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Register a new userError Estimates for Arnoldo-Tikhonov Regularization for Ill-Posed Operator Equations
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Most of the literature on the solution of linear ill-posed operator equations, or their discretization, focuses only on the infinite-dimensional setting or only on the solution of the algebraic linear system of equations obtained by discretization. This paper discusses the influence of the discretization error on the computed solution. We consider the situation when the discretization used yields an algebraic linear system of equations with a large matrix. An approximate solution of this system is computed by first determining a reduced system of fairly small size by carrying out a few steps of the Arnoldi process. Tikhonov regularization is applied to the reduced problem and the regularization parameter is determined by the discrepancy principle. Errors incurred in each step of the solution process are discussed. Computed examples illustrate the error bounds derived.
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Many applications in science and engineering require the solution of large linear discrete ill-posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space-dimensions. The matrix that defines these problems is very ill-conditioned and generally numerically singular, and the right-hand side, which represents measured data, typically is contaminated by measurement error. Straightforward solution of these problems generally is not meaningful due to severe error propagation. Tikhonov regularization seeks to alleviate this difficulty by replacing the given linear discrete ill-posed problem by a penalized least-squares problem, whose solution is less sensitive to the error in the right-hand side and to round-off errors introduced during the computations. This paper discusses the construction of penalty terms that are determined by solving a matrix-nearness problem. These penalty terms allow partial transformation to standard form of Tikhonov regularization problems that stem from the discretization of integral equations on a cube in several space-dimensions.
Let $phi$ be a nontrivial function of $L^1(RR)$. For each $sgeq 0$ we put begin{eqnarray*} p(s)=-log int_{|t|geq s}|phi (t)|dt. end{eqnarray*} If $phi$ satisfies begin{equation} lim_{sto infty}frac{p(s)}{s}=infty ,label{170506.1} end{equation} we obtain asymptotic estimates of the size of small-valued sets $B_{epsilon}={xinRR : |hat{phi}(x)|leq epsilon, |x|leq R_{epsilon}}$ of Fourier transform begin{eqnarray*} hat{phi}(x)=int_{-infty}^{infty}e^{-ixt}phi (t)dt, xin RR, end{eqnarray*} in terms of $p(s)$ or in terms of its Young dual function begin{eqnarray*} p^{*}(t)=sup_{sgeq 0}[st-p(s)], tgeq 0. end{eqnarray*} Applying these results, we give an explicit estimate for the error of Tikhonovs regularization for the solution $f$ of the integral convolution equation begin{eqnarray*} int_{-infty}^{infty}f(t-s)phi (s)ds =g(t), end{eqnarray*} where $f,g in L^2(RR)$ and $phi$ is a nontrivial function of $L^1(RR)$ satisfying condition (ref{170506.1}), and $g,phi$ are known non-exactly. Also, our results extend some results of cite{tld} and cite{tqd}.
Based on the joint bidiagonalization process of a large matrix pair ${A,L}$, we propose and develop an iterative regularization algorithm for the large scale linear discrete ill-posed problems in general-form regularization: $min|Lx| mbox{{rm subject to}} xinmathcal{S} = {x| |Ax-b|leq tau|e|}$ with a Gaussian white noise $e$ and $tau>1$ slightly, where $L$ is a regularization matrix. Our algorithm is different from the hybrid one proposed by Kilmer {em et al.}, which is based on the same process but solves the general-form Tikhonov regularization problem: $min_xleft{|Ax-b|^2+lambda^2|Lx|^2right}$. We prove that the iterates take the form of attractive filtered generalized singular value decomposition (GSVD) expansions, where the filters are given explicitly. This result and the analysis on it show that the method must have the desired semi-convergence property and get insight into the regularizing effects of the method. We use the L-curve criterion or the discrepancy principle to determine $k^*$. The algorithm is simple and effective, and numerical experiments illustrate that it often computes more accurate regularized solutions than the hybrid one.
The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications.Due to the singularity of the logarithmic function, it introducestremendous difficulties in establishing mathematical theories, as well asin designing and analyzing numerical methods for PDEs with such nonlinearity. Here we take the logarithmic Schrodinger equation (LogSE)as a prototype model. Instead of regularizing $f(rho)=ln rho$ in theLogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE byfirst regularizing $F(rho)=rholn rho -rho$ locally near $rho=0^+$ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schrodinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter $0<epll1$. Moreover, the conserved energy of the ERLogSE converges to that of LogSE quadratically, which significantly improvesthe linear convergence rate of the regularization method in the literature. Error estimates are alsopresented for solving the ERLogSE by using Lie-Trotter splittingintegrators. Numerical results are reported to confirm our errorestimates of the LER and of the time-splitting integrators for theERLogSE. Finally our results suggest that the LER performs better than regularizing the logarithmic nonlinearity in the LogSE directly.
This paper is concerned with solving ill-posed tensor linear equations. These kinds of equations may appear from finite difference discretization of high-dimensional convection-diffusion problems or when partial differential equations in many dimensions are discretized by collocation spectral methods. Here, we propose the Tensor Golub--Kahan bidiagonalization (TGKB) algorithm in conjunction with the well known Tikhonov regularization method to solve the mentioned problems. Theoretical results are presented to discuss on conditioning of the Stein tensor equation and to reveal that how the TGKB process can be exploited for general tensor equations. In the last section, some classical test problems are examined to numerically illustrate the feasibility of proposed algorithms and also applications for color image restoration are considered.
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