5 Convexification for ill-posed Cauchy problems for quasi-linear PDEs

2021 ◽  
pp. 93-120
Keyword(s):  
Cauchy Problems ◽  
Ill Posed ◽  
Linear Pdes

Global stability result for parabolic Cauchy problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mourad Choulli ◽  
Masahiro Yamamoto
Keyword(s):  
New York ◽  
Partial Differential Equations ◽  
Global Stability ◽  
Classical Problem ◽  
Stability Result ◽  
Cauchy Problems ◽  
Smooth Solutions ◽  
Linear Partial Differential Equations ◽  
Ill Posed ◽  
The Stability

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

Linear regularizing algorithms for positive solutions of linear inverse problems

1988 ◽  
Vol 415 (1849) ◽  
pp. 257-275 ◽  
Keyword(s):  
Regularization Parameter ◽  
Numerical Differentiation ◽  
Approximate Solutions ◽  
Cauchy Problems ◽  
Negative Part ◽  
Fredholm Equations ◽  
Linear Inverse Problems ◽  
Ill Posed ◽  
Laplace Transform Inversion ◽  
Convolution Equations

Linear-regularization methods provide a simple technique for deter­mining stable approximate solutions of linear ill-posed problems such as Fredholm equations of the first kind, Cauchy problems for elliptic equations and backward solution of forward parabolic equations. In most of these problems the solution must be positive to satisfy physical plausibility. In this paper we consider ill-posed first-kind convolution equations and related problems such as numerical differentiation, Radon transform and Laplace-transform inversion. We investigate several linear regularization algorithms which provide positive approximate solutions for these problems at least in the absence of errors on the data. For noisy data the solution is not necessarily positive. Because the appearance of negative values can then only be an effect of the noise, the negative part of the solution should be negligible with a suitable choice of the regularization parameter. A price to pay for ensuring positivity is always, however, a reduction in resolution.

scholarly journals Regularization for a class of ill-posed Cauchy problems

2005 ◽  
Vol 133 (10) ◽  
pp. 3005-3012 ◽  
Author(s):  
Yongzhong Huang ◽  
Quan Zheng
Keyword(s):  
Cauchy Problems ◽  
Ill Posed
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