Stability estimates and regularization for an inverse heat conduction prolem in semi - infinite and finite time intervals

Engl, Heinz W.; Manselli, Paolo

doi:10.1080/01630568908816316

Cited by 30 publications

(9 citation statements)

References 11 publications

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“…This is an example of a severly ill-posed problem. For the numerical solution of this and the closely related problem of recovering the temperature see Carasso (1982), Elden (1983), Engl and Manselli (1989), King and Neubauer (1988), Plato andVainikko (1990), Murio (1981), and Murio and Guo (1990) The minimal eigenvalues tend to zero much more rapidly for this problem in comparison to our first example. Using the same method as before we list the theoretical and computed reduction factors for Example 5.2 in Table 5 for the parameter choices 2 = 103 ha 4 and ~j = (2 +hi_4 1)-1.…”

Section: W(1 T)=g(t) W(xt) Boundedas X~ W(x 0)= O Wx(o T)= U(t)mentioning

confidence: 89%

Multilevel algorithms for ill-posed problems

King

1992

Numer. Math.

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Summary.In this paper new multilevel algorithms are proposed for the numerical solution of first kind operator equations. Convergence estimates are established for multilevel algorithms applied to Tikhonov type regularization methods. Our theory relates the convergence rate of these algorithms to the minimal eigenvalue of the discrete version of the operator and the regularization parameter. The algorithms and analysis are presented in an abstract setting that can be applied to first kind integral equations.

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Section: W(1 T)=g(t) W(xt) Boundedas X~ W(x 0)= O Wx(o T)= U(t)mentioning

confidence: 89%

Multilevel algorithms for ill-posed problems

King

1992

Numer. Math.

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“…Using old design set {v i = i+1 11 } 6 i=0 we have observed comparatively large differences between corresponding values σ (v i ) in (9). It suggests to switch to more fine set {v i } (see Remark 2 in Sect.…”

Section: Consistent and Non-symmetric Interior Penalty Schemementioning

confidence: 97%

“…In Table 2 we present the results of numerical experiments with the non-symmetric interior penalty Galerkin method based on the formulation (35), where a penalty parameter σ is chosen in accordance with the balancing principle (13). These results correspond to the case when L 2 -norm is used within the framework of the procedure generating the sequence (9). The errors there are also measured in L 2 -norm.…”

Section: Consistent and Non-symmetric Interior Penalty Schemementioning

confidence: 98%

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On the balancing principle for some problems of Numerical Analysis

2007

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We discuss a choice of weight in penalization methods. The motivation for the use of penalization in computational mathematics is to improve the conditioning of the numerical solution. One example of such improvement is a regularization, where a penalization substitutes an ill-posed problem for a well-posed one. In modern numerical methods for PDEs a penalization is used, for example, to enforce a continuity of an approximate solution on non-matching grids. A choice of penalty weight should provide a balance between error components related with convergence and stability, which are usually unknown. In this paper we propose and analyze a simple adaptive strategy for the choice of penalty weight which does not rely on a priori estimates of above mentioned components. It is shown that under natural assumptions the accuracy provided by our adaptive strategy is worse only by a constant factor than one could achieve in the case of known stability and convergence rates. Finally, we successfully apply our strategy for self-regularization of Volterra-type severely ill-posed problems, such as the sideways heat equation, and for the choice of a weight in interior penalty discontinuous approximation on non-matching grids. Numerical experiments on a series of model problems support theoretical results.

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“…Hence, to get the heating history in the body, we want to use temperature measured in the interior of the body. In the simplest model, the heat-conducting body is assumed to have a constant conductivity and represented by the half-line x > 0 (see, e.g., [3,4,7,9]). While giving many useful results, this model is not suitable for the case of a body having a series of superposed layers, each of which has a constant conductivity.…”

Section: Introductionmentioning

confidence: 99%