Linear Compliant Elliptic Problems

A space \(Z\) is a linear or vector space if, for any \(\alpha \in \mathbb{R}\) , \(w,v \in Z\), \(\alpha w+v \in Z\)

An inner product space (or Hilbert space) \(Z\) is a linear space equipped with

Inner Product : An inner product \(w,v \in Z \rightarrow (w,v)_Z \in \mathbb{R}\) has to satisfy

Cauchy-Schwarz inequality: \((w,v)_Z \leq \|w\|_Z\|v\|_Z,\forall w, v \in Z\)

A norm is a map \(\| \cdot \| : Z \rightarrow \mathbb{R}\) such that

Equivalence of norms \(\| \cdot \|_Z\) and \(\| \cdot \|_Y\) : there exist positive constants \(C_1\), \(C_2\) such that \(C_1\|v\|_Z \leq \|v\|_Y \leq C_2\|v\|_Z\).

Given two inner product spaces \(Z_1\) and \(Z_2\), we define \(Z = Z_1 \times Z_2 \equiv \{(w_1,w_2)\ | \ w_1 \in Z_1,\ w_2 \in Z_2\}\) and given \(w = (w_1,w_2) \in Z, v = (v_1,v_2) \in Z\), we define \(w + v \equiv (w_1 + v_1, w_2 + v_2).\).

We also equip \(Z\) with the inner product \((w,v)_Z =(w_1,v_1)_{Z_1} +(w_2,v_2)_{Z_2}\) and induced norm \(\|w\|_Z = (w,w)_Z\).

A functional \(g : Z \rightarrow \mathbb{R}\) is a linear functional if, for any \(\alpha \in \mathbb{R}, w, v \in Z\) \(g(\alpha w + v) = \alpha g(w) + g(v)\)

A linear form is bounded, or continuous, over \(Z\) if \(|g(v)| \leq C \|v\|_Z, \forall v \in Z,\) for some finite real constant \(C\).

Given \(Z\), we define the dual space \(Z'\) as the space of all bounded linear functionals over \(Z\). We associate to \(Z'\) the dual norm \(\|g\|_{Z'} = \displaystyle\sup_{v \in Z} \frac{g(v)}{\|v\|_Z} , \forall g \in Z'\).

For any \(g \in Z'\), there exists a unique \(w_g \in Z\) such that \((w_g, v)_Z =g(v), \forall v \in Z\).

It directly follows that \(\|g\|_{Z'} = \|w_g\|_Z\)

A form \(b:Z_1 \times Z_2 \rightarrow \mathbb{R} \) is bilinear if, for any \(\alpha \in R\),

The bilinear form \(b : Z \times Z \rightarrow \mathbb{R}\) is

We also define, for a general bilinear form \(b : Z \times Z \rightarrow \mathbb{R}\), the

The bilinear form \(b : Z \times Z \rightarrow \mathbb{R}\) is

We introduce

We shall say that

Concepts of symmetry directly extend to the parametric case.

The parametric bilinear form \(b : Z \times Z \times D \rightarrow \mathbb{R}\) is

We also define

We have \(\alpha (\mu) \equiv \inf_{w \in Z} \frac{b_S(w,w;\mu)}{\|w\|^2_Z}\)

THe associated generalized eigenproblem is :

Given \(\mu \in D\), find \((\chi^{co},\nu^{co})_i(\mu) \in Z \times \mathbb{R}, 1 \leq i \leq \dim(Z),\) such that \(b_S(\chi_i^{co}(\mu), v; \mu) = \nu_i^{co}(\mu)(\chi_i^{co}(\mu), v)_Z\) and \(\|\chi_i^{co}(\mu)\|_Z=1\).

