An Introduction to Inverse Scattering and Inverse Spectral Problems

Chadan, K.; Colton, David; Päivärinta, Lassi; Rundell, William

doi:10.1137/1.9780898719710

Cited by 176 publications

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“…It is well known that eigenvalues contain valuable information about the boundary value problem. For example it is known that the sequence of Dirichlet eigenvalues can uniquely determine a potential q symmetric with respect to the point x = 1/2, and together with additional spectral information, one can uniquely determine a general potential q; see [12,86] for an overview of results on the classical inverse Sturm-Liouville problem. In the fractional case, the eigenvalues are generally genuinely complex, and a complex number may carry more information than a real one.…”

Section: Inverse Problems For Space Fractional Diffusionmentioning

confidence: 99%

An inverse problem for a one-dimensional time-fractional diffusion problem

Jin¹,

Rundell²

2012

Inverse Problems

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148

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Abstract. Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weakly singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest.In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of ill-posedness of several "classical" inverse problems for fractional differential equations involving a Djrbashian-Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm-Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of "fractional" inverse problems, and a partial list of surprising observations is given below.(a) Classical backward diffusion is exponentially ill-posed, whereas time fractional backward diffusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more effective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm-Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suffice. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the preci...

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Section: Inverse Problems For Space Fractional Diffusionmentioning

confidence: 99%

An inverse problem for a one-dimensional time-fractional diffusion problem

Jin¹,

Rundell²

2012

Inverse Problems

Self Cite

148

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“…If = 1 and = N − 1, then Ψ satisfies the following boundary conditions (5) and the matching conditions (6):…”

Section: +1mentioning

confidence: 99%

“…Some aspects of the inverse problem theory for noncompact graphs were considered in [5,7,10,11,18,34]. We note that inverse spectral problems for second-order and for higher-order ordinary differential operators on an interval (finite or infinite) have been studied by many authors (see the monographs [2,6,8,15,17,24,26,27] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Inverse problems on star-type graphs: differential operators of different orders on different edges

Yurko

2013

Open Mathematics

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We study inverse spectral problems for ordinary differential equations on compact star-type graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness. MSC:34A55, 34L05, 47E05

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“…If we compare results in [18] and [21,22] with what we are presenting below, we observe the following distinctions. First, our results concern the IP on the semi-axis, though in those papers the IP on an interval is considered.…”

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confidence: 72%

“…[21,22]. In [21] a kind of successive approximation method is developed. In [23,24], the Sturm-Liouville problem is replaced by its finite-dimensional approximation and a regularization procedure is developed based on the known asymptotic representation of the spectrum as the potential is small.…”

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confidence: 99%

Inverse problem on the semi-axis: local approach

Avdonin

Belinskiy

Matthews³

2011

Tamkang J. Math.

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Abstract. We consider the problem of reconstruction of the potential for the wave equation on the semi-axis. We use the local versions of the Gelfand-Levitan and Krein equations, and the linear version of Simon's approach. For all methods, we reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand-side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples verify that the algorithms can be used to reconstruct an unknown potential accurately. The practicality of each approach is briefly discussed. Accurate data preparation is described and implemented.

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An Introduction to Inverse Scattering and Inverse Spectral Problems

Cited by 176 publications

References 0 publications

An inverse problem for a one-dimensional time-fractional diffusion problem

An inverse problem for a one-dimensional time-fractional diffusion problem

Inverse problems on star-type graphs: differential operators of different orders on different edges

Inverse problem on the semi-axis: local approach

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