Fenglian Yang

A meshless method for solving an inverse spacewise-dependent heat source problem

In this paper an effective meshless and integration-free numerical scheme for solving an inverse spacewise-dependent heat source problem is proposed. Due to the use of the fundamental solution as basis functions, the method leads to a global approximation scheme in both spatial and time domains. The standard Tikhonov regularization technique with the generalized cross-validation criterion for choosing the regularization parameter is adopted for solving the resulting ill-conditioned system of linear algebraic equations. The effectiveness of the algorithm is illustrated by several numerical examples.

A Bayesian inference approach to identify a Robin coefficient in one-dimensional parabolic problems

This paper investigates a nonlinear inverse problem associated with the heat conduction problem of identifying a Robin coefficient from boundary temperature measurement. A Bayesian inference approach is presented for the solution of this problem. The prior modeling is achieved via the Markov random field (MRF). The use of a hierarchical Bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed. The Markov chain Monte Carlo (MCMC) algorithm is used to explore the posterior state space. Numerical results indicate that MRF provides an effective prior regularization, and the Bayesian inference approach can provide accurate estimates as well as uncertainty quantification to the solution of the inverse problem.

Reconstruction of part of a boundary for the Laplace equation by using a regularized method of fundamental solutions

In this article, the identification of part of a boundary for the two-dimensional Laplace equation is investigated. One regularized method of fundamental solutions is used for determining an unknown portion of the boundary from the Cauchy data specified on a part of the boundary. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing the Tikhonov regularization technique, while the regularization parameter for the regularization method is provided by the generalized cross-validation criterion. The numerical results show that the proposed method produces a convergent and stable solution.

An adaptive greedy technique for inverse boundary determination problem

In this paper, the method of fundamental solutions (MFS) is employed for determining an unknown portion of the boundary from the Cauchy data specified on parts of the boundary. We propose a new numerical method with adaptive placement of source points in the MFS to solve the inverse boundary determination problem. Since the MFS source points placement here is not trivial due to the unknown boundary, we employ an adaptive technique to choose a sub-optimal arrangement of source points on various fictitious boundaries. Afterwards, the standard Tikhonov regularization method is used to solve ill-conditional matrix equation, while the regularization parameter is chosen by the L-curve criterion. The numerical studies of both open and closed fictitious boundaries are considered. It is shown that the proposed method is effective and stable even for data with relatively high noise levels.

Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method

In this paper, we consider the identification of a corrosion boundary for the two-dimensional Laplace equation. A boundary collocation method is proposed for determining the unknown portion of the boundary from the Cauchy data on a part of the boundary. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing the Tikhonov regularization technique, while the regularization parameter is provided by the generalized cross-validation criterion. Numerical examples show that the proposed method is reasonable and feasible.

The method of approximate particular solutions for the time-fractional diffusion equation with a non-local boundary condition

In this paper, we consider the numerical solution of the time-fractional diffusion equation with a non-local boundary condition. The method of approximate particular solutions (MAPS) using multiquadric radial basis function (MQ-RBF) is employed for this equation. Due to the accuracy of the MQ-based meshless methods is severely influenced by the shape parameter, we adopt a leave-one-out cross validation (LOOCV) algorithm proposed by Rippa 34] to enhance the performance of the MAPS. The numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time-fractional diffusion equation with a non-local boundary condition.

Efficient Kansa-type MFS algorithm for time-fractional inverse diffusion problems

In this study we propose an efficient Kansa-type method of fundamental solutions (MFS-K) for the numerical solution of time-fractional inverse diffusion problems. By approximating the time-fractional derivative through a finite difference scheme, the time-fractional inverse diffusion problem is transformed into a sequence of Cauchy problems associated with inhomogeneous elliptic-type equations, which can be conveniently solved using the MFS-K. Since the matrix arising from the MFS-K discretization is highly ill-conditioned, a regularized solution is obtained by employing the Tikhonov regularization method, while the choice of the regularization parameter is based on the generalized cross-validation criterion. Numerical results are presented for several examples with smooth and piecewise smooth boundaries. The stability of the method with respect to the noise in the data is investigated.

Doubly stochastic radial basis function methods

Abstract We propose a doubly stochastic radial basis function (DSRBF) method for function recoveries. Instead of a constant, we treat the RBF shape parameters as stochastic variables whose distribution were determined by a stochastic leave-one-out cross validation (LOOCV) estimation. A careful operation count is provided in order to determine the ranges of all the parameters in our methods. The overhead cost for setting up the proposed DSRBF method is O ( n 2 ) for function recovery problems with n basis. Numerical experiments confirm that the proposed method not only outperforms constant shape parameter formulation (in terms of accuracy with comparable computational cost) but also the optimal LOOCV formulation (in terms of both accuracy and computational cost).