Let \(\nu_1^{co}(\mu) \leq \nu_2^{co}(\mu) \leq \ldots \leq \nu_{\dim{Z}}^{co} (\mu)\) and \(b\) coercive, then \(\alpha (\mu) = \nu_1^{co}(\mu) > 0.\)

We assume \(g(v;\mu)= \displaystyle\sum_{q=1}^{Q_g} \theta^q_g(\mu)g^q(v), \forall v \in Z,\) where, for \(1 \leq q \leq Q_g\) (finite),

and \(b(w,v;\mu)= \displaystyle\sum_{q=1}^{Q_b} \theta^q_b(\mu) b^q(w,v),\quad \forall w,v \in Z,\) where, for \(1 \leq q \leq Q_b\) (finite),

The coercive bilinear form \(b : Z \times Z \times D \rightarrow \mathbb{R}\) \(b(w,v;\mu)= \displaystyle\sum_{q=1}^{Q_b} \theta^q_b(\mu) b^q(w,v),\quad \forall w,v \in Z,\) is parametrically corecive if \(c\equiv b_S\) is affine \(c(w,v;\mu)= \displaystyle\sum_{q=1}^{Q_c} \theta^q_c(\mu) c^q(w,v),\quad \forall w,v \in Z,\) and satisfies and

We consider (real)

and

And we define the canonical basis vectors as \(e_i, 1 \leq i \leq d.\)

Given a scalar (or one component of a vector)

where

Let \(m \in N_0\), the space \(C^m(\Omega )\) is defined as \(C^m(\Omega )\equiv \{w | D^\sigma w \in C^0(\Omega ), \forall \sigma \in I^{d,m}\},\) and \(C^0(\Omega )\) is the space of continuous functions over \(\Omega \in \mathbb{R}^d\).

We denote by \(C^\infty (\Omega )\) the space of functions \(w\) for which \(D^\sigma\) exists and is continuous for any order \(|\sigma |.\)

We define, for \(1 \leq p < \infty\), the Lebesgue space \(L^p(\Omega )\) as \(L^p(\Omega )\equiv \{ w \text{ measurable } | \|w\|_{L^p(\Omega )} < \infty\}\) where

Let \(m \in \mathbb{N}_0\), the space \(H^m(\Omega )\) is then defined as \(H^m(\Omega )\equiv \{w | D^\sigma w \in L^2(\Omega ), \forall \sigma \in I^{d,m}\},\) with associated inner product \((w,v)_{H^m(\Omega )}\equiv \displaystyle\sum_{\sigma \in I^{d,m}}\int_\Omega D^\sigma w D^\sigma v dx,\) and induced norm \(\|w\|_{H^m(\Omega )} \equiv \sqrt{(w, w)_{H^m(\Omega )}}.\)

Since we only consider second-order PDEs, we require mostly

\(\Rightarrow\) Space of all functions \(w : \Omega \rightarrow \mathbb{R}\) square-integrable over \(\Omega\) .

Note that, for any \(v \in H_0^1(\Omega )\), we have \(C_{PF} \|v\|_{H^1(\Omega )} \leq |v|_{H^1(\Omega )} \leq \|v\|_{H^1(\Omega )},\) and thus \(\|v\|_{H^1(\Omega)} = 0 \, \Rightarrow v = 0\) \(\Rightarrow |v|_{H^1(\Omega )}\) constitutes a norm for \(v \in H_0^1(\Omega ).\)

Given Hilbert Spaces \(Y\) and \(Z \subset Y\) , the projection, \(\Pi : Y \rightarrow Z\), of \(y \in Y\) onto \(Z\) is defined as

Properties:

Given an orthonormal basis \(\{ \varphi_i\}_{i=1, N = \dim(Z)}\), then \(\Pi y= \sum_{i=1}^{\dim(Z)} ( \varphi_i,y)_Y \varphi_i, \forall y \in Y\)

In these notes \(\Omega\) is considered

We consider the following \(\mu-\)PDE

\(a(\cdot,\cdot;\mu)\) is :

\(f(\cdot;\mu), \ell(\cdot;\mu)\) are :

and in particular, to start, the compliant case

We require for the RB methodology \(a(u,v;\mu) = \displaystyle\sum_{q=1}^{Q_a} \Theta^q_a(\mu)\ a^q( u, v )\) where for \(q=1,...,Q_a\)

Remark :

