Reduced Basis PMOR method

Description

The Reduced Basis Method, (RBM) we present here is a Projection based MOR method, 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 \( l(\cdot; \mu) \) of the field variable \(u(\mu)\).

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

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

Through spatial discretization, e.g. finite element method, we consider the discretized system

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

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

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

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

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

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

in a greedy sampling process. 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 \( V_N \)

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

The greedy sampling uses an error estimator ot error indicator \( \Delta_{N}(\mu) \) for the approximation error \( \| u(\mu) - u_N(\mu) \| \).

Steps of the greedy sampling process:

1. Let \( \Xi \) denote a finite sample of \( \mathcal{D} \) and set \( S_1 = \{\mu^1\} \text{ and } V_1 = span\{ u(\mu^1) \} \).

2. For \( N = 2 , ... , N_{max} \), find \( \mu^N = \text{arg max}_{ \mu \in \Xi } \Delta_{N-1}(\mu) \),

3. Set \( S_N = S_{N-1} \cup \mu^N , \quad V_N = V_{N-1} + span\{u(\mu^N)\} \).

This method is used in the following models:

Coplanar_Waveguide

Branchline Coupler

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 . Then the input-output response can be presented as follows, through 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 the empirical interpolation method can be exploited for offline-online decomposition.

This method has been used for Batch Chromatography, where the empirical interpolation method was used for treating the nonaffinity.

Contact

Martin Hess

Yongjin Zhang