# Linear Compliant Elliptic Problems

## 1. Notations, Definitions, Problem Statement, Example

### 1.1. Inner Product Spaces

#### 1.1.1. Definitions

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

 $\mathbb{R}$ denotes the real numbers, and $\mathbb{N}$ and $\mathbb{C}$ shall denote the natural and complex numbers, respectively.

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

• an inner product $(w,v)_Z, \forall w,v \in Z$, and

• induced norm $\|w\|_Z = (w,w)_Z, \forall w \in Z$.

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

• Bilinearity :

• $(\alpha w+v,z)_Z =\alpha(w,z)_Z +(v,z)_Z \forall \alpha\in R,w,v,z\in Z$

• $(z,\alpha w+v)_Z =\alpha(z,w)_Z +(z,v)_Z, \forall \alpha\in R, w,v,z\in Z$

• Symmetry : $(w,v)_Z = (v,w)_Z, \forall w,v \in Z$

• Positivity :

• $(w,w)_Z >0, \forall w \in Z, w \neq 0$

• $(w,w)_Z =0$ only if $w=0$

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

#### 1.1.2. Norm

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

• $\|w\|_Z > 0\quad \forall w\in Z,w\neq 0,$

• $\|\alpha w\|_Z = |\alpha |\|w\|_Z\quad \forall \alpha \in \mathbb{R},\ \forall w\in Z,$

• $\|w+v\|_Z \leq \|w\|_Z +\|v\|_Z\quad \forall w\in Z,\ \forall v\in Z.$

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

#### 1.1.3. Cartesian Product Space

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 $\|w\|_Z = (w,w)_Z$.

### 1.2. Linear and Bilinear Forms

#### 1.2.1. Linear Forms

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 $C$.

#### 1.2.2. Dual Spaces

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 $\|g\|_{Z'} = \|w_g\|_Z$

#### 1.2.3. Bilinear Forms

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

• $b(\alpha w + v,z) = \alpha b(w,z) + b(v,z), \forall w,v \in Z_1, z \in Z_2$

• $b(z,\alpha w + v) =\alpha b(z,w) + b(z,v), \forall z \in Z_1, w,v \in Z_2$

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

• symmetric, if $b(w,v) = b(v,w),$

• skew-symmetric, if $b(w,v) = -b(v,w),$

• positive definite, if $b(v,v) \geq 0\text{ , with equality only for } v = 0.$

• positive semidefinite, if $b(v,v) \geq 0, \forall v \in Z.$

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

• symmetric part as $b_S(w,v) = 1/2 (b(w,v) + b(v,w)), \forall w,v \in Z;$

• the skew-symmetric part as $b_{SS}(w,v) = 1/2 (b(w,v) - b(v,w)), \forall w,v \in Z.$

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

• coercive over $Z$ if $\alpha \equiv \inf_{w\in Z} \frac{b(w,w)}{\|w\|^2_Z}$ is positive;

• continuous over $Z$ if $\gamma \equiv \sup_{w\in Z} \sup_{v\in Z} \frac{b(w, v)}{\|w\|_Z \|v\|_Z}$ is finite.

#### 1.2.4. Parametric Linear and Bilinear Forms

We introduce

• $D \subset \mathbb{R}^P$ : closed bounded parameter domain;

• $\mu = (\mu_1,\ldots,\mu_P) \in D$ : parameter vector.

We shall say that

• $g:Z\times D\rightarrow \mathbb{R}$ is a parametric linear form if, for all $\mu \in D, g( \cdot ; \mu) : Z \rightarrow \mathbb{R}$ is a linear form;

• $b:Z\times Z\times D\rightarrow \mathbb{R}$ is a parametric bilinear form if,for all $\mu \in D, b( \cdot , \cdot ; \mu) : Z \times Z \rightarrow \mathbb{R}$ is a bilinear form.

Concepts of symmetry directly extend to the parametric case.

