Fourier regularization for a backward heat equation

In this paper a simple and convenient new regularization method for solving backward heat equation—Fourier regularization method is given. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.     

Frozen Landweber Iteration for Nonlinear Ill-Posed Problems

In this paper we propose a modification of the Landweber iteration termed frozen Landweberiteration for nonlinear ill-posed problems.A convergence analysis for this iteration is presented.The numericalperformance of this frozen Landweber iteration for a nonlinear Hammerstein integral equation is compared withthat of the Landweber iteration.We obtain a shorter running time of the frozen Landweber iteration based onthe same convergence accuracy.     

Parameter choice by discrepancy principles for ill-posed problems leading to optimal convergence rates

Schock (Ref. 1) considered a general a posteriori parameter choice strategy for the Tikhonov regularization of the ill-posed operator equationTx=y which provides nearly the optimal rate of convergence if the minimal-norm least-squares solution belongs to the range of the operator (T ^* T) ^v , o<v1. Recently, Nair (Ref. 2) improved the result of Schock and also provided the optimal rate ifv=1. In this note, we further improve the result and show in particular that the optimal rate can be achieved for 1/2v1.The final version of this work was written while M. T. Nair was a Visiting Fellow at the Centre for Mathematics and Its Applications, Australian National University, Canberra, Australia. The work of S. George was supported by a Senior Research Fellowship from CSIR, India.     

A new method for numerical differentiation based on direct and inverse problems of partial differential equations

In this paper, we mainly study a numerical differentiation problem which aims to approximate the second order derivative of a single variable function from its noise data. By transforming the problem into a combination of direct and inverse problems of partial differential equations (heat conduction equations), a new method that we call the PDEs-based numerical differentiation method is proposed. By means of the finite element method and the Tikhonov regularization, implementations of the proposed PDEs-based method are presented with a posterior strategy for choosing regularization parameters. Numerical results show that the PDEs-based numerical differentiation method is highly feasible and stable with respect to data noise.     

Two numerical methods for a Cauchy problem for modified Helmholtz equation

We consider a Cauchy problem for the modified Helmholtz equation. A conditional stability estimate is proved. Two order optimal regularization methods and error estimates are investigated. Numerical experiment shows that these methods work well.     

A new L-curve for ill-posed problems

The truncated singular value decomposition is a popular method for the solution of linear ill-posed problems. The method requires the choice of a truncation index, which affects the quality of the computed approximate solution. This paper proposes that an L-curve, which is determined by how well the given data (right-hand side) can be approximated by a linear combination of the first (few) left singular vectors (or functions), be used as an aid for determining the truncation index.     

具有变号非线性项的奇异二阶三点边值问题的三个非零正解

该文研究一类具有变号非线性项的奇异二阶三点边值问题多个正解的存在性. 运用Leggett-Williams不动点定理在$f$满足一定的增长条件下, 获得了至少三个非零正解存在的结果.     

Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space

An approach for solving Fredholm integral equations of the first kind is proposed for in a reproducing kernel Hilbert space (RKHS). The interest in this problem is strongly motivated by applications to actual prospecting. In many applications one is puzzled by an ill-posed problem in space C[a,b] or L²[a,b], namely, measurements of the experimental data can result in unbounded errors of solutions of the equation. In this work, the representation of solutions for Fredholm integral equations of the first kind is obtained if there are solutions and the stability of solutions is discussed in RKHS. At the same time, a conclusion is obtained that approximate solutions are also stable with respect to _∞ or _L² in RKHS. A numerical experiment shows that the method given in the work is valid.     

Modified quasi-reversibility method for final value problems in Banach spaces

This paper is concerned with the final value problem associated with a linear operator A in a Banach space, where −A is the generator of a uniformly bounded analytic semigroup. Based on the deLaubenfels' functional calculus, we use new quasi-reversibility method, introduced by Boussetila and Rebbani recently, to form an approximate problem. We obtain some results in a Banach space similar to those in a Hilbert space.     

Corrigendum to “Discrepancy principle for the dynamical systems method” [Communications in Nonlinear Science and Numerical Simulation 10 (2005) 95–101]

Consider an operator equation B(u) − f = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator B′(u) is not boundedly invertible, and well-posed otherwise. The dynamical systems method (DSM) for solving this equation consists of a construction of a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends to infinity, (3) the limit is the minimal-norm solution to the equation B(u) = f. A global convergence theorem is proved for DSM for equation B(u) − f = 0 with monotone operators B.