Oversmoothing Tikhonov regularization in Banach spaces
                  <sup>*</sup>

Chen, De-Han; Hofmann, Bernd; Yousept, Irwin

doi:10.1088/1361-6420/abcea0

Cited by 9 publications

(17 citation statements)

References 31 publications

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“…Recently, the validity of VSC was shown for the ill-posed backward nonlinear Maxwell's equations in [10], by means of the semigroup theory and extrapolation of Hilbert spaces. The authors of this work verified VSC and obtained a Hölder type convergence rate in [7,9,11] for the Tikhonov regularized solutions of the inverse elliptic and parabolic radiativity problems. It is important to emphasize that all the above results were established under the full measurement data.…”

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confidence: 59%

Variational source conditions for inverse Robin and flux problems by partial measurements

Chen¹,

Jiang²,

Yousept³

et al. 2022

IPI

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confidence: 59%

Variational source conditions for inverse Robin and flux problems by partial measurements

Chen¹,

Jiang²,

Yousept³

et al. 2022

IPI

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“…Note that Proposition 3 also yields the existence of minimizers in (4) under Assumptions 1 and 2 and Eqs. (7).…”

Section: Existence and Uniqueness Of Minimizersmentioning

confidence: 99%

“…The special case of diagonal operators in 1 -regularization has been discussed in [20]. In a very recent work, Chen et al [7] have studied oversmoothing for finitely smoothing operators in scales of Banach spaces generated by sectorial operators.…”

Section: Introductionmentioning

confidence: 99%

Maximal spaces for approximation rates in $$\ell ^1$$-regularization

Miller

Hohage

2021

Numer. Math.

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We study Tikhonov regularization for possibly nonlinear inverse problems with weighted $$\ell ^1$$ ℓ 1 -penalization. The forward operator, mapping from a sequence space to an arbitrary Banach space, typically an $$L^2$$ L 2 -space, is assumed to satisfy a two-sided Lipschitz condition with respect to a weighted $$\ell ^2$$ ℓ 2 -norm and the norm of the image space. We show that in this setting approximation rates of arbitrarily high Hölder-type order in the regularization parameter can be achieved, and we characterize maximal subspaces of sequences on which these rates are attained. On these subspaces the method also converges with optimal rates in terms of the noise level with the discrepancy principle as parameter choice rule. Our analysis includes the case that the penalty term is not finite at the exact solution (’oversmoothing’). As a standard example we discuss wavelet regularization in Besov spaces $$B^r_{1,1}$$ B 1 , 1 r . In this setting we demonstrate in numerical simulations for a parameter identification problem in a differential equation that our theoretical results correctly predict improved rates of convergence for piecewise smooth unknown coefficients.

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“…Therefore, the concept of variational source conditions does not apply here and other ideas are needed to tackle this question. Situations where the true solution does not admit a finite penalty are referred to as oversmoothing in the literature and we refer to the papers [60,62,86,17] for convergence rates theory in this case. In Chapter 5 we provide a general convergence rates theory for oversmoothing in Banach space regularization.…”

Section: Remark 323 (Limiting Case: R = S) In the Setting Of Theorem ...mentioning

confidence: 99%

“…The special case of diagonal operators in 1 -regularization has been discussed in [46]. In a very recent work, which has been drafted in parallel, Chen, Hofmann & Yousept [17] have studied oversmoothing for finitely smoothing operators in scales of Banach spaces generated by sectorial operators. Although the aims and the type of results of this paper are similar to ones presented in this chapter we believe that our approach is considerably simpler and more flexible.…”

Section: Chapter Fivementioning

confidence: 99%

Variational regularization theory for sparsity promoting wavelet regularization

Miller¹

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ω ∈ ∂R(f ) for some ω ∈ Y.(2.4) We refer to [61] for a comprehensive treatment of the modulus of continuity for linear operators in Hilbert spaces. We finish this section with a practicable criterion to verify order optimality.Corollary 2.7 (Order optimality via the modulus of continuity). In the setting of Proposition 2.6 suppose φ : (0, ∞) → [0, ∞) is non-decreasing and that there exists a constant c ω > 0 and δ 0 > 0 such thatThen R is an order optimal reconstruction method on K. Literature on convergence rates for sparsity promoting regularizationWe give a brief overview of the literature on convergence rate theory for 1 -regularization.In the early paper [78] from 2008 the rate O(δ 1/2 ) in the 1-norm is shown assuming that the unknown solution is sparse (i.e. has only finitely many non-vanishing entries) and that the forward operator is linear. The paper [52] provides the rate O(δ 1/2 ) for nonlinear operators under a source condition that coincides with (2.4) in the linear case. Furthermore, by additionally requiring sparsity of the unknown the authors achieve the linear rate O(δ) and discuss that in contrast to classical Tikhonov regularization, which has the highest possible rate O(δ 2/3 ), in 1 -regularization no saturation effect occurs. To the best of the author's knowledge the linear rate O(δ) was first proven in [14] for a regularization scheme similar to (2.1), which is called residual method in [52]. In [50] a linear convergence rate is shown in the more general setting of positively homogeneous functionals under the source condition (2.4) and a mild injectivity type assumption. Furthermore, in [53] it is proven (again under a mild injectivity type assumption) that the condition (2.4) is not only sufficient but even necessary for a linear convergence rate of 1 -regularization. The phenomenon of exact recovery, i.e. the question whether the support of the estimator equals the support of a sparse exact solution, is affirmatively treated in [79].However, it is usually more realistic to assume that the true solution is only approximately sparse in the sense that it can well be approximated by sparse vectors. Using a variational source condition, convergence rates are shown for non-sparse solutions in [11] for linear forward operators. Therein the analysis is based on the assumption that the unit vectors belong to the range of the adjoint operator. The rates are characterized in terms of the growth of the norms of the preimages of the unit vectors and the speed of decay of the true solution. In [3] the range condition is further discussed and the convergence rate results are extended to q -regularization with q < 1. We will discuss the latter range condition in more detail in Section 3.2.In [41] a relaxation of the condition on the unit vectors is introduced, and it is shown

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Oversmoothing Tikhonov regularization in Banach spaces ^*

Cited by 9 publications

References 31 publications

Variational source conditions for inverse Robin and flux problems by partial measurements

Variational source conditions for inverse Robin and flux problems by partial measurements

Maximal spaces for approximation rates in $$\ell ^1$$-regularization

Variational regularization theory for sparsity promoting wavelet regularization

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Oversmoothing Tikhonov regularization in Banach spaces *

Cited by 9 publications

References 31 publications

Variational source conditions for inverse Robin and flux problems by partial measurements

Variational source conditions for inverse Robin and flux problems by partial measurements

Maximal spaces for approximation rates in $$\ell ^1$$-regularization

Variational regularization theory for sparsity promoting wavelet regularization

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Oversmoothing Tikhonov regularization in Banach spaces ^*