#### 1.2.5. Parametric Linear and Bilinear Forms

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

• coercive over Z if $\alpha(\mu) \equiv \inf_{w \in Z} \frac{b(w,w;\mu)}{\|w\|^2_Z}$ is positive for all $\mu \in D$;

• continuous over $Z$ if $\gamma(\mu)\equiv \sup_{w \in Z} \sup_{v \in Z} \frac{b(w, v; \mu)}{\|w\|_Z\|v\|_Z}$ is finite for all $\mu \in D.$

We also define

\begin{align} (0 <) \alpha _0 & \equiv \min_{\mu \in D} \alpha (\mu)\\ \gamma_0 & \equiv \max_{\mu \in D} \gamma (\mu) (< \infty ). \end{align}

#### 1.2.6. Coercivity EigenProblem

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 $\alpha (\mu) = \nu_1^{co}(\mu) > 0.$

#### 1.2.7. Parameter affine Dependence

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),

• parameter-dependant functions $\theta^q_g : D \rightarrow \mathbb{R}$,

• parameter-independant forms $g^q : Z \rightarrow \mathbb{R};$

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),

• parameter-dependant functions $\theta^q_b : D \rightarrow \mathbb{R}$,

• parameter-independant forms $b^q : Z \times Z \rightarrow \mathbb{R}$.

#### 1.2.8. Parametric Coercivity

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

• $\theta^q_c(\mu)>0, \forall \mu \in D, 1\leq q\leq Q_c,$

• $c^q(v,v)\geq 0,\forall v \in Z, 1\leq q\leq Q_c.$

### 1.3. Classes of Functions

#### 1.3.1. Scalar and Vector Fields

We consider (real)

• scalar-valued field variables (e.g., temperature, pressure) $w : \Omega \rightarrow \mathbb{R}^{d=1}$

• vector-valued field variables (e.g., displacement, velocity) $\mathbf{w} : \Omega \rightarrow \mathbb{R}^d$ , where $\mathbf{w}(x) = (w_1(x), \ldots , w_d (x));$

and

• $\Omega \in \mathbb{R}^d, d=1, 2, \text{or } 3$ is an open bounded domain

• $x = (x_1,...,x_d) \in \Omega$;

• $\Omega$ has Lipschitz continuous boundary $\partial \Omega$

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

#### 1.3.2. Multi-Index Derivative

Given a scalar (or one component of a vector)

• field $w : \Omega \rightarrow \mathbb{R}$ (SPATIAL DERIVATIVE)

$(D^{\sigma} w)(x) = \frac{\partial^\sigma w}{\partial x_1^{\sigma_1} ...\partial x_d^{\sigma d}}$
• parametric field $w : \Omega \times D \rightarrow \mathbb{R}$ (SENSITIVITY DERIVATIVE)

$(D_{\sigma} w)(x) = \frac{\partial^\sigma w}{\partial \mu_1^{\sigma_1} ...\partial \mu_d^{\sigma d}}$

where

• $\sigma = (\sigma_1,\ldots,\sigma_d)$, $\sigma_i, 1 \leq i \leq d$, non-negative integers;

• $|\sigma| = \sum_{j=1}^{d} \sigma_j$ is the order of the derivative; and

• $I^{d,n}$ is set of all index vectors $\sigma \in N^d_0$ such that $|\sigma | \leq n.$

#### 1.3.3. Function Spaces

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 $|\sigma |.$

#### 1.3.4. Lebesgue Spaces

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

• $\|w\|_{L^p(\Omega )} \equiv \left( \int_\Omega |w|^pdx\right)^{1/p} , 1\leq p<\infty,$

• $\|w\|_{L^\infty (\Omega )} \equiv \mathrm{ess} \sup_{x\in\Omega} |w(x)|, p = \infty .$

#### 1.3.5. Hilbert Space

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 $\|w\|_{H^m(\Omega )} \equiv \sqrt{(w, w)_{H^m(\Omega )}}.$

#### 1.3.6. Special (most important) cases

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

• $L^2(\Omega ) = H^0(\Omega )$: Lebesgue Space $p = 2$

• $(w,v)_{L^2(\Omega)} = \int_\Omega w v \quad \forall w, v \in L^2(\Omega )$

• $\|w\|_{L^2(\Omega)} = \sqrt{(w,w)_{L^2(\Omega)}} \forall w \in L^2(\Omega ),$

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

• $H^1(\Omega)$ $H^1(\Omega ) \equiv \{w \in L^2(\Omega )| \frac{\partial w}{ \partial xi} \in L^2(\Omega ), 1\leq i\leq d\}$ with inner product and induced norm $(w,v)_{H^1(\Omega )} \equiv \int_\Omega \nabla w \cdot \nabla v + wv\quad \forall w,v \in H^1(\Omega ),$, $\|w\|_{H^1(\Omega )} \equiv \sqrt{(w,w)_{H^1(\Omega)}}\quad \forall w \in H^1(\Omega ),$ and seminorm $|w|_{H^1(\Omega )} \equiv \int_\Omega \nabla w \cdot \nabla w,\quad \forall w \in H^1(\Omega ).$

