Cubically convergent methods for selecting the regularization parameters in linear inverse problems

AbstractWe present three cubically convergent methods for choosing the regularization parameters in linear inverse problems. The detailed algorithms are given and the convergence rates are estimated. Our basic tools are Tikhonov regularization and Morozov's discrepancy principle. We prove that, in comparison with the standard Newton method, the computational costs for our cubically convergent methods are nearly the same, but the number of iteration steps is even less. Numerical experiments for an elliptic boundary value problem illustrate the efficiency of the proposed algorithms

Numerical analysis of a time discretized method for nonlinear filtering problem with L\'evy process observations

In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is a unnormalized probability density function of the filter solution. Then we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis

ON COMPUTING HETEROCLINIC TRAJECTORIES OF NON-AUTONOMOUS MAPS

Hüls T, Zou Y. ON COMPUTING HETEROCLINIC TRAJECTORIES OF NON-AUTONOMOUS MAPS. Discrete and Continuous Dynamical Systems - Series B. 2012;17(1):79-99.We propose an adequate notion of a heteroclinic trajectory in non-autonomous systems that generalizes the notion of a heteroclinic orbit of an autonomous system. A heteroclinic trajectory connects two families of semi-bounded trajectories that are bounded in backward and forward time. We apply boundary value techniques for computing one representative of each family. These approximations allow the construction of projection boundary conditions that enable the calculation of a heteroclinic trajectory with high accuracy. The resulting algorithm is applied to non-autonomous toy models as well as to an example from mathematical biology

The Spectral Method for the Cahn-Hilliard Equation with Concentration-Dependent Mobility

This paper is concerned with the numerical approximations of the Cahn-Hilliard-type equation with concentration-dependent mobility. Convergence analysis and error estimates are presented for the numerical solutions based on the spectral method for the space and the implicit Euler method for the time. Numerical experiments are carried out to illustrate the theoretical analysis

On the existence of transversal heteroclinic orbits in discretized dynamical systems

Zou Y, Beyn W-J. On the existence of transversal heteroclinic orbits in discretized dynamical systems. Nonlinearity. 2004;17(6):2275-2292.In this paper we prove the existence of transversal heteroclinic orbits for maps that are obtained from one-step methods applied to a continuous dynamical system. It is assumed that the continuous system exhibits a heteroclinic orbit at a specific value of a parameter. While it is known that analytic vector fields lead to exponentially small splittings of separatrices in the discrete system, we analyse here the case of a continuous system that is smooth of finite order only. Assuming that a certain derivative has a jump discontinuity at a specific hyperplane we show that discretized systems have transversal heteroclinic orbits. The essential step in deriving such a result is a refinement of a previously developed error analysis that applies exponential dichotomy and Fredholm techniques to the discretized system

Generalized Hopf bifurcation for planar Filippov systems continuous at the origin

Zou Y, Küpper T, Beyn W-J. Generalized Hopf bifurcation for planar Filippov systems continuous at the origin. Journal of Nonlinear Science. 2006;16(2):159-177.In this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical method to investigate the existence of periodic orbits that are obtained by searching for the fixed points of return maps

Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j+1 with j≥0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis

Numerical analysis of degenerate connecting orbits for maps

Beyn W-J, Hüls T, Kleinkauf JM, Zou Y. Numerical analysis of degenerate connecting orbits for maps. International Journal of Bifurcation and Chaos. 2004;14(10):3385-3407.This paper contains a survey of numerical methods for connecting orbits in discrete dynamical systems. Special emphasis is put on degenerate cases where either the orbit. loses transversality or one of its endpoints loses hyperbolicity. Numerical methods that, approximate the connecting orbits by finite orbit sequences are described in detail and theoretical results on the error analysis are provided. For most of the degenerate cases we present examples and numerical results that, illustrate the applicability of the methods and the validity of the error estimates