Reducing computational time via order reduction of a class of reaction–diffusion systems

In this paper, we consider a class of reaction– diffusion PDEs. For this class, a suitable state transformation allows conversion to a heat equation together with a lower order PDE set. By giving an explicit solution to the heat equation we are able to obtain a complete solution to the original PDE. By focusing on the computational load, we give a comparison of the pure numerical, analytical/numerical, analytical/approximated, and approximated methods of solving the PDE. In some examples, we note an almost order of magnitude improvement in computational load

Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics

We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics

A class of high-order Runge-Kutta-Chebyshev stability polynomials

The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC) stability polynomials of arbitrary order $N$ is presented. Roots of FRKC stability polynomials of degree $L=MN$ are used to construct explicit schemes comprising $L$ forward Euler stages with internal stability ensured through a sequencing algorithm which limits the internal amplification factors to $\sim L^2$. The associated stability domain scales as $M^2$ along the real axis. Marginally stable real-valued points on the interior of the stability domain are removed via a prescribed damping procedure. By construction, FRKC schemes meet all linear order conditions; for nonlinear problems at orders above 2, complex splitting or Butcher series composition methods are required. Linear order conditions of the FRKC stability polynomials are verified at orders 2, 4, and 6 in numerical experiments. Comparative studies with existing methods show the second-order unsplit FRKC2 scheme and higher order (4 and 6) split FRKCs schemes are efficient for large moderately stiff problems.Comment: 24 pages, 5 figures. Accepted for publication in Journal of Computational Physics, 22 Jul 2015. Revise

Goal-oriented sensitivity analysis for lattice kinetic Monte Carlo simulations

In this paper we propose a new class of coupling methods for the sensitivity analysis of high dimensional stochastic systems and in particular for lattice Kinetic Monte Carlo. Sensitivity analysis for stochastic systems is typically based on approximating continuous derivatives with respect to model parameters by the mean value of samples from a finite difference scheme. Instead of using independent samples the proposed algorithm reduces the variance of the estimator by developing a strongly correlated-"coupled"- stochastic process for both the perturbed and unperturbed stochastic processes, defined in a common state space. The novelty of our construction is that the new coupled process depends on the targeted observables, e.g. coverage, Hamiltonian, spatial correlations, surface roughness, etc., hence we refer to the proposed method as em goal-oriented sensitivity analysis. In particular, the rates of the coupled Continuous Time Markov Chain are obtained as solutions to a goal-oriented optimization problem, depending on the observable of interest, by considering the minimization functional of the corresponding variance. We show that this functional can be used as a diagnostic tool for the design and evaluation of different classes of couplings. Furthermore the resulting KMC sensitivity algorithm has an easy implementation that is based on the Bortz-Kalos-Lebowitz algorithm's philosophy, where here events are divided in classes depending on level sets of the observable of interest. Finally, we demonstrate in several examples including adsorption, desorption and diffusion Kinetic Monte Carlo that for the same confidence interval and observable, the proposed goal-oriented algorithm can be two orders of magnitude faster than existing coupling algorithms for spatial KMC such as the Common Random Number approach

Kinetic Intersections as a Source of Information on Rate Coefficients

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Multi-Lattice Kinetic Monte Carlo Simulations from First-Principles: Reduction of the Pd(100) Surface Oxide by CO

We present a multi-lattice kinetic Monte Carlo (kMC) approach that efficiently describes the atomistic dynamics of morphological transitions between commensurate structures at crystal surfaces. As an example we study the reduction of a $(\sqrt{5}\times \sqrt{5})R27^{\circ}$ PdO(101) overlayer on Pd(100) in a CO atmosphere. Extensive density-functional theory calculations are used to establish an atomistic pathway for the oxide reduction process. First-principles multi-lattice kMC simulations on the basis of this pathway fully reproduce the experimental temperature dependence of the reduction rate [Fernandes et al., Surf. Sci. 2014, 621, 31-39] and highlight the crucial role of elementary processes special to the boundary between oxide and metal domains.Comment: 19 pages, 10 figure