• the space $H_0^1(\Omega )$ $H^1_0(\Omega) \equiv \{v \in H^1(\Omega )|v_{|\partial \Omega}=0 \}$ where $v = 0$ on the boundary $\partial \Omega .$

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 $v \in H_0^1(\Omega ).$

#### 1.3.7. Projection

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

$(\Pi y,v)_Y = (y,v)_Y , \forall v \in Z$

Properties:

• Orthogonality: $(y - \Pi y, v)_Y = 0$

• Idempotence: $\Pi (\Pi y) = \Pi y$

• Best Approximation $\|y-\Pi y\|^2_Y = \inf_{v \in Z} \|y-v\|^2_Y, \,$

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

### 1.4. Notations

#### 1.4.1. Notations

• $(\cdot)^\mathcal{N}$ finite element approximation

• $(\cdot)_N$ reduced basis approximation

• $\mu$ input parameter (physical, geometrical,…​)

• $s(t;\mu) \approx s^\mathcal{N}(t;\mu)\approx s_N(t;\mu)$ output approximations

• $\mu \rightarrow s(t;\mu)$ input-output relationship

#### 1.4.2. Definitions

• $\Omega \subset \mathbb{R}^d$ spatial domain

• $\mu$ $P$-uplet

• $\mathcal{D}^\mu \subset \mathbb{R}^P$ parameter space

• $s$ output, $\ell, f$ functionals

• $u$ field variable

• $X$ function space $H^1_0(\Omega)^\nu \subset X \subset H^1(\Omega)^\nu$ ($\nu=1$ for simplicity)
$(\cdot,\cdot)_X$ scalar product and $\|\cdot\|_X$ norm associated to $X$

### 1.5. Problem Statement

The formal problem statement reads: Given $\mu \in \mathcal{D}^\mu$, evaluate $s(\mu) = \ell(u(\mu);\mu)$ where $u(x;\mu) \in X$ satisfies $a(u(\mu), v; \mu ) = f(v; \mu), \quad \forall v \in X$

We consider first the case of linear affine compliant elliptic problem and then complexify.

#### 1.5.1. Hypothesis: Reference Geometry

In these notes $\Omega$ is considered

• To apply the reduced basis methodology exposed later, we need to setup a reference spatial domain $\Omega_{\mathrm{ref}}$

• We introduce an affine mapping $\mathcal{T}(\cdot;\mu) : \Omega (\equiv \Omega_{\mathrm{ref}} = \Omega_o(\bar{\mu})) \rightarrow \Omega_o(\mu)$ such that $a(u,v;\mu) = a_o(u_o \circ \mathcal{T}_\mu,v_o \circ \mathcal{T}_\mu;\mu)$

#### 1.5.2. Hypothesis: Continuity, stability, compliance

We consider the following $\mu-$PDE

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

• bilinear

• symmetric

• continuous

• coercive ($\forall \mu \in \mathcal{D}^\mu$)

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

• linear

• bounded ($\forall \mu \in \mathcal{D}^\mu$)

and in particular, to start, the compliant case

• $a$ symmetric

• $f(\cdot;\mu) = \ell(\cdot;\mu)\quad \forall \mu \in \mathcal{D}^\mu$

#### 1.5.3. Hypothesis: Affine dependence in the parameter

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 $q=1,...,Q_a$

\begin{align*} \theta^q_a : & \mathcal{D}^\mu \to \mathbb{R} & \mu\text{-dependant functions}\\ a^q :& X \times X \rightarrow \mathbb{R} & \mu\text{-independent bilinear forms} \end{align*}

Remark :

• similar decomposition is required for $\ell(v;\mu)$ and $f(v;\mu)$, and denote $Q_\ell$ and $Q_f$ the corresponding number of terms

• applicable to a large class of problems including geometric variations

• can be relaxed (see non affine/non linear problems)

