Lipschitz stability in inverse parabolic problems by the Carleman estimate

Imanuvilov, Oleg Yu.; Yamamoto, Masahiro

doi:10.1088/0266-5611/14/5/009

Cited by 228 publications

(257 citation statements)

References 19 publications

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“…For the parabolic case with t-dependent principal part, a similar stability was already proved in Imanuvilov and Yamamoto [12]. In fact, in the parabolic case, the proof for the t-dependent case is the same as the t-independent case because of the character of the parabolic Carleman estimate.…”

Section: Theorem 23supporting

confidence: 71%

Inverse source problem for the hyperbolic equation with a time-dependent principal part

Jiang

Liu

Yamamoto

2017

Journal of Differential Equations

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In this paper, we investigate the inverse problem on determining the spatial component of the source term in a hyperbolic equation with time-dependent principal part. Based on a newly established Carleman estimate for general hyperbolic operators, we prove a local stability result of Hölder type in both cases of partial boundary and interior observation data. Numerically, we adopt the classical Tikhonov regularization to transform the inverse problem into an output least-squares minimization, which can be solved by the iterative thresholding algorithm. The proposed algorithm is computationally easy and efficient: the minimizer at each step has explicit solution. Abundant amounts of numerical experiments are presented to demonstrate the accuracy and efficiency of the algorithm.

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Section: Theorem 23supporting

confidence: 71%

Inverse source problem for the hyperbolic equation with a time-dependent principal part

Jiang

Liu

Yamamoto

2017

Journal of Differential Equations

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“…The inverse source problems for the classical diffusion equation have been extensively studied; see e.g., [7,37,9]. Inverse source problems for FDEs have also been numerically studied.…”

Section: In [88 Theorem 44] a Stability Results Was Established Formentioning

confidence: 99%

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

Jin¹,

Rundell²

2012

Inverse Problems

150

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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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“…The problem of recovering sources has been first addressed for hyperbolic equations in [39,40] and for parabolic equations in [27]. The determination of coefficients has also drawn the attention of many authors.…”

Section: Introductionmentioning

confidence: 99%

“…The second step relies on the application of a Carleman estimate whose parameters and weight are crucial to drive to the conclusion. An idea of Imanuvilov and Yamamoto, presented for the first time in [27] (for parabolic equation) and then often used (for other pde's), allows to use the appropriate global Carleman inequality to estimate the unknown coefficient in terms of internal or boundary measurements of the solution. The main drawback of this method, that is not lifted yet is the hypothesis that at least one initial condition, (or some of its derivatives) never vanishes in the domain.…”

Section: Introductionmentioning

confidence: 99%