Wavelet and Fourier Methods for Solving the Sideways Heat Equation

Eldén, Lars; Berntsson, Fredrik; Regińska, Teresa

doi:10.1137/s1064827597331394

Cited by 215 publications

(160 citation statements)

References 22 publications

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“…Of course, for T > T α the situation completely reverses. With J = 10, α = 1/4 and T = 10 T α the growth factor is a possibly workable value of around 600; while for the heat equation it is greater than 10 25 . We reiterate that the apparent contradiction between the theoretical ill-conditioning and numerical stability is due to the spectral cutoff present in any practical reconstruction procedure.…”

Section: Inverse Problems For Time Fractional Diffusionmentioning

confidence: 97%

“…The sideways problem for the classical diffusion has been extensively studied, and many efficient numerical methods have been developed and analyzed [11,8,24,25]. In the fractional case, however, there are only a few works on numerical schemes, mostly for one-dimensional problems, and there seems no theoretical study on stability etc.…”

Section: Inverse Problems For Time Fractional Diffusionmentioning

confidence: 99%

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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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Section: Inverse Problems For Time Fractional Diffusionmentioning

confidence: 97%

Section: Inverse Problems For Time Fractional Diffusionmentioning

confidence: 99%

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

Jin¹,

Rundell²

2012

Inverse Problems

150

View full text Add to dashboard Cite

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“…Proposition 4. Let u be defined by (13) and assume that hypotheses corresponding to those in Theorem 3 hold. Let v k be defined by (17).…”

Section: Accuracy Of the Stabilized Approximationmentioning

confidence: 99%

A numerical solution of a Cauchy problem for an elliptic equation by Krylov subspaces

Eldén

Simoncini

2009

Inverse Problems

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Abstract.We study the numerical solution of a Cauchy problem for a self-adjoint elliptic partial differential equation u zz − Lu = 0 in three space dimensions (x, y, z) , where the domain is cylindrical in z. Cauchy data are given on the lower boundary and the boundary values on the upper boundary are sought. The problem is severely ill-posed. The formal solution is written as a hyperbolic cosine function in terms of the two-dimensional elliptic operator L (via its eigenfunction expansion), and it is shown that the solution is stabilized (regularized) if the large eigenvalues are cut off. We suggest a numerical procedure based on the rational Krylov method, where the solution is projected onto a subspace generated using the operator L −1 . This means that in each Krylov step a well-posed two-dimensional elliptic problem involving L is solved. Furthermore, the hyperbolic cosine is evaluated explicitly only for a small symmetric matrix. A stopping criterion for the Krylov recursion is suggested based on the relative change of an approximate residual, which can be computed very cheaply. Two numerical examples are given that demonstrate the accuracy of the method and the efficiency of the stopping criterion.

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“…Recently, the truncated regularization method has been effectively applied to solve the sideways heat equation [6], [7], a more general sideways parabolic equation [8] and backward heat [9]. This regularization method is rather simple and convenient for dealing with some ill-posed problems.…”

Section: Introductionmentioning

confidence: 99%