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 problems. But more attention should be paid to the dynamics of the system and the stability becomes 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

\( \begin{cases} \text{For } \mu \in \mathcal{D} \subset \mathbb{R}^P, t\in [0,T], \text{ evaluate } \\ s^e(\mu;t) = l(u^e(\mu;t);\mu;t), \\ \text{where } u^e(\mu;t) \in X^e(\Omega) \text{ satisfies } \\ a(u^e(\mu;t),v;\mu;t) = f(v;\mu;t), \forall v \in X^e. \end{cases} \)

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