Reduced Basis PMOR method

The Reduced Basis Method (RBM) we present here is applicable to static and time-dependent linear PDEs.

Time-Independent PDEs

The typical model problem of the RBM consists of a parametrized PDE stated in weak form with bilinear form $a(\cdot ,\cdot ;\mu )$ and linear form $f(\cdot ;\mu )$. The parameter $\mu$ is considered within a domain ${𝒟}$ and we are interested in an output quantity $s(\mu )$ which can be expressed via a linear functional of the field variable $l(\cdot ;\mu )$.

The exact, infinite-dimensional formulation, indicated by the superscript e, is given by

$\{\begin{array}{l}\text{For }\mu \in {𝒟}\subset {\mathbb{ℝ}}^P,\text{ evaluate }\\ s^e(\mu )=l(u^e(\mu );\mu ),\\ \text{where }{u}^e(\mu )\in X^e(\mathrm{\Omega })\text{ satisfies }\\ a(u^e(\mu ),v;\mu )=f(v;\mu ),\mathrm{\forall }v\in X^e.\end{array}$

We assume a large-scale discretization to be given, such that we consider

$\{\begin{array}{l}\text{For }\mu \in {𝒟}\subset {\mathbb{ℝ}}^P,\text{ evaluate }\\ s(\mu )=l(u(\mu );\mu ),\\ \text{where }u(\mu )\in X(\mathrm{\Omega })\text{ satisfies }\\ a(u(\mu ),v;\mu )=f(v;\mu ),\mathrm{\forall }v\in X.\end{array}$

The underlying assumption of the RBM is that the parametrically induced manifold ${ℳ}=\{u(\mu )|\mu \in {𝒟}\}$ can be approximated by a low dimensional space $V_N$.

It also applies the concept of an offline-online decomposition, in that a large pre-processing offline cost is acceptable in view of a very low online cost (of a reduced order model) for each input-output evaluation, when in a many-query or real-time context.

The essential assumption which allows the offline-online decomposition is that there exists an affine parameter dependence

$a(w,v;\mu )=\sum _{q=1}^Q{\mathrm{\Theta }}_a^q(\mu )a^q(w,v)$

$f(v;\mu )=\sum _{q=1}^{Q^f}{\mathrm{\Theta }}_f^q(\mu )f^q(v).$

The Lagrange Reduced Basis space is established by iteratively choosing Lagrange parameter samples

$S_N=\{{\mu }^1,...,{\mu }^N\}$

and considering the associated Lagrange RB spaces

$V_N=\text{span}\{u^{{𝒩}}({\mu }^n),1\le n\le N\}$

in a greedy sampling. This leads to hierarchical RB spaces: $V_1\subset V_2\subset ...\subset V_{N_{max}}$.

We then consider the galerkin projection onto the RB-space $V_N$

$\{\begin{array}{l}\text{For }\mu \in {𝒟}\subset {\mathbb{ℝ}}^P,\text{ evaluate }\\ s_N(\mu )=f(u_N(\mu )),\\ \text{where }{u}_N(\mu )\in V_N\text{ satisfies }\\ a(u_N(\mu ),v;\mu )=f(v),\mathrm{\forall }v\in V_N\end{array}$

The greedy sampling uses an error estimator ${\mathrm{\Delta }}_N(\mu )$ which estimates (even rigorously, in some cases) the approximation error $\Vert u(\mu )-u_N(\mu )\Vert$.

Let $\mathrm{\Xi }$ denote a finite sample of ${𝒟}$ and set $S_1=\{{\mu }^1\}\text{ and }{V}_1=span\{u({\mu }^1)\}$. For $N=2,...,N_{max}$, find ${\mu }^N={\text{arg max}}_{\mu \in \mathrm{\Xi }}{\mathrm{\Delta }}_{N-1}(\mu )$, and then set $S_N=S_{N-1}\cup {\mu }^N,V_N=V_{N-1}+span\{u({\mu }^N)\}$.

Time-Dependent PDEs

When time is involved, it can be roughly considered as an usual parameter just as time-independent case. But more attention should be paid to the dynamics of the system and the stability is also a major concern, especially for the nonlinear case. Mostly, we use the same notation as time-independent case except the variable $t$ is added explicitly.

The exact, infinite-dimensional formulation, indicated by the superscript e, is given by

$\{\begin{array}{l}\text{For }\mu \in {𝒟}\subset {\mathbb{ℝ}}^P,t^k\in [0,T]\text{ evaluate }\\ s^e(\mu ,t^k)=l(u^e(\mu ,t);\mu ),\\ \text{where }{u}^e(\mu ,t)\in X^e(\mathrm{\Omega })\text{ satisfies }\\ m(u^e(\mu ,t^k),v;\mu )+\mathrm{\Delta }ta(u^e(\mu ,t^k),v;\mu )=m(u^e(\mu ,t^{k-1}),v;\mu )+\mathrm{\Delta }tf(v;\mu )u^e(\mu ,t^k),\mathrm{\forall }v\in X^e.\end{array}$

Here $m(\cdot ,\cdot ;\mu )$ is also a bilinear form.

Assume a reference discretization form is given as follows,

$\{\begin{array}{l}\text{For }\mu \in {𝒟}\subset {\mathbb{ℝ}}^P,t^k\in [0,T]\text{ evaluate }\\ s(\mu ,t^k)=l(u(\mu ,t);\mu ),\\ \text{where }u(\mu ,t)\in X_{{𝒩}}(\mathrm{\Omega })\text{ satisfies }\\ m(u(\mu ,t^k),v;\mu )+\mathrm{\Delta }ta(u(\mu ,t^k),v;\mu )=m(u(\mu ,t^{k-1}),v;\mu )+\mathrm{\Delta }tf(v;\mu )u(\mu ,t^k),\mathrm{\forall }v\in X_{{𝒩}}.\end{array}$

The underlying assumption of the RBM is that the parametrically induced manifold ${ℳ}=\{u(\mu ,t)|\mu \in {𝒟}\}$ can be approximated by a low dimensional space $V_N$.

To apply the offline-online decomposition, we assume they are affine parameter-dependent, i.e.

$m(w,v;\mu )=\sum _{q=1}^{Q_m}{\mathrm{\Theta }}_m^q(\mu ,t)m^q(w,v)$

$a(w,v;\mu )=\sum _{q=1}^{Q_a}{\mathrm{\Theta }}_a^q(\mu ,t)a^q(w,v)$

$f(v;\mu )=\sum _{q=1}^{Q_f}{\mathrm{\Theta }}_f^q(\mu ,t)f^q(v).$

The Lagrange Reduced Basis space $V_N$ is usually established by POD-Greedy algorithm [3]. Then the following reduced model can be obtained by using Galerkin projection.

$\{\begin{array}{l}\text{For }\mu \in {𝒟}\subset {\mathbb{ℝ}}^P,t^k\in [0,T]\text{ evaluate }\\ s(\mu ,t^k)=l(u_N(\mu ,t);\mu ),\\ \text{where }{u}_N(\mu ,t)\in X_N(\mathrm{\Omega })\text{ satisfies }\\ m(u_N(\mu ,t^k),v;\mu )+\mathrm{\Delta }ta(u_N(\mu ,t^k),v;\mu )=m(u_N(\mu ,t^{k-1}),v;\mu )+\mathrm{\Delta }tf(v;\mu )u_N(\mu ,t^k),\mathrm{\forall }v\in X_N.\end{array}$

Note that the assumption of affine form can be relaxed in practice, then empirical interpolation [1] methods can be exploited for offline-online decomposition.

References

[1] M. Barrault, Y. Maday, N. Nguyen, and A. Patera, An `empirical interpolation' method: application to effcient reduced-basis discretization of partial differential equations, C. R. Math. Acad. Sci. Paris Series I, 339 (2004), 667-672.

[2] M. Grepl, Reduced--basis approximations and posteriori error estimation for parabolic partial differential equations, PhD thesis, MIT, 2005.

[3] B. Haasdonk and M. Ohlberger, Reduced basis method for finite volume approximations of parameterized linear evolution equations, Mathematical Modeling and Numerical Analysis, 42 (2008), 277-302.

[4] G. Rozza, D.B.P. Huynh, A.T. Patera Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations, Arch Comput Methods Eng (2008) 15: 229–275.

Contact information:

Martin Hess

Yongjin Zhang