Stefan problems with nonlinear diffusion and convection

We consider a class of Stefan-type problems having a convection term and a pseudomonotone nonlinear diffusion operator. Assuming data in L¹, we prove existence, uniqueness and stability in the framework of renormalized solutions. Existence is established from compactness and monotonicity arguments which yield stability of solutions with respect to L¹ convergence of the data. Uniqueness is proved through a classical L¹-contraction principle, obtained by a refinement of the doubling variable technique which allows us to extend previous results to a more general class of nonlinear possibly degenerate operators.     

On the one-dimensional two-phase inverse Stefan problems

New formulations of the inverse nonstationary Stefan problems are considered: (a) forx [0,1] (the inverse problem IP₁; (b) forx [0, (t)] with a degenerate initial condition (the inverse problem IP). Necessary conditions for the existence and uniqueness of a solution to these problems are formulated. On the first phase {x [0, y(t)]{, the solution of the inverse problem is found in the form of a series; on the second phase {x [y(t), 1] orx [y(t), (t)]{, it is found as a sum of heat double-layer potentials. By representing the inverse problem in the form of two connected boundary-value problems for the heat conduction equation in the domains with moving boundaries, it can be reduced to the integral Volterra equations of the second kind. An exact solution of the problem IP is found for the self similar motion of the boundariesx=y(t) andx=(t).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1058–1065, August, 1993.     

A Haar wavelets based approximation for nonlinear inverse problems influenced by unknown heat source

In this discussion, a new numerical algorithm focused on the Haar wavelet is used to solve linear and nonlinear inverse problems with unknown heat source. The heat source is dependent on time and space variables. These types of inverse problems are ill-posed and are challenging to solve accurately. The linearization technique converted the nonlinear problem into simple nonhomogeneous partial differential equation. In this Haar wavelet collocation method (HWCM), the time part is discretized by using finite difference approximation, and space variables are handled by Haar series approximation. The main contribution of the proposed method is transforming this ill-posed problem into well-conditioned algebraic equation with the help of Haar functions, and hence, there is no need to implement any sort of regularization technique. The results of numerical method are efficient and stable for this ill-posed problems containing different noisy levels. We have utilized the proposed method on several numerical examples and have valuable efficiency and accuracy.     

Newton methods for nonlinear inverse problems with random noise

We study the convergence of regularized Newton methods applied to nonlinear operator equations in Hilbert spaces if the data are perturbed by random noise. We show that under certain conditions it is possible to achieve the minimax rates of the corresponding linearized problem if the smoothness of the solution is known. If the smoothness is unknown and the stopping index is determined by Lepskij's balancing principle, we show that the rates remain the same up to a logarithmic factor due to adaptation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On some nonlinear elliptic problems with an unknown measure data

We prove the existence of a solution of some nonlinear elliptic problems with a Radon measure data. In contrast with the usual elliptic problem, this measure will be an unknown of the problem depending on the solution. We shall use a Minimax Ambrosetti–Rabinowitz argument. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quasilinearization and inverse problems of nonlinear dynamics

In this paper, we describe a new method for solving the state identification problem associated with a set of ordinary nonlinear differential equations. It is proved that the method has quadratic convergence. We present the results of numerical experiments carried out on two classical models: the Lotka-Volterra system and the chaotic Lorenz model.     

A two-stage method for nonlinear inverse problems

In this work we are interested in the solution of nonlinear inverse problems of the form F(x)=y$F(x)=y$. We consider a two-stage method which is third order convergent for well-posed problems. Combining the method with Levenberg–Marquardt regularization of the linearized problems at each stage and using the discrepancy principle as a stopping criterion, we obtain a regularization method for ill-posed problems. Numerical experiments on some parameter identification and inverse acoustic scattering problems are presented to illustrate the performance of the method.     

Solutions to nonlinear Neumann problems with an inverse square potential

Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem

Quasilinearization and inverse problems for nonlinear control systems

In this paper, we describe a new method for solving the inverse problem associated with a set of ordinary nonlinear differential equations. It is proved that the method has quadratic convergence.     

Proximal algorithm for minimization problems in l0-regularization for nonlinear inverse problems

Numerical Algorithms - In this paper, we study a proximal method for the minimization problem arising from l0-regularization for nonlinear inverse problems. First of all, we prove the existence of...     

Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems

We develop an abstract framework and convergence theory for Galerkin approximation for inverse problems involving the identification of nonautonomous, in general nonlinear, distributed parameter systems. We provide a set of relatively easily verified conditions which are sufficient to guarantee the existence of optimal solutions and their approximation by a sequence of solutions to a sequence of approximating finite-dimensional identification problems. Our approach is based upon the theory of monotone operators in Banach spaces and is applicable to a reasonably broad class of nonlinear distributed systems. Operator theoretic and variational techniques are used to establish a fundamental convergence result. An example involving evolution systems with dynamics described by nonstationary quasi-linear elliptic operators along with some applications and numerical results are presented and discussed.Part of this research was carried out while the first and third authors were visiting scientists at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center. Hampton, VA, which is operated under NASA Contracts NAS1-17070 and NAS1-18107. Also, a portion of this research was carried out with computational resources made available through a grant to the second and third authors from the San Diego Supercomputer Center operated for the National Science Foundation by General Atomics, San Diego, CA. The research of the first author was supported in part under Grants NSF MCS-8504316, NASA NAG-1-517, AFOSR-84-0398, and AFOSR-F49620-86-C-0111. The second author's research was supported in part by the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund. The research of the third author was supported in part under Grants AFOSR-84-0393 and AFOSR-87-0356.