#### 1.5.4. Inner Products and Norms

• energy inner product and associated norm (parameter dependant) \begin{aligned} (((w,v)))_\mu &= a(w,v;\mu) &\ \forall u,v \in X\\ |||v|||_\mu &= \sqrt{a(v,v;\mu)} &\ \forall v \in X \end{aligned}

• $X$-inner product and associated norm (parameter independant) \begin{aligned} (w,v)_X &= (((w,v)))_{\bar{\mu}} \ (\equiv a(w,v;\bar{\mu})) &\ \forall u,v \in X\\ ||v||_X &= |||v|||_{\bar{\mu}} \ (\equiv \sqrt{a(v,v;\bar{\mu})}) & \ \forall v \in X \end{aligned}

#### 1.5.5. Coercivity and Continuity Constants

We assume $a$ coercive and continuous

Recall that

• coercitivy constant $(0 < ) \alpha(\mu) \equiv \inf_{v\in X}\frac{a(v,v;\mu)}{||v||^2_X}$

• continuity constant $\gamma(\mu) \equiv \sup_{w\in X} \sup_{v\in X}\frac{a(w,v;\mu)}{\|w\|_X \|v\|_X} ( < \infty)$

#### « Truth » FEM Approximation

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 $\ell \in X',\qquad\displaystyle ||\ell||_{X'} \equiv \operatorname{sup}_{v\in X}\frac{\ell(v)}{||v||_X}$.

#### 1.5.6. Purpose

• Equate $u(\mu)$ and $u_{\mathcal{N}}(\mu)$ in the sense that $||u(\mu)-u_{\mathcal{N}}(\mu)||_X \leq \mathrm{tol}\quad\forall \mu \in \mathcal{D}^\mu$

• Build the reduced basis approximation using the FEM approximation

• Measure the error associated with the reduced basis approximation relative to the FEM approximation

$\Rightarrow u^{\mathcal{N}} (\mu)$ 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 $\varepsilon^\mathcal{N} \ll \varepsilon_{\mathrm{tol,min}}$

## 2. Reduced Basis Approximation

### 2.1. Reduced Basis Objectives

For any given accuracy $\epsilon$, evaluate

$\mu \in \mathcal{D}^\mu\mapsto s_N(\mu) (\approx s^\mathcal{N}(\mu)) \text{ and } \Delta^s_N(\mu)\qquad\text{Accuracy}$

that probably achieves the desired accuracy

$|s^\mathcal{N}(\mu)-s_N(\mu)| \leq \Delta^s_N(\mu) \leq \epsilon \qquad\text{Reliability}$

for a very low cost $t_{\text{comp}}$ Efficiency

$\textbf{Independant}\text{ of} \mathcal{N} \text{ as } \mathcal{N} \rightarrow \infty \qquad\text{Efficiency}$

where $t_{\text{comp}}$ is the time to perform the input-output relationship $\mu \mapsto (s_N(\mu),\Delta^s_N(\mu))$.

### 2.2. Rapid Convergence

Build a rapidly convergent approximation of $s_N(\mu) \in \mathbb{R}$ and $u_N(\mu) \in X^N \subset X^{\mathcal{N}} \subset X$ such that for all $\mu$, we have $s_N(\mu) \rightarrow s^{\mathcal{N}}(\mu)$ and $u_N(\mu) \rightarrow u^{\mathcal{N}}(\mu)$ rapidly as $N = \operatorname{dim}{X_N} \rightarrow \infty (= 10-200)$ (and of $\mathcal{N}$)

### 2.3. Reliability and Sharpness

Provide a posteriori error bound $\Delta_N(\mu)$ and $\Delta^s_N(\mu)$ :

$1 (\text{rigor}) \leq \dfrac{\Delta_N(\mu)}{\|u^{\mathcal{N}}(\mu) - u_N(\mu)\|_X} \leq \ E (\text{sharpness})$

and

$1 (\text{rigor}) \leq \frac{\Delta^s_N(\mu)}{|s^{\mathcal{N}}(\mu) - s_N(\mu)|} \leq \ E (\text{sharpness})$

for all $N = 1 \ldots N_{\text{max}}$ and $\mu \in \mathcal{D}^\mu$.

