Nonlinear Reaction-diffusion Partial Differential Equation Research Articles

Parallel Newton--Krylov BDDC and FETI-DP Deluxe Solvers for Implicit Time discretizations of the Cardiac Bidomain Equations

Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A finite element discretization in space and a segregated implicit discretization in time, based on decoupling the PDEs from the ODEs, yields at each time step the solution of a non-linear algebraic system. The Jacobian linear system at each Newton iteration is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced {\em deluxe} scaling of the dual variables. A polylogarithmic convergence rate bound is proven for the resulting parallel Bidomain solvers. Extensive numerical experiments on linux clusters up to two thousands processors confirm the theoretical estimates, showing that the proposed parallel solvers are scalable and quasi-optimal.

Uncertain Linear Stationary Model of Population Density in a Bounded Habitat

The purpose of this article is to identify an uncertain linear time-invariant (LTI) model of the spatial distribution of a specimen over 1-D habitual space. Population density is described by a nonlinear reaction-diffusion partial differential equation (PDE) following the population dynamics theory. The nonlinear PDE is approximated with a linear parameter varying state-space model from the 120th order with two inputs and 100 outputs, which describe the spatial distribution where the manipulated variable in the model is the length of the habitat. A random excitation signal is applied corresponding to expansion or contraction of the habitual space to assess the dynamical response of the population density. The collected data are used to estimate 100 Box and Jenkins (BJ) models, which describe the reaction of population density across the habitat. Then, a single uncertain LTI model is identified from the magnitude frequency response of the BJ models to cover all of them.

Stabilized hybrid discontinuous Galerkin finite element method for the cardiac monodomain equation.

Numerical methods for solving the cardiac electrophysiology model, which describes the electrical activity in the heart, are proposed. The model problem consists of a nonlinear reaction-diffusion partial differential equation coupled to systems of ordinary differential equations that describes electrochemical reactions in cardiac cells. The proposed methods combine an operator splitting technique for the reaction-diffusion equation with primal hybrid methods for spatial discretization considering continuous or discontinuous approximations for the Lagrange multiplier. A static condensation is adopted to form a reduced global system in terms of the multiplier only. Convergence studies exhibit optimal rates of convergence and numerical experiments show that the proposed schemes can be more efficient than standard numerical techniques commonly used in this context when preconditioned iterative methods are used for the solution of linear systems.

Exponential Stability for the Schl\xf6gl System by Pyragas Feedback

The Schlögl system is governed by a nonlinear reaction-diffusion partial differential equation with a cubic nonlinearity. In this paper, feedback laws of Pyragas-type are presented that stabilize the system in a periodic state with a given period and given boundary traces. We consider the system both with boundary feedback laws of Pyragas type and distributed feedback laws of Pyragas and classical type. Stabilization to periodic orbits is important for medical applications that concern Parkinson’s disease. The exponential stability of the closed loop system with respect to the L2-norm is proved. Numerical examples are provided.

Challenges and opportunities for the simulation of calcium waves on modern multi-core and many-core parallel computing platforms.

State-of-the-art distributed-memory computer clusters contain multicore CPUs with 16 and more cores. The second generation of the Intel Xeon Phi many-core processor has more than 60cores with 16GB of high-performance on-chip memory. We contrast the performance of the second-generation Intel Xeon Phi, code-named Knights Landing (KNL), with 68computational cores to the latest multicore CPU Intel Skylake with 18cores. A special-purpose code solving a system of nonlinear reaction-diffusion partial differential equations with several thousands of point sources modeled mathematically by Dirac delta distributions serves as realistic test bed. The system is discretized in space by the finite volume method and advanced by fully implicit time-stepping, with a matrix-free implementation that allows the complex model to have an extremely small memory footprint. The sample application is a seven variable model of calcium-induced calcium release (CICR) that models the interplay between electrical excitation, calcium signaling, and mechanical contraction in a heart cell. The results demonstrate that excellent parallel scalability is possible on both hardware platforms, but that modern multicore CPUs outperform the specialized many-core Intel Xeon Phi KNL architecture for a large class of problems such as systems of parabolic partial differential equations.

