A simple regularization method for stable analytic continuation

Fu, Chu-Li; Dou, Fangfang; Feng, Xiaoli; Qian, Zhi

doi:10.1088/0266-5611/24/6/065003

Cited by 38 publications

(24 citation statements)

References 12 publications

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“…Even for one particle moving in a potential well, the direct back-rotation leads to unphysical large oscillations in the wave function [17,18]. To prevent this, a proper regularization procedure needs to be applied [57,58]. The radial density is defined as the mean value of the operator: …”

Section: Back Rotation: From Complex Scaling To Gamow Statesmentioning

confidence: 99%

Nuclear three-body problem in the complex energy plane: Complex-scaling Slater method

et al. 2014

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Background:The physics of open quantum systems is an interdisciplinary area of research. The nuclear "openness" manifests itself through the presence of the many-body continuum representing various decay, scattering, and reaction channels. As the radioactive nuclear beam experimentation extends the known nuclear landscape toward the particle drip lines, the coupling to the continuum space becomes exceedingly more important. Of particular interest are weakly bound and unbound nuclear states appearing around particle thresholds. Theories of such nuclei must take into account their open quantum nature. Purpose:To describe open quantum systems, we introduce a complex-scaling (CS) approach in the Slater basis. We benchmark it with the complex-energy Gamow shell model (GSM) by studying energies and wave functions of the bound and unbound states of the two-neutron halo nucleus 6 He viewed as an α + n + n cluster system. Methods: Both CS and GSM approaches are applied to a translationally invariant Hamiltonian with the two-body interaction approximated by the finite-range central Minnesota force. In the CS approach, we use the Slater basis, which exhibits the correct asymptotic behavior at large distances. To extract particle densities from the back-rotated CS solutions, we apply the Tikhonov regularization procedure, which minimizes the ultraviolet numerical noise. Results: We show that the CS-Slater method is both accurate and efficient. Its equivalence to the GSM approach has been demonstrated numerically for both energies and wave functions of 6 He. One important technical aspect of our calculation was to fully retrieve the correct asymptotic behavior of a resonance state from the complex-scaled (square-integrable) wave function. While standard applications of the inverse complex transformation to the complex-rotated solution provide unstable results, the stabilization method fully reproduces the GSM benchmark. We also propose a method to determine the smoothing parameter of the Tikhonov regularization. Conclusions:The combined suite of CS-Slater and GSM techniques has many attractive features when applied to nuclear problems involving weakly bound and unbound states. While both methods can describe energies, total widths, and wave functions of nuclear states, the CS-Slater method-if it can be applied-can provide additional information about partial energy widths associated with individual thresholds.

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Section: Back Rotation: From Complex Scaling To Gamow Statesmentioning

confidence: 99%

Nuclear three-body problem in the complex energy plane: Complex-scaling Slater method

et al. 2014

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“…Problems of this type have been considered, e. g., in [13,17,18,80]. They arise in different important applications such as in medical imaging, see [15].…”

Section: Proof (Sketch)mentioning

confidence: 99%

“…Step 1 (Operator equation formulation of the problem): From [17,18] we have that the operator equation formulation of the above problem is given by A f = g with A f := e ξy f (ξ)…”

Section: Proof (Sketch)mentioning

confidence: 99%

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

Tautenhahn

Hämarik

Hofmann

et al. 2013

Numerical Functional Analysis and Optimization

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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.

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“…the even indices satisfy φ(2k) = (k + 1) 2 , while the subsequence of odd indices is the ordered sequence of non-quadratic numbers (including 1). Since φ(N) = N the associted subspaces X ′ n satisfy the discretization condition (9). The sequence of j(A, X ′ n ) can be calculated as…”

Section: Now We Choose a Different Discretizationmentioning

confidence: 99%

“…For their stable approximate solution such equations require regularization when the given data are noisy. The mathematical theory and practice of regularization (see, e.g., the textbooks [1,4,7,11,20,25] and the papers [2,5,9,22,24,26,28,33,35]) takes advantage of some knowledge concerning the nature of ill-posedness of the underlying problem. This nature regards available a priori information and the degree of ill-posedness from which conclusions with respect to appropriate regularization methods and efficient regularization parameter choices can be drawn.…”

Section: Introductionmentioning

confidence: 99%

On the Degree of Ill-posedness for Linear Problems with Noncompact Operators

Hofmann¹,

Kindermann²

2010

Methods and Applications of Analysis

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Abstract. In inverse problems it is quite usual to encounter equations that are ill-posed and require regularization aimed at finding stable approximate solutions when the given data are noisy. In this paper, we discuss definitions and concepts for the degree of ill-posedness for linear operator equations in a Hilbert space setting. It is important to distinguish between a global version of such degree taking into account the smoothing properties of the forward operator, only, and a local version combining that with the corresponding solution smoothness. We include the rarely discussed case of non-compact forward operators and explain why the usual notion of degree of ill-posedness cannot be used in this case.Key words. Degree of ill-posedness, regularization, linear operator equation, Hilbert space, modulus of continuity, spectral distribution, source condition.AMS subject classifications. 47A52, 65J20.1. Introduction. It is an intrinsic property of a wide class of inverse problems that small perturbations in the data may lead to arbitrarily large errors in the solution. Hence, abstract models of inverse problems are frequently associated with operator equations formulated in infinite dimensional spaces that are ill-posed in the sense of Hadamard. For their stable approximate solution such equations require regularization when the given data are noisy. The mathematical theory and practice of regularization (see, e.g., the textbooks [1,4,7,11,20,25] and the papers [2,5,9,22,24,26,28,33,35]) takes advantage of some knowledge concerning the nature of ill-posedness of the underlying problem. This nature regards available a priori information and the degree of ill-posedness from which conclusions with respect to appropriate regularization methods and efficient regularization parameter choices can be drawn.We restrict our considerations here on ill-posed linear operator equations

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A simple regularization method for stable analytic continuation

Cited by 38 publications

References 12 publications

Nuclear three-body problem in the complex energy plane: Complex-scaling Slater method

Nuclear three-body problem in the complex energy plane: Complex-scaling Slater method

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

On the Degree of Ill-posedness for Linear Problems with Noncompact Operators

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