scholarly journals Goal Oriented Time Adaptivity Using Local Error Estimates

Algorithms ◽  
2020 ◽  
Vol 13 (5) ◽  
pp. 113
Author(s):  
Peter Meisrimel ◽  
Philipp Birken
Keyword(s):  
Error Estimates ◽  
Initial Value Problems ◽  
Error Propagation ◽  
Adaptive Methods ◽  
Local Error ◽  
A Posteriori ◽  
Initial Value ◽  
Numerical Tests ◽  
Performance Predictions ◽  
Local Error Estimate

We consider initial value problems (IVPs) where we are interested in a quantity of interest (QoI) that is the integral in time of a functional of the solution. For these, we analyze goal oriented time adaptive methods that use only local error estimates. A local error estimate and timestep controller for step-wise contributions to the QoI are derived. We prove convergence of the error in the QoI for tolerance to zero under a controllability assumption. By analyzing global error propagation with respect to the QoI, we can identify possible issues and make performance predictions. Numerical tests verify these results. We compare performance with classical local error based time-adaptivity and a posteriori based adaptivity using the dual-weighted residual (DWR) method. For dissipative problems, local error based methods show better performance than DWR and the goal oriented method shows good results in most examples, with significant speedups in some cases.

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.

Adaptive Methods for Second Order Initial Value Problems

2010 ◽  
Author(s):  
Sesappa A. Rai ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Initial Value Problems ◽  
Adaptive Methods ◽  
Second Order ◽  
Initial Value

EXPLICIT EIGHTH ORDER NUMEROV-TYPE METHODS WITH REDUCED NUMBER OF STAGES FOR OSCILLATORY IVPs

2008 ◽  
Vol 19 (06) ◽  
pp. 957-970 ◽  
Author(s):  
I. Th. FAMELIS
Keyword(s):  
Initial Value Problems ◽  
Computational Cost ◽  
Phase Lag ◽  
Initial Value ◽  
Eighth Order ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
New Methods ◽  
Oscillating Solutions ◽  
Algebraic Order

Using a new methodology for deriving hybrid Numerov-type schemes, we present new explicit methods for the solution of second order initial value problems with oscillating solutions. The new methods attain algebraic order eight at a cost of eight function evaluations per step which is the most economical in computational cost that can be found in the literature. The methods have high amplification and phase-lag order characteristics in order to suit to the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

Variational Multiscale A Posteriori Error Estimation for Quantities of Interest

2009 ◽  
Vol 76 (2) ◽  
Author(s):  
Guillermo Hauke ◽  
Daniel Fuster
Keyword(s):  
Error Estimates ◽  
Posteriori Error ◽  
Local Error ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Variational Multiscale ◽  
Stabilized Methods ◽  
Global And Local ◽  
Quantities Of Interest

This paper applies the variational multiscale theory to develop an explicit a posteriori error estimator for quantities of interest and linear functionals of the solution. The method is an extension of a previous work on global and local error estimates for solutions computed with stabilized methods. The technique is based on approximating an exact representation of the error formulated as a function of the fine-scale Green function. Numerical examples for the multidimensional transport equation confirm that the method can provide good local error estimates of quantities of interest both in the diffusive and the advective limit.

NUMEROV-TYPE METHODS FOR OSCILLATORY LINEAR INITIAL VALUE PROBLEMS

2009 ◽  
Vol 20 (03) ◽  
pp. 383-398 ◽  
Author(s):  
I. TH. FAMELIS
Keyword(s):  
Initial Value Problems ◽  
New Method ◽  
Phase Lag ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
Type Method ◽  
Numerical Tests ◽  
Oscillating Solutions ◽  
Algebraic Order ◽  
High Phase

We present a new explicit Numerov-type method for the solution of second-order linear initial value problems with oscillating solutions. The new method attains algebraic order seven at a cost of six function evaluations per step. The method has the characteristic of zero dissipation and high phase-lag order making it suitable for the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

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