Applications in Process Engineering and Molecular Simulations

Process engineering and molecular simulations are among the important applications with large practical relevance, especially to industries. To validate their effectivity, model order reduction (MOR) algorithms are applied to some typical applications.

PMOR for Poisson-Boltzmann Equation

Electrostatics plays an important role in virtually all processes that involve biomolecules in solution, for example, proteins, nucleic acids and their ligands. The Poisson-Boltzmann equation (PBE) is one of the most fundamental approaches for treating the effects of electrostatics in biomolecular solutions.

Electrostatic Interactions
In computer simulations of Brownian motion of biomolecular systems, the PBE is used to determine the electrostaic potentials of interacting proteins. Currently, the PB solution is pre-computed on a three-dimensional grid and held fixed during simulations due to the high complexity of re-computing the full solution subjected to the parameter change. Therefore, higher level iterations may converge slowly, and a loss of accuracy is expected. The aim of this project is to develop a reduced basis method (RBM) that yields a parametric low-order model allowing the fast solution of the PBE in a many-query context since the solution can be re-computed for the changed parameter with little effort. Consequently, the Brownian dynamics computations can be accelerated significantly.

We have also applied successfully a solution decomposition technique based on range-separated (RS) canonical/Tucker tensor format to the PBE. The RS tensor format provides efficient numerical treatment of the long-range electrostatic potentials in many-particle systems using large 3D Cartesian grids. In this study, we modify the PBE by replacing the Dirac delta distribution function by its smooth long-range approximation obtained from the RS tensor format. This is aimed at eliminating the numerical approximation to the short-range component of the potential which corresponds to the highly singular data from the delta function. Thus, the resultant long-range component is calculated numerically from the modified PBE and added to the short-range component which is calculated apriori from the RS tensor format. The resulting solution is of high accuracy compared to the original model. 


F. Fogolari, A. Brigo, and H. Molinari
The Poisson-Boltzmann equation for biomolecular electrostatics: a tool for structural biology
J. Mol. Recognit., vol. 15, pp. 377-392, 2002.
M. J. Holst
Multilevel methods for the Poisson-Boltzmann equation
J.S. Hesthaven, G. Rozza, and B. Stamm
Certified Reduced Basis Methods for Paramtertized Partial Differential Equations
SpringerBriefs in Mathematics, ISBN: 978-3-319-22469-5, Springer Cham, 2016.
P. Benner, V. Khoromskaia, and B.N. Khoromskij
Range-separated tensor formats for numerical modeling of many-particle interaction potentials

MOR for nonlinear fluidized bed crystallizer

We consider model order reduction (MOR) for population balance system of a fluidized bed crystallizer model. Finite volume discretization of the partial differential equations which describe the crystallizer model results in a very large-scale nonlinear system. MOR constructs reduced-order model (ROM) of the large-scale system using proper orthogonal decomposition (POD) method and empirical interpolation (EI). We employ a posteriori error estimation to derive a reliable ROM. In addition, we seek to construct the basis for POD and the basis for EI automatically by using the error estimation, so that the final ROM can be generated automatically.


S. Chaturantabut and D. C. Sorensen
Nonlinear model reduction via discrete empirical interpolation

SIAM Journal on Scientific Computing, 32 (5): 2737-2764, 2010.
M.Mangold, L.Feng, D.Khlopov, S.Palis, P.Benner, D.Binev and A.Seidel-Morgenstern
Nonlinear model reduction of a continuous fluidized bed crystallizer
Journal of Computational and Applied Mathematics 289, pp. 253 - 266, 2015.
Y. Zhang, L. Feng, S. Li, and P. Benner
An efficient output error estimation for model order reduction of parametrized evolution equations
SIAM Journal on Scientific Computing 2015 37:6, B910-B936, 2015.
L. Feng, M. Mangold, and P. Benner.
Adaptive POD-DEIM basis construction and its application to a nonlinear population balance system
AIChE J., 63: 3832–3844, 2017.
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