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 $\mathcal{D}$ 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

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

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

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

The underlying assumption of the RBM is that the parametrically induced manifold $\mathcal{M} = \{u(\mu) | \mu \in \mathcal{D}\}$ 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

$a(w,v;\mu) = \sum_{q=1}^Q \Theta_a^q(\mu) a^q(w,v)$

$f(v;\mu) = \sum_{q=1}^{Q^f} \Theta_f^{q}(\mu) f^q(v).$

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

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

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 $V_N$

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

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

Let $\Xi$ denote a finite sample of $\mathcal{D}$ and set $S_1 = \{\mu^1\} \text{ and } V_1 = span\{ u(\mu^1) \}$. For $N = 2 , ... , N_{max}$, find $\mu^N = \text{arg max}_{ \mu \in \Xi } \Delta_{N-1}(\mu)$, and then set $S_N = S_{N-1} \cup \mu^N , \quad V_N = V_{N-1} + span\{u(\mu^N)\}$.

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

$\begin{cases} \text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, t^k \in [0,T] \text{ evaluate } \\ s^e(\mu,t^k) = l(u^e(\mu,t);\mu), \\ \text{where } u^e(\mu,t) \in X^e(\Omega) \text{ satisfies } \\ m(u^e(\mu,t^k),v;\mu) + \Delta t a(u^e(\mu,t^k),v;\mu) = m(u^e(\mu,t^{k-1}),v;\mu) + \Delta t f(v;\mu)u^e(\mu, t^k), \forall v \in X^e. \end{cases}$ Here $m(\cdot,\cdot;\mu)$ is also a bilinear form.

Assume a reference discretization form is given as follows,

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

The underlying assumption of the RBM is that the parametrically induced manifold $\mathcal{M} = \{u(\mu,t) | \mu \in \mathcal{D}\}$ 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.

$m(w,v;\mu) = \sum_{q=1}^{Q_m} \Theta_m^q(\mu,t) m^q(w,v)$

$a(w,v;\mu) = \sum_{q=1}^{Q_a} \Theta_a^q(\mu,t) a^q(w,v)$

$f(v;\mu) = \sum_{q=1}^{Q_f} \Theta_f^{q}(\mu,t) f^q(v).$

The Lagrange Reduced Basis space $V_N$ is usually established by POD-Greedy algorithm [2]. Then the following reduced model can be obtained by using Galerkin projection.

$\begin{cases} \text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, t^k \in [0,T] \text{ evaluate } \\ s(\mu,t^k) = l(u_N(\mu,t);\mu), \\ \text{where } u_N(\mu,t) \in X_{N}(\Omega) \text{ satisfies } \\ m(u_N(\mu,t^k),v;\mu) + \Delta t a(u_N(\mu,t^k),v;\mu) = m(u_N(\mu,t^{k-1}),v;\mu) + \Delta t f(v;\mu)u_N(\mu, t^k), \forall v \in X_N. \end{cases}$

Note that the assumption of affine form can be relaxed in practice, then empirical interpolation [3] 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