Optimal spatiotemporal reduced order modeling, Part I: proposed framework

Abstract

An optimal spatiotemporal reduced order modeling framework is proposed for nonlinear dynamical systems in continuum mechanics. In this paper, Part I, the governing equations for a general system are modified for an under-resolved simulation in space and time with an arbitrary discretization scheme. Basic filtering concepts are used to demonstrate the manner in which subgrid-scale dynamics arise with a coarse computational grid. Models are then developed to account for the underlying spatiotemporal structure via inclusion of statistical information into the governing equations on a multi-point stencil. These subgrid-scale models are designed to provide closure by accounting for the interactions between spatiotemporal microscales and macroscales as the system evolves. Predictions for the modified system are based upon principles of mean-square error minimization, conditional expectations and stochastic estimation, thus rendering the optimal solution with respect to the chosen resolution. Practical methods are suggested for model construction, appraisal, error measure and implementation. The companion paper, Part II, is devoted to demonstrating the methodology through a computational study of a nonlinear beam.

Appendices

Appendix 1: conditional average

From a statistical perspective [67], the purpose of the subgrid-scale model $\varvec{\mathcal{M }}$ is to estimate one random variable $\varvec{\tau }$ in terms of another random variable $\tilde{\mathbf{u}}$. We seek a function of $\tilde{\mathbf{u}}$, written as $\varvec{\mathcal{M }}\left(\tilde{\mathbf{u}}\right)$, to approximate $\varvec{\tau }$ such that the mean-square error is minimal; hence,

$$\begin{aligned} \mathbf{e}&= \left<\left[\varvec{\tau }-\varvec{\mathcal{M }}\left(\tilde{\mathbf{u}}\right)\right]^2\right> \\&= \int \limits ^{\infty }_{-\infty }\int \limits ^{\infty }_{-\infty }\left[\varvec{\tau }-\varvec{\mathcal{M }}\left(\tilde{\mathbf{u}}\right)\right]^2 \varvec{f}\left(\varvec{\tau },\tilde{\mathbf{u}}\right)d\varvec{\tau }d\tilde{\mathbf{u}},\nonumber \end{aligned}$$

(27)

where $\varvec{f}\left(\varvec{\tau },\tilde{\mathbf{u}}\right)$ is the joint PDF of $\tilde{\mathbf{u}}$ and $\varvec{\tau }$. Knowledge of $\varvec{f}$ is not required, but can be found given $\mathbf{u}$ and $\tilde{\mathbf{u}}$. From the definition of a conditional PDF, $\varvec{f}\left(\varvec{\tau },\tilde{\mathbf{u}}\right)=\varvec{f}\left(\varvec{\tau }|\tilde{\mathbf{u}}\right)\varvec{f}\left(\tilde{\mathbf{u}}\right)$, and (27) becomes

$$\begin{aligned} \mathbf{e}=\int \limits ^{\infty }_{-\infty }\varvec{f}\left(\tilde{\mathbf{u}}\right)\int \limits ^{\infty }_{-\infty }\left[\varvec{\tau }-\varvec{\mathcal{M }}\left(\tilde{\mathbf{u}}\right)\right]^2 \varvec{f}\left(\varvec{\tau }|\tilde{\mathbf{u}}\right)d\varvec{\tau }d\tilde{\mathbf{u}}. \end{aligned}$$

(28)

Since $\varvec{f}$ must be positive everywhere, both integrands are positive. The mean-square error is minimized with respect to $\tilde{\mathbf{u}}$ if

$$\begin{aligned} \frac{\partial \mathbf{e}}{\partial \varvec{\mathcal{M }}}=0 =\int \limits ^{\infty }_{-\infty }\varvec{f}\left(\tilde{\mathbf{u}}\right)\int \limits ^{\infty }_{-\infty }2\left[\varvec{\tau }-\varvec{\mathcal{M }}\left(\tilde{\mathbf{u}}\right)\right] \varvec{f}\left(\varvec{\tau }|\tilde{\mathbf{u}}\right)d\varvec{\tau }d\tilde{\mathbf{u}}, \nonumber \\ \end{aligned}$$

(29)

which holds if the inner integral evaluates to zero. That is, if

$$\begin{aligned} \varvec{\mathcal{M }}\left(\tilde{\mathbf{u}}\right)=\int \limits ^{\infty }_{-\infty }\varvec{\tau }\varvec{f}\left(\varvec{\tau }|\tilde{\mathbf{u}}\right)d\varvec{\tau }=\left<\varvec{\tau }|\tilde{\mathbf{u}}\right>. \end{aligned}$$

(30)

Hence, out of all possible representations for $\varvec{\mathcal{M }}\left(\tilde{\mathbf{u}}\right)$, the mean-square error is minimized when the model is equal to the mean of $\varvec{\tau }$ conditional on $\tilde{\mathbf{u}}$.

