# American Institute of Mathematical Sciences

June  2017, 4(1&2): 1-19. doi: 10.3934/jcd.2017001

## Kernel methods for the approximation of some key quantities of nonlinear systems

 1 Laboratory for Computational and Statistical Learning, Massachusetts Institute of Technology, Cambridge, MA, USA 2 Department of Mathematics, AlFaisal University, Riyadh, KSA

* Corresponding author:Boumediene Hamzi

Published  April 2017

Fund Project: BH thanks the European Commission and the Scientific and the Technological Research Council of Turkey (Tubitak) for financial support received through a Marie Curie Fellowship, and JB gratefully acknowledges support under NSF contracts NSF-IIS-08-03293 and NSF-CCF-08-08847 to M. Maggioni.
Parts of this work were done while the authors were at the Department of Mathematics of Duke University and then while the second author was with the Departments of Mathematics of Imperial College London, Yildiz Technical University and Ko¸c University for Marie Curie Fellowships and at the Fields Institute.

We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems -with a reasonable expectation of success -once the nonlinear system has been mapped into a high or infinite dimensional feature space. In particular, we embed a nonlinear system in a reproducing kernel Hilbert space where linear theory can be used to develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems. In all cases the relevant quantities are estimated from simulated or observed data. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.

Citation: Jake Bouvrie, Boumediene Hamzi. Kernel methods for the approximation of some key quantities of nonlinear systems. Journal of Computational Dynamics, 2017, 4 (1&2) : 1-19. doi: 10.3934/jcd.2017001
##### References:

show all references

##### References:
Comparison between the exact (red), the kernel-based (green) and the empirical (black) steady-state distribution
 [1] Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845 [2] Luis Almeida, Federica Bubba, Benoît Perthame, Camille Pouchol. Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations. Networks & Heterogeneous Media, 2019, 14 (1) : 23-41. doi: 10.3934/nhm.2019002 [3] Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017 [4] Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485 [5] Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016 [6] José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 [7] Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056 [8] Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008 [9] Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028 [10] Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028 [11] Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250 [12] Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215 [13] Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165 [14] Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic & Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009 [15] Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65 [16] Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079 [17] Simon Plazotta. A BDF2-approach for the non-linear Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2893-2913. doi: 10.3934/dcds.2019120 [18] Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028 [19] Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337 [20] Florian Schneider, Andreas Roth, Jochen Kall. First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic & Related Models, 2017, 10 (4) : 1127-1161. doi: 10.3934/krm.2017044

Impact Factor: