Meth. Appl. Mech. Engrg.}, \textbf{194}, 4135--4195, 2005.
\bibitem{CHB} J.A. Cottrell, T.J.R. Hughes and Y. Bazilevs. Isogeometric analysis: toward integration of CAD and FEA. John Wiley \& Sons, 2009.
\bibitem{LMS} T. Lyche, C. Manni, H. Speleers, Tchebycheffian B-splines revisited: an introductory exposition, in: Advanced Methods for Geometric Modeling and Numerical Simulation, in: Springer INdAM Ser., vol. 35, 2019, pp. 179—216.
\bibitem{CLM} P. Costantini, T. Lyche, C. Manni, On a class of weak Tchebycheff systems, \textit{Numer. Math.}, \textbf{101} (2005) 333–354.
\bibitem{M1} M.-L. Mazure, On a general new class of quasi-Chebyshevian splines, \textit{Numer. Algorithms}, \textbf{58} (2011) 399–438.
\bibitem{M2} M.-L. Mazure, How to build all Chebyshevian spline spaces good for Geometric Design, \textit{Numer. Math.} \textbf{119} (2011) 517–556
\bibitem{MPS1} C. Manni, F. Pelosi, M.L. Sampoli, Generalized B-splines as a tool in isogeometric analysis, \textit{Comput. Methods Appl. Mech. Engrg.}
\textbf{200} (2011) 867–881.
\bibitem{MPS2} C. Manni, F. Pelosi, M.L. Sampoli, Isogeometric analysis in advection—diffusion problems: tension splines approximation, \textit{J. Comput.
Appl. Math.} \textbf{236} (2011) 511–528.
\bibitem{MPSp} C. Manni, F. Pelosi, H. Speleers, Local hierarchical h-refinements in IgA based on generalized B-splines, in: M. Floater, et al. (Eds.),
Mathematical Methods for Curves and Surfaces 2012, in: Lecture Notes in Computer Science, vol. 8177, 2014, pp. 341–363.
\bibitem{Cho1} D. Cho, Overlapping Schwarz methods for isogeometric analysis based on generalized B-splines, \textit{Comput. Methods Appl. Mech. Engrg.} \textbf{372} (2020), 113430, 16 pp.
\bibitem{Cho2} D. Cho, Isogeometric Schwarz preconditioners with generalized B-splines for the biharmonic problem, (in revision), 2023.
\end{thebibliography}
\end{document}
[04469] Spectral Element discretizations in cardiac electrophysiology: a matrix-free approach
Author(s) :
Pasquale Claudio Africa (mathLab, SISSA International School for Advanced Studies, Trieste)
Matteo Salvador ( Stanford University)
Paola Gervasio (Università degli Studi di Brescia)
Abstract : We propose a high-order Spectral Element Method (SEM) matrix-free solver for the numerical solution of cardiac electrophysiology. We compare it to SEM with Numerical Integration and demonstrate that increasing the local polynomial degree leads to improved accuracy and faster computations than reducing the mesh size. Our matrix-free approach, enhanced by a suitable implementation of a Geometric Multigrid (GMG) preconditioner, yields up to 45\(\times\) speed-up compared to a conventional matrix-based solver. Several numerical experiments are analyzed.
[04801] Multi-scale modelling and simulation: EMI models, 3D-1D transport, and DG methods
Author(s) :
Rami Masri (Simula Research Laboratory)
Marius Zeinhofer (Simula Research Laboratory)
Miroslav Kuchta (Simula Research Laboratory)
Marie Rognes (Simula Research Laboratory )
Abstract : In this presentation, we discuss several of our findings on a variety of multi-scale models and their discretizations. First for the EMI equations which are used to model excitable tissue, we formulate and analyse discontinuous Galerkin interior penalty formulations. The practical advantages of such an approach are that (i) it can be implemented in any finite element library without additional multimesh/mixed-dimensional features, and that (ii) black box multigrid solvers perform well. Second, we formulate coupled time dependent 3D-1D models of transport used to model a variety of phenomena. The modeling and the discretisation errors for finite element approximations are discussed.
[04831] A nonlinear preconditioning strategy for solving phase-field fracture problems in a constrained minimization framework
Author(s) :
Hardik Kothari (Università della Svizzera Italiana)
Alena Kopaničáková (Brown Universitty)
Rolf Krause (Università della Svizzera Italiana)
Abstract : The phase-field approach to fracture allows one to model crack propagation, branching, and merging. Despite its robust modeling properties, solving this problem is computationally challenging due to the non-convex, non-smooth, highly nonlinear, and ill-conditioned nature of the underlying energy function. We propose a field-split-based additive/multiplicative Schwarz preconditioned Newton method to solve the fracture problem by employing a right preconditioner that can handle inequality constraints. The robustness of the method will be shown using numerical examples.
[04846] SCALABLE SOLVERS FOR BULK-SURFACE MATERIALS UNDERGOING SPINODAL DECOMPOSITION
Author(s) :
stefano zampini (KAUST)
Luis Espath (University of Nottingham)
Luca Heltai (SISSA)
Hector Gomez (Purdue University)
Abstract : In this work, we present numerical results for two- and three-dimensional bulk-surface materials undergoing spinodal decomposition. The emphasis will be on the numerical implementation using the deal.II framework, and on the solution of the nonlinear equations using the PETSc library.
[04869] Platform Portable Distributed Solvers and Preconditioners in Cardiac Simulations
Author(s) :
Fritz Goebel (Karlsruhe Institute of Technology)
Hartwig Anzt (Karlsruhe Institute of Technology)
Terry Cojean (Karlsruhe Institute of technology)
Marcel Koch (Karlsruhe Institute of Technology)
Abstract : In the European MICROCARD project we work on simulating the electrophysiology of the human heart with a new Cell-by-Cell model. The model's high resolution and the resulting linear systems pose a computational challenge. In this talk we report on recent developments on scaleable Krylov Methods and Preconditioners in the open source library Ginkgo that we aim to leverage in these simulations.