Appendix 2: stochastic estimate

A stochastic estimate [51] for the model $\varvec{\mathcal{M }}$ can be found by expanding the conditional average in the form of a multivariate power series about the event $\tilde{\mathbf{u}}=0$. Since the expansion must be truncated at some level, terms up to quadratic are retained in (15). The unknown coefficients $\varvec{\mathcal{A }}$, $\varvec{\mathcal{B }}$ and $\varvec{\mathcal{C }}$ are determined by minimizing the mean-square error between the power series and the conditional average:

$$\begin{aligned} \left(e_i\right)_j^n&= \left<\left[ \left(\mathcal{M }_i\right)_j^n - \left(\mathcal{A }_i\right)_j^n -\sum _\alpha ^{M_S}\sum _\xi ^{M_N}\sum _\mu ^{M_T} \left(\mathcal{B }_{i\alpha }\right)_{j\xi }^{n\mu } \left(\tilde{u}_\alpha \right)_\xi ^\mu \right.\right.\nonumber \\&- \left.\left. \sum _{\alpha ,\beta }^{M_S}\sum _{\xi ,\eta }^{M_N}\sum _{\mu ,\nu }^{M_T} \left(\mathcal{C }_{i\alpha \beta }\right)_{j\xi \eta }^{n\mu \nu } \left(\tilde{u}_\alpha \right)_\xi ^\mu \left(\tilde{u}_\beta \right)_\eta ^\nu \right]^2\right>. \end{aligned}$$

(31)

The orthogonality principle [67] states that each term in the mean-square error (31) must be statistically uncorrelated with the known data in the domain of interest. For the quadratic estimation, the following inner products must be orthogonal:

$$\begin{aligned}&\left<\left[ \left(\mathcal{M }_i\right)_j^n- \left(\mathcal{A }_i\right)_j^n- \sum _\alpha ^{M_S}\sum _\xi ^{M_N}\sum _\mu ^{M_T} \left(\mathcal{B }_{i\alpha }\right)_{j\xi }^{n\mu } \left(\tilde{u}_\alpha \right)_\xi ^\mu \right.\right.\\&\quad - \left.\left. \sum _{\alpha ,\beta }^{M_S}\sum _{\xi ,\eta }^{M_N}\sum _{\mu ,\nu }^{M_T} \left(\mathcal{C }_{i\alpha \beta }\right)_{j\xi \eta }^{n\mu \nu } \left(\tilde{u}_\alpha \right)_\xi ^\mu \left(\tilde{u}_\beta \right)_\eta ^\nu \right], 1 \right> =0\nonumber \\&\left<\left[ \left(\mathcal{M }_i\right)_j^n- \left(\mathcal{A }_i\right)_j^n- \sum _\alpha ^{M_S}\sum _\xi ^{M_N}\sum _\mu ^{M_T} \left(\mathcal{B }_{i\alpha }\right)_{j\xi }^{n\mu } \left(\tilde{u}_\alpha \right)_\xi ^\mu \right.\right.\nonumber \\&\quad - \left.\left. \sum _{\alpha ,\beta }^{M_S}\sum _{\xi ,\eta }^{M_N}\sum _{\mu ,\nu }^{M_T} \left(\mathcal{C }_{i\alpha \beta }\right)_{j\xi \eta }^{n\mu \nu } \left(\tilde{u}_\alpha \right)_\xi ^\mu \left(\tilde{u}_\beta \right)_\eta ^\nu \right], \left(\tilde{u}_\gamma \right)_\phi ^\lambda \right> =0\nonumber \\&\left<\left[ \left(\mathcal{M }_i\right)_j^n- \left(\mathcal{A }_i\right)_j^n- \sum _\alpha ^{M_S}\sum _\xi ^{M_N}\sum _\mu ^{M_T} \left(\mathcal{B }_{i\alpha }\right)_{j\xi }^{n\mu } \left(\tilde{u}_\alpha \right)_\xi ^\mu \right.\right.\nonumber \\&\quad - \left.\left. \sum _{\alpha ,\beta }^{M_S}\sum _{\xi ,\eta }^{M_N}\sum _{\mu ,\nu }^{M_T} \left(\mathcal{C }_{i\alpha \beta }\right)_{j\xi \eta }^{n\mu \nu } \left(\tilde{u}_\alpha \right)_\xi ^\mu \left(\tilde{u}_\beta \right)_\eta ^\nu \right], \left(\tilde{u}_\gamma \right)_\phi ^\lambda \left(\tilde{u}_\delta \right)_\psi ^\rho \right> =0 \nonumber \end{aligned}$$

(32)

The system in (32) can be simplified to the form in (16) first by letting $\varvec{\mathcal{M }}=\left<\varvec{\tau }|\tilde{\mathbf{u}}\right>$, then by expanding the inner products, commuting the mean operator, assuming constant coefficients and rearranging terms. The stochastic estimation coefficients can be found with knowledge of the moments amongst $\varvec{\tau }$ and $\tilde{\mathbf{u}}$, assuming they form a linearly independent system in (16).