A parallel Newton-Krylov method for optimal control of the monodomain model in cardiac electrophysiology

This work addresses an optimal control approach for a model problem in cardiac electrophysiology with the goal of extinction of a reentry phenomenon. After the introduction of the mathematical model, the derivation of the optimality system, the description of its discretization and a numerical feasibility study in a parallel environment are provided.

[1]  Rodrigo Weber dos Santos,et al.  Parallel multigrid preconditioner for the cardiac bidomain model , 2004, IEEE Transactions on Biomedical Engineering.

[2]  George Biros,et al.  Parallel Lagrange-Newton-Krylov-Schur Methods for PDE-Constrained Optimization. Part I: The Krylov-Schur Solver , 2005, SIAM J. Sci. Comput..

[3]  P. C. Franzone,et al.  A PARALLEL SOLVER FOR REACTION-DIFFUSION SYSTEMS IN COMPUTATIONAL ELECTROCARDIOLOGY , 2004 .

[4]  Karl Kunisch,et al.  Second Order Methods for Optimal Control of Time-Dependent Fluid Flow , 2001, SIAM J. Control. Optim..

[5]  H. ChenT,et al.  A Quasi-Infinite Horizon Nonlinear Model Predictive Control Scheme with Guaranteed Stability * , 1998 .

[6]  Joakim Sundnes,et al.  Computing the electrical activity in the heart , 2006 .

[7]  Jens Lang,et al.  Konrad-zuse-zentrum F ¨ Ur Informationstechnik Berlin Adaptivity in Space and Time for Reaction-diffusion Systems in Electrocardiology Adaptivity in Space and Time for Reaction-diffusion Systems in Electrocardiology , 2022 .

[8]  K. Kunisch,et al.  Numerical Solutions for Optimal Control of Monodomain Equations in Cardiac Electrophysiology , 2010 .

[9]  A. McCulloch,et al.  A collocation-Galerkin finite element model of cardiac action potential propagation , 1994, IEEE Transactions on Biomedical Engineering.

[10]  Wim Michiels,et al.  Recent Advances in Optimization and its Applications in Engineering , 2010 .

[11]  K. Kunisch,et al.  Higher order optimization and adaptive numerical solution for optimal control of monodomain equations in cardiac electrophysiology , 2011 .

[12]  K. Kunisch,et al.  Optimal control of the bidomain system (I): The monodomain approximation with the Rogers–McCulloch model , 2012 .

[13]  Gernot Plank,et al.  Near-real-time simulations of biolelectric activity in small mammalian hearts using graphical processing units , 2009, 2009 Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[14]  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.

[15]  N Trayanova,et al.  Termination of Spiral Waves with Biphasic Shocks: , 2000, Journal of cardiovascular electrophysiology.

[16]  Y. Bourgault,et al.  Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology , 2009 .

[17]  J. Trangenstein,et al.  Operator splitting and adaptive mesh refinement for the Luo-Rudy I model , 2004 .

[18]  Aslak Tveito,et al.  Optimal monodomain approximations of the bidomain equations , 2007, Appl. Math. Comput..

[19]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[20]  Alvin Shrier,et al.  Optical mapping of pacemaker interactions , 1999 .

[21]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[22]  A. Winfree Heart Muscle as a Reaction–Diffusion Medium: The Roles of Electric Potential Diffusion, Activation Front Curvature, and Anisotropy , 1997 .

[23]  Andreas Dedner,et al.  A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework , 2008, Computing.

[24]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[25]  Boris Vexler,et al.  Adaptive Space-Time Finite Element Methods for Parabolic Optimization Problems , 2007, SIAM J. Control. Optim..

[26]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[27]  Jens Lang,et al.  Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems - Theory, Algorithm, and Applications , 2001, Lecture Notes in Computational Science and Engineering.

[28]  Michael Hintermüller,et al.  Goal-oriented adaptivity in control constrained optimal control of partial differential equations , 2008, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[29]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[30]  M. Heinkenschloss,et al.  Real-Time PDE-Constrained Optimization , 2007 .

[31]  Andreas Dedner,et al.  A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE , 2008, Computing.

[32]  Luca F. Pavarino,et al.  Multilevel Additive Schwarz Preconditioners for the Bidomain Reaction-Diffusion System , 2008, SIAM J. Sci. Comput..

[33]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[34]  M. Rojas-Medar,et al.  Theoretical analysis and control results for the Fitzhugh-Nagumo equation , 2008 .

[35]  Karl Kunisch,et al.  Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology , 2011, Comput. Optim. Appl..

[36]  Marco Veneroni,et al.  Reaction–diffusion systems for the macroscopic bidomain model of the cardiac electric field , 2009 .

[37]  G Plank,et al.  Solvers for the cardiac bidomain equations. , 2008, Progress in biophysics and molecular biology.

[38]  Ivan P. Gavrilyuk,et al.  Lagrange multiplier approach to variational problems and applications , 2010, Math. Comput..

[39]  K. Kunisch,et al.  Receding horizon optimal control for infinite dimensional systems , 2002 .

[40]  M. Allessie,et al.  Circus Movement in Rabbit Atrial Muscle as a Mechanism of Tachycardia , 1973, Circulation research.

[41]  Rodrigo Weber dos Santos,et al.  Algebraic Multigrid Preconditioner for the Cardiac Bidomain Model , 2007, IEEE Transactions on Biomedical Engineering.