Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations

Peter Benner; Tobias Breiten; Carsten Hartmann; Burkhard Schmidt

doi:10.3934/jcd.2020001

Article Contents

2020, Volume 7, Issue 1: 1-33. Doi: 10.3934/jcd.2020001

This issue Previous Article Next Article Evaluating the accuracy of the dynamic mode decomposition

Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations

1.
Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, D-39106 Magdeburg, Germany
2.
Institut für Mathematik, Karl-Franzens-Universität, Heinrichstr. 36/Ⅲ, A-8010 Graz, Austria
3.
Institut für Mathematik, Brandenburgische Technische Universität Cottbus-Senftenberg, Konrad-Wachsmann-Allee 1, D-03046 Cottbus, Germany
4.
Institut für Mathematik, Freie Universität Berlin, Arnimallee 6, D-14195 Berlin, Germany

^* Corresponding author
^* Corresponding author

Received: May 2018

Revised: June 2019

Published: June 2020

This work was partly supported by the Einstein Center for Mathematics Berlin (ECMath)

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Abstract

We study and compare two different model reduction techniques for bilinear systems, specifically generalized balancing and $ \mathcal{H}_2 $-based model reduction, and apply it to semi-discretized controlled Fokker-Planck and Liouville–von Neumann equations. For this class of transport equations, the control enters the dynamics as an advection term that leads to the bilinear form. A specific feature of the systems is that they are stable, but not asymptotically stable, and we discuss aspects regarding structure and stability preservation in some depth as these aspects are particularly relevant for the equations of interest. Another focus of this article is on the numerical implementation and a thorough comparison of the aforementioned model reduction methods.

Keywords:

Mathematics Subject Classification: Primary: 93A15, 93B40; Secondary: 93C10, 93C70.

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Figure 1. Periodically perturbed quadruple-well potential (36) used in our FPE example

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Figure 2. Eigenvectors of the discretization matrix $ A $ for our FPE example, associated with the first four right eigenvalues $ \lambda $ for $ \beta = 4 $. Note that the eigenvector for $ \lambda = 0 $ (upper left panel) corresponds to the canonical density (35)

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Figure 3. $ {\mathcal H}_2 $ error versus reduced dimension for the FPE example for $ \beta = 4 $. Comparison of BT method, SP method, and H2 method. Values that are not shown are those for which the computed error has dropped below machine precision (see Sec. 6.4)

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Figure 4. Time evolution of observables for the FPE example for $ \beta = 4 $ and for the control field given by Eq. (41) with $ t_0 = 150 $, $ \tau = 100 $, and $ a = 0.5 $: populations of the four quadrants of the $ x_1 $-$ x_2 $ plane for full ($ n = 2401 $) versus reduced dimensionality. From left to right: BT method, SP method, and H2 method

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Figure 5. Spectrum of the $ A $ matrix for the LvNE example for full versus reduced dimensionality. For relaxation rate $ \Gamma = 0.1 $ and temperature $ \Theta = 0.1 $. From left to right: BT method, SP method, and H2 method

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Figure 6. $ {\mathcal H}_2 $ error versus reduced dimensionality $ d $ for the LvNE example. Simulation results for which the error has dropped below machine precision are considered numerical artifact and thus are not shown. Left: For various values of the relaxation rate $ \Gamma $ (for constant temperature, $ \Theta = 0.1 $). Right: For various values of the temperature $ \Theta $ (for constant relaxation, $ \Gamma = 0.1 $)

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Figure 7. Time evolution of observables for the LvNE example for relaxation rate $ \Gamma = 0.1 $ and temperature $ \Theta = 0.1 $. The control field is given by Eq. (41) with $ a = 3 $, $ t_0 = 15 $, and $ \tau = 10 $. Populations of states localized in the left well, in the right well, and delocalized states over the barrier, for full ($ n = 441 $) versus reduced dimensionality. From left to right: BT method, SP method, and H2 method

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Table 1. Lowest twelve eigenvalues (in magnitude) of the discretization matrix $ A $ of the FPE example for inverse temperature $ \beta = 4 $ showing three clusters of four members each. Comparison of full versus reduced dynamics using the BT and H2 method. Results for the SP method (not shown) are very close to those for the BT method. For all practical purposes, the reduced systems for $ d>100 $ are virtually indistinguishable from the full-rank system

	BT method			H2 method
full	$ d=100 $	$ d=50 $	$ d=25 $	$ d=100 $	$ d=50 $	$ d=25 $
-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000	-0.0000
-0.0037	-0.0037	-0.0037	-0.0037	-0.0037	-0.0037	-0.0037
-0.0073	-0.0073	-0.0073	-0.0074	-0.0073	-0.0073	-0.0074
-0.0118	-0.0118	-0.0118	-0.0118	-0.0118	-0.0118	-0.0118
-0.3266	-0.3265	-0.3260	-0.3264	-0.3265	-0.3255	-0.3263
-0.3303	-0.3297	-0.3294	-0.3504	-0.3298	-0.3298	-0.3629
-0.3358	-0.3353	-0.3423	-0.5455	-0.3349	-0.3432	-0.5450
-0.3447	-0.3447	-0.3432	-0.5582	-0.3445	-0.3432	-0.6058
-0.5421	-0.5422	-0.5434	-0.6083	-0.5412	-0.5435	-0.6336
-0.5453	-0.5455	-0.5606	-0.6487	-0.5450	-0.5622	-0.6676
-0.5666	-0.5657	-0.5867	-0.7683	-0.5663	-0.5888	-0.7791
-0.5948	-0.5951	-0.6107	-0.8003	-0.5951	-0.6192	-0.8052

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