The quasi-potential is a key function in the Large Deviation Theory. It characterizes the difficulty of the escape from the neighborhood of an attractor of a stochastic non-gradient dynamical system due to the influence of small white noise. It also gives an estimate of the invariant probability distribution in the neighborhood of the attractor up to the exponential order. We present a new family of methods for computing the quasi-potential on a regular mesh named the Ordered Line Integral Methods (OLIMs). In comparison with the first proposed quasi-potential finder based on the Ordered Upwind Method (OUM) (Cameron, 2012), the new methods are 1.5 to 4 times faster, can produce error two to three orders of magnitude smaller, and may exhibit faster convergence. Similar to the OUM, OLIMs employ the dynamical programming principle. Contrary to it, they (i) have an optimized strategy for the use of computationally expensive triangle updates leading to a notable speed-up, and (ii) directly solve local minimization problems using quadrature rules instead of solving the corresponding Hamilton-Jacobi-type equation by the first order finite difference upwind scheme. The OLIM with the right-hand quadrature rule is equivalent to OUM. The use of higher order quadrature rules in local minimization problems dramatically boosts up the accuracy of OLIMs. We offer a detailed discussion on the origin of numerical errors in OLIMs and propose rules-of-thumb for the choice of the important parameter, the update factor, in the OUM and OLIMs. Our results are supported by extensive numerical tests on two challenging 2D examples.

Nongradient stochastic differential equations (SDEs) with position-dependent and anisotropic diffusion are often used in biological modeling. The quasi-potential is a crucial function in the Large Deviation Theory that allows one to estimate transition rates between attractors of the corresponding ordinary differential equation and find the maximum likelihood transition paths. Unfortunately, the quasi-potential can rarely be found analytically. It is defined as the solution to a certain action minimization problem. In this work, the recently introduced Ordered Line Integral Method (OLIM) is extended for computing the quasi-potential for 2D SDEs with anisotropic and position-dependent diffusion scaled by a small parameter on a regular rectangular mesh. The presented solver employs the dynamical programming principle. At each step, a local action minimization problem is solved using straight line path segments and the midpoint quadrature rule. The solver is tested on two examples where analytic formulas for the quasi-potential are available. The dependence of the computational error on the mesh size, the update factor K (a key parameter of OLIMs), as well as the degree and the orientation of anisotropy is established. The effect of anisotropy on the quasi-potential and the maximum likelihood paths is demonstrated on the Maier-Stein model. The proposed solver is applied to find the quasi-potential and the maximum likelihood transition paths in a model of the genetic switch in Lambda Phage between the lysogenic state where the phage reproduces inside the infected cell without killing it, and the lytic state where the phage destroys the infected cell.

No abstract

The fast marching method is computationally efficient in approximating the viscosity solution of the eikonal equation in the case of unidirectional wavefront propagation through a medium at rest. The main assumption of this method is that the front propagates only in its normal direction, which is the case when the medium of propagation is at rest. In many real-time applications, the medium may be occupied with a moving fluid. In such cases, the governing equation is a generalized (anisotropic) eikonal equation. The main assumption of the fast marching method may not hold in this case, since the front may propagate in both the tangential and the normal direction. This leads to instability in the fast marching method due to violation of the upwind criterion. In this work, we develop a fast marching method for the generalized eikonal equation, called the characteristic fast marching method, where the upwind criterion is achieved using the characteristic direction of the propagating wavefront at each grid point. We suitably modify the narrow band algorithm of the fast marching method so that the anisotropic nature of the medium is incorporated in the method. We compare the numerical results obtained from our method with the solution obtained using the ray theory (geometrical optics theory) to show that the method accurately captures the viscosity solution of the generalized eikonal equation. We apply the method to study the propagation of a wavefront in a medium with a cavity and also study the merging of two wavefronts from different sources. The method can easily be generalized to higher order approximations. We develop a method with second order finite difference approximation and study the rate of convergence numerically.

No abstract