One new strategy for
                    <i>a priori</i>
                    choice of regularizing parameters in Tikhonov's regularization

Cheng, Jin; Yamamoto, Masahiro

doi:10.1088/0266-5611/16/4/101

Cited by 155 publications

(127 citation statements)

References 24 publications

Supporting

Mentioning

126

Contrasting

Order By: Relevance

“…A number of regularization parameter choice techniques have been developed for numerical differentiation. They yield satisfactory results when the smoothness of the function to be differentiated is given very precisely [1], [2], [22], [26]. However, in applications this smoothness is usually unknown, as one can see it from the following example.…”

Section: How Do We Approximate a Derivative Y (T) Of A Smooth Functiomentioning

confidence: 99%

Numerical differentiation from a viewpoint of regularization theory

Lü

Pereverzev

2006

Math. Comp.

View full text Add to dashboard Cite

Abstract. In this paper, we discuss the classical ill-posed problem of numerical differentiation, assuming that the smoothness of the function to be differentiated is unknown. Using recent results on adaptive regularization of general ill-posed problems, we propose new rules for the choice of the stepsize in the finite-difference methods, and for the regularization parameter choice in numerical differentiation regularized by the iterated Tikhonov method. These methods are shown to be effective for the differentiation of noisy functions, and the order-optimal convergence results for them are proved.

show abstract

Section: How Do We Approximate a Derivative Y (T) Of A Smooth Functiomentioning

confidence: 99%

Numerical differentiation from a viewpoint of regularization theory

Lü

Pereverzev

2006

Math. Comp.

View full text Add to dashboard Cite

show abstract

“…Stability estimates for inverse problems are not only important from the theoretical viewpoint, but also useful for numerical algorithms. In particular, by Cheng and Yamamoto [10] for example, a stability estimate gives convergence rates of Tikhonov regularized solutions, which are widely used as approximating solutions to the inverse problems.…”

Section: )∂ I Y(t X) + C(x)y(t X) + H(t X) (T X) ∈ Qmentioning

confidence: 99%

Lipschitz stability in the determination of the principal part of a parabolic equation

Yuan

Yamamoto²

2008

ESAIM: COCV

View full text Add to dashboard Cite

Abstract. Let y(h) (t, x) be one solution towith a non-homogeneous term h, and y| (0,T )×∂Ω = 0, where Ω ⊂ R n is a bounded domain. We discuss an inverse problem of determining n(n + 1)/2 unknown functions aij.., h 0 suitably, where Γ0 is an arbitrary subboundary, ∂ν denotes the normal derivative, 0 < θ < T and 0 ∈ N. In the case of 0 = (n + 1) 2 n/2, we prove the Lipschitz stability in the inverse problem if we choose (h1, ..., h 0 ) from a set H ⊂ {C ∞ 0 ((0, T ) × ω)} 0 with an arbitrarily fixed subdomain ω ⊂ Ω. Moreover we can take 0 = (n + 3)n/2 by making special choices for h , 1 ≤ ≤ 0. The proof is based on a Carleman estimate.Mathematics Subject Classification. 35R30, 35K20.

show abstract

“…The derivation of such functions β for given compact subsets M ⊂ X has attracted attention, see, e.g., [1,67] and the reference cited therein. For its application in regularization, see, e. g., [12,33,34,35].…”

Section: Introductionmentioning

confidence: 99%

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

Tautenhahn

Hämarik

Hofmann

et al. 2013

Numerical Functional Analysis and Optimization

View full text Add to dashboard Cite

Abstract. The focus of this paper is on conditional stability estimates for illposed inverse problems in partial differential equations. Conditional stability estimates have been obtained in the literature by a couple different methods. In this paper we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We are going to work out the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capability of our method is illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation.

show abstract

One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization

Cited by 155 publications

References 24 publications

Numerical differentiation from a viewpoint of regularization theory

Numerical differentiation from a viewpoint of regularization theory

Lipschitz stability in the determination of the principal part of a parabolic equation

Conditional Stability Estimates for Ill-Posed PDE Problems by Using Interpolation

Contact Info

Product

Resources

About