Reduced Basis PMOR method: Difference between revisions

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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 $a (\cdot, \cdot; μ)$ and linear form $f (\cdot; μ)$ . The parameter $μ$ is considered within a domain $𝒟$ and we are interested in an output quantity $s (μ)$ which can be expressed via a linear functional of the field variable $l (\cdot; μ)$ .

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

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

Time-Dependent PDEs

References

@@ Line 2: / Line 2: @@
 [[Category:parametric method]]
-The Reduced Basis Method we present here is applicable to static and time-dependent linear PDEs.
+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 <math> a(\cdot, \cdot; \mu) </math> and linear form <math> f(\cdot; \mu) </math>.
+The parameter <math> \mu </math> is considered within a domain <math> \mathcal{D} </math>
+and we are interested in an output quantity <math> s(\mu) </math> which can be
+expressed via a linear functional of the field variable <math> l(\cdot; \mu) </math>.
+The exact, infinite-dimensional formulation, indicated by the superscript e, is given by
 <math>
@@ Line 11: / Line 20: @@
 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
+a(u^e(\mu),v;\mu) = f(v;\mu), \forall v \in X^e.
 \end{cases}
 </math>