scholarly journals Convergent numerical methods for parabolic equations with reversed time via a new Carleman estimate

2019 ◽  
Vol 35 (11) ◽  
pp. 115012 ◽  
Author(s):  
Michael V Klibanov ◽  
Anatoly G Yagola
Keyword(s):  
Numerical Methods ◽  
Parabolic Equations ◽  
Carleman Estimate ◽  
Reversed Time

scholarly journals Carleman estimates for time-discrete parabolic equations and applications to controllability

2020 ◽  
Vol 26 ◽  
pp. 12
Author(s):  
Franck Boyer ◽  
Víctor Hernández-Santamaría
Keyword(s):  
Parabolic Equations ◽  
Linear Time ◽  
Parabolic Systems ◽  
Carleman Estimate ◽  
Carleman Estimates ◽  
Discretization Scheme ◽  
Parabolic Operator ◽  
Time Step ◽  
Coupled Parabolic Systems ◽  
Observability Estimates

In this paper, we prove a Carleman estimate for a time-discrete parabolic operator under some condition relating the large Carleman parameter to the time step of the discretization scheme. This estimate is then used to obtain relaxed observability estimates that yield, by duality, some controllability results for linear and semi-linear time-discrete parabolic equations. We also discuss the application of this Carleman estimate to the controllability of time-discrete coupled parabolic systems.

Inverse problems for linear degenerate parabolic equations by “time-like” Carleman estimate

2015 ◽  
Vol 23 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Atsushi Kawamoto
Keyword(s):  
Inverse Problems ◽  
Parabolic Equations ◽  
Fixed Time ◽  
Carleman Estimate ◽  
Coupled Systems ◽  
Degenerate Parabolic Equations ◽  
Strongly Coupled ◽  
Degenerate Parabolic ◽  
Inverse Source Problems ◽  
Linear Degenerate

AbstractIn this paper, we study inverse problems for multi-dimensional linear degenerate parabolic equations and strongly coupled systems. In particular we discuss the Lipschitz type stability results for the inverse source problems which determine a source term by boundary data on an appropriate sub-boundary and the data on any fixed time. Our arguments are based on the Carleman estimate. Here we prove and use the Carleman estimate with the

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