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.