Space-time multilevel quadrature methods and their application for cardiac electrophysiology

We present a novel approach which aims at high-performance uncertainty quantification for cardiac electrophysiology simulations. Employing the monodomain equation to model the transmembrane potential inside the cardiac cells, we evaluate the effect of spatially correlated perturbations of the heart fibers on the statistics of the resulting quantities of interest. Our methodology relies on a close integration of multilevel quadrature methods, parallel iterative solvers and space-time finite element discretizations, allowing for a fully parallelized framework in space, time and stochastics. Extensive numerical studies are presented to evaluate convergence rates and to compare the performance of classical Monte Carlo methods such as standard Monte Carlo (MC) and quasi-Monte Carlo (QMC), as well as multilevel strategies, i.e. multilevel Monte Carlo (MLMC) and multilevel quasi-Monte Carlo (MLQMC) on hierarchies of nested meshes. We especially also employ a recently suggested variant of the multilevel approach for non-nested meshes to deal with a realistic heart geometry. 1 ar X iv :2 10 5. 02 00 7v 1 [ m at h. N A ] 5 M ay 2 02 1

[1]  Helmut Harbrecht,et al.  Efficient approximation of random fields for numerical applications , 2015, Numer. Linear Algebra Appl..

[2]  Michael Griebel,et al.  Multilevel Quadrature for Elliptic Parametric Partial Differential Equations in Case of Polygonal Approximations of Curved Domains , 2015, SIAM J. Numer. Anal..

[3]  Andrea Barth,et al.  Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients , 2011, Numerische Mathematik.

[4]  H. Harbrecht,et al.  On the low-rank approximation by the pivoted Cholesky decomposition , 2012 .

[5]  Nejib Zemzemi,et al.  Decoupled time-marching schemes in computational cardiac electrophysiology and ECG numerical simulation. , 2010, Mathematical biosciences.

[6]  Helmut Harbrecht,et al.  On Multilevel Quadrature for Elliptic Stochastic Partial Differential Equations , 2012 .

[7]  Will Light,et al.  Approximation Theory in Tensor Product Spaces , 1985 .

[8]  Stefan Heinrich,et al.  Multilevel Monte Carlo Methods , 2001, LSSC.

[9]  C. Schwab,et al.  Multilevel Quasi-Monte Carlo Uncertainty Quantification for Advection-Diffusion-Reaction , 2018 .

[10]  G. Plank,et al.  A Novel Rule-Based Algorithm for Assigning Myocardial Fiber Orientation to Computational Heart Models , 2012, Annals of Biomedical Engineering.

[11]  H. Harbrecht,et al.  Multilevel methods for uncertainty quantification of elliptic PDEs with random anisotropic diffusion , 2017, Stochastics and Partial Differential Equations: Analysis and Computations.

[12]  D. Hurtado,et al.  Gradient flows and variational principles for cardiac electrophysiology: Toward efficient and robust numerical simulations of the electrical activity of the heart , 2014 .

[13]  H. HARBRECHT,et al.  Uncertainty Quantification for PDEs with Anisotropic Random Diffusion , 2016, SIAM J. Numer. Anal..

[14]  R. G. Cooke.  Functional Analysis and Semi-Groups , 1949, Nature.

[15]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[16]  M. Griebel,et al.  A Note on the Construction of L-Fold Sparse Tensor Product Spaces , 2013, Constructive Approximation.

[17]  Mark Potse,et al.  A Comparison of Monodomain and Bidomain Reaction-Diffusion Models for Action Potential Propagation in the Human Heart , 2006, IEEE Transactions on Biomedical Engineering.

[18]  Andrew J. Wathen,et al.  A Simple Proposal for Parallel Computation Over Time of an Evolutionary Process with Implicit Time Stepping , 2015, ENUMATH.

[19]  Miguel A. Fernández,et al.  Mathematical Modeling of Electrocardiograms: A Numerical Study , 2010, Annals of Biomedical Engineering.

[20]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[21]  Stefan Heinrich.  The Multilevel Method of Dependent Tests , 2000 .

[22]  Stefano Serra Capizzano,et al.  Space-Time FE-DG Discretization of the Anisotropic Diffusion Equation in Any Dimension: The Spectral Symbol , 2018, SIAM J. Matrix Anal. Appl..

[23]  Martin J. Gander,et al.  50 Years of Time Parallel Time Integration , 2015 .

[24]  David B. Geselowitz,et al.  Simulation Studies of the Electrocardiogram , 2017 .

[25]  Mark Potse,et al.  On Sampling Spatially-Correlated Random Fields for Complex Geometries , 2019, FIMH.

[26]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[27]  Thomas Gerstner,et al.  Dimension- and Time-Adaptive Multilevel Monte Carlo Methods , 2012 .

[28]  Rolf Krause,et al.  Space-time multilevel Monte Carlo methods and their application to cardiac electrophysiology , 2019, J. Comput. Phys..

[29]  Markus Siebenmorgen,et al.  Quadrature methods for elliptic PDEs with random diffusion , 2015 .

[30]  D. Geselowitz,et al.  Simulation Studies of the Electrocardiogram: I. The Normal Heart , 1978, Circulation research.