scholarly journals On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems

2013 ◽  
Vol 126 (4) ◽  
pp. 679-701 ◽  
Author(s):  
Takehiko Kinoshita ◽  
Takuma Kimura ◽  
Mitsuhiro T. Nakao
Keyword(s):  
Parabolic Equations ◽  
Nonlinear Problems ◽  
A Posteriori ◽  
Linear Parabolic Equations ◽  
A Posteriori Estimates

scholarly journals A posteriori snapshot location for POD in optimal control of linear parabolic equations

2018 ◽  
Vol 52 (5) ◽  
pp. 1847-1873 ◽  
Author(s):  
Alessandro Alla ◽  
Carmen Grässle ◽  
Michael Hinze
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Parabolic Equations ◽  
Error Control ◽  
Order Reduction ◽  
A Posteriori ◽  
Parabolic Pdes ◽  
Linear Parabolic Equations ◽  
The Right

In this paper we study the approximation of an optimal control problem for linear parabolic PDEs with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the basis functions are obtained upon information contained in time snapshots of the parabolic PDE related to given input data. In the present work we show that for POD-MOR in optimal control of parabolic equations it is important to have knowledge about the controlled system at the right time instances. We propose to determine the time instances (snapshot locations) by an a posteriori error control concept. The proposed method is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system which is approximated by a space-time finite element method. Finally, we present numerical tests to illustrate our approach and show the effectiveness of the method in comparison to existing approaches.

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

scholarly journals Smoothing properties in multistep backward difference method and time derivative approximation for linear parabolic equations

2005 ◽  
Vol 2005 (4) ◽  
pp. 523-536
Author(s):  
Yubin Yan
Keyword(s):  
Difference Method ◽  
Parabolic Equations ◽  
Parabolic Problem ◽  
Approximation Scheme ◽  
Time Derivative ◽  
Nonsmooth Data ◽  
Smoothing Property ◽  
Linear Parabolic Equations ◽  
Linear Parabolic Problem ◽  
Data Error

A smoothing property in multistep backward difference method for a linear parabolic problem in Hilbert space has been proved, where the operator is selfadjoint, positive definite with compact inverse. By using the solutions computed by a multistep backward difference method for the parabolic problem, we introduce an approximation scheme for time derivative. The nonsmooth data error estimate for the approximation of time derivative has been obtained.

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