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Stochastic Mean-Field Dynamics and Applications to Life Sciences Read chapter Read first chapter

2019 | OriginalPaper | Chapter

On Optimal Decay Estimates for ODEs and PDEs with Modal Decomposition

Authors : Franz Achleitner, Anton Arnold, Beatrice Signorello

Published in: Stochastic Dynamics Out of Equilibrium

Publisher: Springer International Publishing

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Abstract

We consider the Goldstein-Taylor model, which is a 2-velocity BGK model, and construct the “optimal” Lyapunov functional to quantify the convergence to the unique normalized steady state. The Lyapunov functional is optimal in the sense that it yields decay estimates in \(L^2\)-norm with the sharp exponential decay rate and minimal multiplicative constant. The modal decomposition of the Goldstein-Taylor model leads to the study of a family of 2-dimensional ODE systems. Therefore we discuss the characterization of “optimal” Lyapunov functionals for linear ODE systems with positive stable diagonalizable matrices. We give a complete answer for optimal decay rates of 2-dimensional ODE systems, and a partial answer for higher dimensional ODE systems.

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previous chapter Stochastic Solutions to Hamilton-Jacobi Equations
next chapter Adaptive Importance Sampling with Forward-Backward Stochastic Differential Equations
Footnotes
1
An eigenvalue is defective if its geometric multiplicity is strictly less than its algebraic multiplicity.
 
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Metadata
Title
On Optimal Decay Estimates for ODEs and PDEs with Modal Decomposition
Authors
Franz Achleitner
Anton Arnold
Beatrice Signorello
Copyright Year
2019
Book
Stochastic Dynamics Out of Equilibrium

Print ISBN: 978-3-030-15095-2

Electronic ISBN: 978-3-030-15096-9

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-15096-9_6