Thamban M Nair

Regularized versions of continuous analogues of Newton's method and modified Newton's method for obtaining approximate solutions to a nonlinear ill-posed operator equation of the form F(u)=f, where F is a monotone operator defined from a Hilbert space H into itself, have been studied in the literature. For such methods, error estimates are available only under Holder-type source conditions on the solution. In this paper, presenting the background materials systematically, we derive error estimates under a general source condition. For the special case of the regularized modified Newton's method under a Holder-type source condition, we also carry out error analysis by replacing the monotonicity of F by a weaker assumption. This analysis facilitates inclusion of certain examples of parameter identification problems, which was not possible otherwise. Moreover, an a priori stopping rule is considered when we have a noisy data f instead of f. This rule yields not only convergence of the regularized approximations to the exact solution as the noise level tends to zero but also provides convergence rates that are optimal under the source conditions considered.

In order to obtain regularized approximations for the solution q of the parameter identification problem -∇.(q∇u) = f in Ω along with the Neumann boundary condition q ∂u/∂v = g on ∂Ω, which is an ill-posed problem, we consider its weak formulation as a linear operator equation with operator as a function of the data u ∈ W1,∞(Ω), and then apply the Tikhonov regularization and a finite-dimensional approximation procedurewhen the data is noisy. Here, Ω is a bounded domain inRd with Lipschitz boundary, f ∈ L2(Ω) and g ∈ H-1/2(∂ Ω). This approach is akin to the equation error method of Al-Jamal and Gockenback (2012) wherein error estimates are obtained in terms of a quotient norm, whereas our procedure facilitates to obtain error estimates in terms of the regularization parameters and data errors with respect to the norms of the spaces under consideration. In order to obtain error estimates when the noisy data belongs to L2(Ω) instead of W1,∞(Ω), we shall make use of a smoothing procedure using the Clement operator under additional assumptions of Ω and u.

In this paper we consider a collocation method for solving Fredholm integral equations of the first kind, which is known to be an ill-posed problem. An "unregularized" use of this method can give reliable results in the case when the rate at which smallest singular values of the collocation matrices decrease is known a priori. In this case the number of collocation points plays the role of a regularization parameter. If the a priori information mentioned above is not available, then a combination of collocation with Tikhonov regularization can be the method of choice. We analyze such regularized collocation in a rather general setting, when a solution smoothness is given as a source condition with an operator monotone index function. This setting covers all types of smoothness studied so far in the theory of Tikhonov regularization. One more issue discussed in this paper is an a posteriori choice of the regularization parameter, which allows us to reach an optimal order of accuracy for deterministic noise model without any knowledge of solution smoothness. © 2006 Elsevier Inc. All rights reserved.

We consider the spectral truncation as the regularization for an ill-posed non-homogeneous parabolic final value problem, and obtain error estimates under a general source condition when the data, which consist of the non-homogeneous term as well as the final value, are noisy. The resulting error estimate is compared with the corresponding estimate under the Lavrentieve method, and showed that the truncation method has no index of saturation.

The non homogeneous backward Cauchy problem ut+Au=f(t), u(τ)=φ for 0≤t<τ is considered, where A is a densely defined positive self-adjoint unbounded operator on a Hilbert space H with f∈L1([0, τ], H) and φ∈H is known to be an ill-posed problem. A truncated spectral representation of the mild solution of the above problem is shown to be a regularized approximation, and error analysis is considered when both φ and f are noisy. Error estimates are derived under appropriate choice of the regularization parameter. The results obtained unify and generalize many of the results available in the literature.

This paper is devoted to the problem of determining the initial data for the backward non-homogeneous time fractional heat conduction problem by the Fourier truncation method. The exact solution for the forward and backward fractional heat problems is expressed in terms of eigen function expansion and Mittag–Leffler function. Due to the instability of determining initial data, a regularized truncated solution is considered. Further, the stability estimate for the exact solution and the convergence estimates for the regularized solution using an á-priori choice rule and an á-posteriori choice rule are derived.

For solving linear ill-posed problems with noisy data regularization methods are required. We analyze a simplified regularization scheme in Hilbert scales for operator equations with nonnegative self-adjoint operators. By exploiting the operator monotonicity of certain functions, order-optimal error bounds are derived that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothness conditions. © 2008, Institute of Mathematics, NAS of Belarus. All rights reserved.

A multilevel augmentation method is considered to solve parameter identification problems in elliptic systems. With the help of the natural linearization technique, the identification problems can be transformed into a linear ill-posed operation equation, where noise exists not only in RHS data but also in operators. Based on multiscale decomposition in solution space, the multilevel augmentation method leads to a fast algorithm for solving discretized ill-posed problems. Combining with Tikhonov regularization, in the implementation of the multilevel augmentation method, one only needs to invert the same matrix with a relatively small size and perform a matrix-vector multiplication at the linear computational complexity. As a result, the computation cost is dramatically reduced. The a posteriori regularization parameter choice rule and the convergence rate for the regularized solution are also studied in this work. Numerical tests illustrate the proposed algorithm and the theoretical estimates. © 2013 Institute of Mathematics.

In this paper we study linear ill-posed problems Ax = y in a Hilbert space setting where instead of exact data y noisy data yδ are given satisfying

Let X1 and X2 be complex Banach spaces, and let A1 ∈ BL(X1), A2 ∈ BL(X2), A3 ∈ BL(X1, X2) and A4 ∈ BL(X2, X1). We propose an iterative procedure which is a modified form of Newton's iterations for obtaining approximations for the solution R ∈ BL(X1, X2) of the Riccati equation A2R - RA1 = A3 + RA4R, and show that the convergence of the method is quadratic. The advantage of the present procedure is that the conditions imposed on the operators A1, A2, A3, A4 are weaker than the corresponding conditions for Newton's iterations, considered earlier by Demmel (1987), Nair (1989) and Nair (1990) in the context of obtaining error bounds for approximate spectral elements. Also, we discuss an application of the procedure to spectral approximation under perturbations of the operator.