Global Stabilization of a Class of Nonlinear Reaction-Diffusion Partial Differential Equations by Boundary Feedback

This paper provides global exponential stabilization results by means of boundary feedback control for one dimensional nonlinear unstable reaction-diffusion partial differential equations (PDEs) with nonlinearities of superlinear growth. The class of systems studied are parabolic PDEs with nonlinear reaction terms that provide “damping” when the norm of the state is large (the class includes reaction-diffusion PDEs with polynomial nonlinearities). The case of Dirichlet actuation at one end of the domain is considered and a Control-Lypunov Functional construction is applied in conjunction with Stampacchia's truncation method. The paper also provides several important auxiliary results, among which is an extension of Wirtinger's inequality, used here for the construction of the Control-Lyapunov functional.

Gradient and Laplacian-Based Hyperspectral Anisotropic Diffusion

To improve accuracy and efficiency of object detection and classification with hyperspectral imagery (HSI), we propose a novel smoothing algorithm by coupling of a Laplacian-based reaction term to a classical divergence-based anisotropic diffusion partial differential equation (PDE). In addition, an adaptive parameter is introduced to regularize this nonlinear reaction-diffusion PDE by explicitly integrating the interband correlations with the noise level of each band in HSI. It is also well-known that the interband correlations can be implicitly embedded into the diffusion coefficient of the divergence-based PDE, to allow a selective smoothing that reduces the local homogeneous area variability while preventing smoothing across edges. Therefore, the interband correlations in HSI are exploited in the proposed method in both explicit and implicit ways. As a result, our algorithm is more effective at controlling the behavior of the diffusion evolution when compared to previous multi/hyperspectral diffusion algorithms. The simulations based on both synthetic data and real hyperspectral remote sensing data show that our algorithm can improve the hyperspectral data quality in terms of both visual inspection and image quality indices.

Boundary feedback stabilization of the Schlögl system

The Schlögl system is governed by a nonlinear reaction–diffusion partial differential equation with a cubic nonlinearity that determines three constant equilibrium states. It is a classical example of a chemical reaction system that is bistable. The constant equilibrium that is enclosed by the other two constant equilibrium points is unstable.In this paper, Robin boundary feedback laws are presented that stabilize the system in a given stationary state or more generally in a given time-dependent desired system orbit. The exponential stability of the closed loop system with respect to the L2-norm is proved. In particular, it is shown that with the boundary feedback law the unstable constant equilibrium point can be stabilized.

ADER finite volume schemes for nonlinear reaction–diffusion equations

We construct finite volume schemes of arbitrary order of accuracy in space and time for solving nonlinear reaction–diffusion partial differential equations. The numerical schemes, written in conservative form, result from extending the Godunov and the ADER frameworks, both originally developed for approximating solutions to hyperbolic equations. The task is to define numerical fluxes and numerical sources. In the ADER approach, numerical fluxes are computed from solutions to the Derivative Riemann Problem (DRP) (or generalized Riemann problem, or high-order Riemann problem), the Cauchy problem in which the initial conditions either side of the interface are smooth functions, polynomials of arbitrary degree, for example. We propose, and systematically asses, a general DRP solver for nonlinear reaction–diffusion equations and construct corresponding finite volume schemes of arbitrary order of accuracy. Schemes of 1st to 10-th order of accuracy in space and time are implemented and systematically assessed, with particular attention paid to their convergence rates. Numerical examples are also given.

On the modelling of solid state reactions.Synthesis of YAG

There is a model of yttrium aluminium garnet (YAG) synthesis presented in this article. The developed model is based on nonlinear reaction–diffusion partial differential equations. The solution was carried out numerically using finite difference techniques. We got dependability curves for diffusion and reaction rates and offered possible method to localize values of diffusion and reaction rate constants precisely enough.

Periodic travelling waves in nonlinear reaction–diffusion equations via multiple Hopf bifurcation

We keep track of periodic wave trains for some classes of one dimensional nonlinear reaction–diffusion partial differential equations. These periodic wave trains can be seen as limit cycles of some planar differential systems. We use standard techniques within the multiple Hopf bifurcation framework.