We assume \(a\) coercive and continuous

Recall that

Let \(\mu \in \mathcal{D}^{\mu}\), evaluate \(\displaystyle s^{\mathcal{N}} (\mu) = \ell (u^{\mathcal{N}} (\mu))\), where \(u^{\mathcal{N}} (\mu) \in X^{\mathcal{N}}\) satisfies \(a (u^{\mathcal{N}} (\mu), v; \mu ) = f (v), \quad \forall \: v \in X^{\mathcal{N}}\). Here \(X^{\mathcal{N}} \subset X\) is a truth finite element approximation of dimension \(\mathcal{N}\gg 1\) equiped with an inner product \((\cdot,\cdot)_X\) and induced norm \(||\cdot||_X\). Denote also \(X'\) and associated norm \(\ell \in X',\qquad\displaystyle ||\ell||_{X'} \equiv \operatorname{sup}_{v\in X}\frac{\ell(v)}{||v||_X}\).

\(\Rightarrow u^{\mathcal{N}} (\mu)\) is a calculable surrogate for \(u(\mu).\) \(\|u(\mu)-u^\mathcal{N}(\mu)\|_{X} \leq \underbrace{\|u(\mu)-u^\mathcal{N}(\mu)\|_{X}}_{\leq \varepsilon^\mathcal{N}} + \underbrace{\|u^\mathcal{N}(\mu)-u^N(\mu)\|_X}_{\varepsilon_{\mathrm{tol,min}}}\)

with \(\varepsilon^\mathcal{N} \ll \varepsilon_{\mathrm{tol,min}}\)

Given \(\xi^n = u(\mu^n), 1 \leq n \leq N_\text{max}\) (Lagrange case) we construct the basis set \(\{\zeta^n, 1 \leq n \leq N_\text{max}\}\), from

Given reduced basis space \(X_N = {\rm span} \: \{ \zeta^n, n = 1, \ldots, N \}, 1 \leq N \leq N_{max}\) we can express any \(w_N \in X_N\) as \(w_N = \displaystyle\sum_{k=1}^N {w_N}_n \zeta^n\) for unique \({w_N}_n \in \mathbb{R}, 1 \leq n \leq N\).

Reduced basis « matrices » \(Z_N \in \mathbb{R}^{\mathcal{N}\times N} , 1 \leq N \leq N_{max}:\) \(Z_N=[\zeta^1,\zeta^2,...,\zeta^N], 1 \leq N \leq N_{max}\) where, from orthogonality, \(Z^T_{N_{max}} X Z^T_{N_{max}} = I_{N_{max}}\), and \(I_M\) is the Identity matrix in \(\mathbb{R}^{M\times M}\).

Given \(\mu \in \mathcal{D}^{\mu}\) evaluate

where \(u_N (\mu) \in X_N\) satisfies

For any \(\mu \in \mathcal{D}^\mu\), we have the following optimality results (thanks to Galerkin)

and

and finally \(0 \leq s(\mu)-s_N(\mu) \leq \gamma(\mu)\inf_{v_N \in X_N} ||u(\mu) - v_N(\mu)||^2_X\)

Expand our RB approximations

Express \(s_N(\mu)\)

where \({u_N}_i(\mu), 1 \leq i \leq N\) satisfies

for \(1 \leq i \leq N\)

Solve \(\underline{A}_N (\mu) \: \underline{u}_N (\mu) = \underline{F}_N\)

where

Offline: independent of \(\mu\)

Online: independent of \(\mathcal{N}\)

Online: \(N << \mathcal{N}\) Online we realize often orders of magnitude computational economies relative to FEM in the context of many \(\mu\)-queries.

Proposition : Thanks to the orthonormalization of the basis function, we have that the condition number of \(A_N(\mu)\) is bounded by the ratio \(\dfrac{\gamma(\mu)}{\alpha(\mu)}\).

Proof : * Write the Rayleigh Quotient \(\dfrac{v_N^T A_N(\mu) v_N}{v_N^T v_N}, \quad \forall v_N \in \mathbb{R}^N\) * Express \(v_N = \sum_{n=1}^N v_{N_n} \zeta^n\) * Use coercivity, continuity and orthonormality.