POD-DEIM Approach on Dimension Reduction of a Model of Chemotactic Bacteria Patterns in Semi-Solid Medium

Last modified: 2014-03-31

#### Abstract

The reduced-order approach is based on projecting the dynamical system onto subspacesconsisting of basis elements that contain characteristics of the expected solution. Currently, Proper Orthogonal Decomposition (POD) is probably the mostly used and most successful model reduction technique, where the basis functions contain information from the solutions of the dynamical system at pre-specified time-instances, so-called{\it snapshots}. Due to a possible linear dependence or almost linear dependence, the snapshotsthemselves are not appropriate as a basis. Hence a singular value decomposition is carried outand the leading generalized eigenfunctions are chosen as a basis, referred to as the POD basis.

A considerable reduction in complexity is achieved by DEIM -- a discrete variation of Empirical Interpolation Method (EIM), proposed by Barrault, Maday, Nguyen and Patera in: {\it An ``empirical interpolation'' method: Application to efficient reduced-basis discretization of partial differential equations}, C. R. Math. Acad. Sci. Paris, 339 (2004), 667--672.According to this method, the evaluation of the approximate nonlinearterm does not require a prolongation of the reduced state variables back to theoriginal high dimensional state approximation required to evaluate the nonlinearityin the POD approximation.

In this study we carry out an application of DEIM combined with POD to provide dimension reduction of a model that focuses on the chemotactic bacteria patterns in semi-solid medium.The model defined by a system of three reaction-diffusion-chemotaxis equations was proposed by Tyson, R., Lubkin, S.R., Murray, J.D.: {\it A minimal mechanism for bacterial pattern formation}. Proc.Roy. Soc. Lond. B 266, 299--304 (1999), which, in dimensionless form, can be written as:

\begin{eqnarray*}& & u_t +\alpha\nabla\cdot\left[\frac{u}{(1+v)^2}\nabla v\right] = d_u\Delta u + \rho u \left(\delta\frac{w^2}{1+w^2}-u\right) \\& & v_t = d_v\Delta v + \beta\frac{wu^2}{\alpha+u^2}-\gamma u v \\& & w_t = d_w\Delta w -\kappa u \frac{w^2}{1+w^2}\end{eqnarray*}The variables $u$ and $v$ denote the cell density and a chemoattractantconcentration, respectively. The variable $w$ represents the nutrientconcentration and the coefficients $\alpha$, $d_u$, $\rho$, $\delta$, $d_v$, $\beta$, $\sigma$, $\gamma$, $d_w$, and $\kappa$ are positive constants.We show DEIM improves the efficiency of the POD approximation and achieves a complexity reductionof the nonlinear terms. Numerical results are presented.

A considerable reduction in complexity is achieved by DEIM -- a discrete variation of Empirical Interpolation Method (EIM), proposed by Barrault, Maday, Nguyen and Patera in: {\it An ``empirical interpolation'' method: Application to efficient reduced-basis discretization of partial differential equations}, C. R. Math. Acad. Sci. Paris, 339 (2004), 667--672.According to this method, the evaluation of the approximate nonlinearterm does not require a prolongation of the reduced state variables back to theoriginal high dimensional state approximation required to evaluate the nonlinearityin the POD approximation.

In this study we carry out an application of DEIM combined with POD to provide dimension reduction of a model that focuses on the chemotactic bacteria patterns in semi-solid medium.The model defined by a system of three reaction-diffusion-chemotaxis equations was proposed by Tyson, R., Lubkin, S.R., Murray, J.D.: {\it A minimal mechanism for bacterial pattern formation}. Proc.Roy. Soc. Lond. B 266, 299--304 (1999), which, in dimensionless form, can be written as:

\begin{eqnarray*}& & u_t +\alpha\nabla\cdot\left[\frac{u}{(1+v)^2}\nabla v\right] = d_u\Delta u + \rho u \left(\delta\frac{w^2}{1+w^2}-u\right) \\& & v_t = d_v\Delta v + \beta\frac{wu^2}{\alpha+u^2}-\gamma u v \\& & w_t = d_w\Delta w -\kappa u \frac{w^2}{1+w^2}\end{eqnarray*}The variables $u$ and $v$ denote the cell density and a chemoattractantconcentration, respectively. The variable $w$ represents the nutrientconcentration and the coefficients $\alpha$, $d_u$, $\rho$, $\delta$, $d_v$, $\beta$, $\sigma$, $\gamma$, $d_w$, and $\kappa$ are positive constants.We show DEIM improves the efficiency of the POD approximation and achieves a complexity reductionof the nonlinear terms. Numerical results are presented.

#### Keywords

chemotactic bacteria pattern; reduced order modeling; proper orthogonal decomposition (POD); discrete empirical interpolation method (DEIM)