2010
DOI: 10.1080/00036810903393809
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About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains

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Cited by 53 publications
(97 citation statements)
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“…There are different techniques for deriving conditional stability functions β including the logarithmic convexity method, weighted energy methods, Gronwall's lemma and maximum principles [2,30,48], or methods based on Carleman estimates [8,37,38,75]. Let us explain by a simple model example the method of logarithmic convexity as one of the most traditional methods to obtain conditional stability functions β.…”
Section: Lemma 11 Let M ⊂ X Be Such That ω(δ M) Defined By (12) Imentioning
confidence: 99%
“…There are different techniques for deriving conditional stability functions β including the logarithmic convexity method, weighted energy methods, Gronwall's lemma and maximum principles [2,30,48], or methods based on Carleman estimates [8,37,38,75]. Let us explain by a simple model example the method of logarithmic convexity as one of the most traditional methods to obtain conditional stability functions β.…”
Section: Lemma 11 Let M ⊂ X Be Such That ω(δ M) Defined By (12) Imentioning
confidence: 99%
“…However, for β = 10 −6 the error could not be decreased further with the finest mesh (I = J = K = 511), hence, a higher mesh resolution should be adopted if we want to obtain a higher accuracy. In addition, Figure 2 shows the graphs of u β (x, y, M) for Examples 1 and 2 with the exact data {ϕ 0 i j } for the coarse mesh resolution I × J × K = 63 3 . For β too small such as 10 −6 , the instability phenomenon is manifested by the strongly oscillating contour lines.…”
Section: Calculation Proceduresmentioning
confidence: 99%
“…For β too small such as 10 −6 , the instability phenomenon is manifested by the strongly oscillating contour lines. In case of noisy data with ǫ > 0, to show the sensitivity of the computational accuracy to noise of the data, we repeated calculations with a variety of noise amplitudes ǫ = 10 −l for l = 2, 6, and illustrated the numerical results only with the finest mesh I × J × K = 511 3 , so that errors from mesh resolution do not contribute to δ ǫ,β . These results are shown in Tables 2 and 3 and Figures Table 3 1.0E-1 1.7E+0 3 and 4 for Examples 1 and 2, respectively.…”
Section: Calculation Proceduresmentioning
confidence: 99%
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“…Nevertheless, the literature devoted to the Cauchy problem for linear homogeneous elliptic equations is very rich, see e.g. [4,5,7,9,12,13,16,21,23,29,33,35] and the references therein. Recently, a linear inhomogeneous version of Helmholtz equation (i.e.…”
Section: Introductionmentioning
confidence: 99%