### 2.4. Efficiency

Develop a two stage strategy : Offline/Online

Offline

very expensive pre-processing, we have typically that for a given $\mu \in \mathcal{D}^\mu$ $t^{\text{offline}}_{\text{comp}} \gg t^{\mu\rightarrow s^{\mathcal{N}}(\mu)}_{\text{comp}}$

Online

very rapid convergent certified reduced basis input-output relationship $t^{\text{online}}_{\text{comp}} \text{ independent of } \mathcal{N}$

$\mathcal{N}$ may/should be chosen conservatively.

### 2.5. Parametric Manifold $\mathcal{M}^\mathcal{N}$

We assume

• the form $a$ is continuous and coercive (or inf-sup stable),

• affine-dependence,

• the $\theta^q(\mu), 1 \leq q \leq Q$, are smooth (i.e., $\theta^q \in C^\infty(\mathcal{D})$

then $\mathcal{M}^\mathcal{N} = \{ u^\mathcal{N}(\mu),\, \mu \in \mathcal{D}\}$ is a smooth $P$-dimensional manifold in $X^\mathcal{N}$, since $\| D_\sigma y^\mathcal{N}(\mu) \| \leq C_\sigma \forall \mu \in \mathcal{D}$, for any order $|\sigma| \in \mathbb{N}_{+0}$

### 2.6. Low-Dimension Manifold

To approximate $u(\mu)$ and thus $s(\mu)$, we need not represent all functions in $Y$. We need only approximate functions in low dimensional manifole $W = \{ u(\mu)\in Y; \mu\in \mathcal{D}^\mu\}$.

We construct the approximation space $W_N = \{u(\mu^i)\in Y, (\mu^i)_{i=1...N} \subset \mathcal{D}^\mu\}$

Figure 1. Construction of the manifold

### 2.7. Spaces & Bases

We define the RB approximation space $X_N =\operatorname*{span}\{\xi^n, 1 \leq n \leq N \},\, 1 \leq N \leq N_{max}$ with linearly independent basis functions $\xi^n \in X,\, 1 \leq n \leq N_{max}$. We thus obtain $X_N \subset X, \, \operatorname{dim}(X_N) = N,\, 1 \leq N \leq N_{max}$ and $X_1 \subset X_2 \subset \ldots X_{N_{max}} (\subset X)$.

We denote non-hierarchical RB spaces as $X^{nh}_N, 1 \leq N \leq Nmax$ $X^{nh}_N \subset X, \, \operatorname{dim}(X^{nh}_N) = N,\, 1 \leq N \leq N_{max}$

Parameter Samples : $S_N = \{ \mu_1 \in \mathcal{D}^{\mu}, \ldots, \mu_N \in \mathcal{D}^{\mu} \}\quad 1 \leq N \leq N_{\mathrm{max}}$, with $S_1 \subset S_2 \ldots S_{N_\mathrm{max}-1} \subset S_{N_\mathrm{max}} \subset \mathcal{D}^\mu$

Lagrangian Hierarchical Space : $W_N = {\rm span} \: \{ \xi^n \equiv \underbrace{ u (\mu^n)}_{u^{\mathcal{N}} (\mu^n)}, n = 1, \ldots, N \}$, with $W_1 \subset W_2 \ldots \subset W_{N_\mathrm{max}} \subset X^{\mathcal{N}} \subset{X}$

Sampling strategies?

• Equidistributed points in $\mathcal{D}^\mu$(curse of dimensionality)

• Log-random distributed points in $\mathcal{D}^\mu$

• See later for more efficient, adaptive strategies

#### 2.7.1. Taylor & Hermite

• Taylor reduced basis spaces: $W^{Taylor}_N = \operatorname*{span}\{D_\sigma u(\mu), \forall \sigma \in I^{P,N-1} \}, 1 \leq N \leq N_{max}$, field variable and sensitivity derivatives at one point in $\mathcal{D}$.

• Hermite reduced basis spaces: $W^{Hermite}_N « = » W^{Lagrangian}_N \cup W^{Taylor}_N$ field variable and sensitivity derivatives at several points in $\mathcal{D}$

#### 2.7.2. Orthogonal Basis

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

Gram-Schmidt algoritmh
1. Input: $\xi^n = u(\mu^n), 1 \leq n \leq N_\text{max}$

2. Set $\zeta^1 := \dfrac{\xi^1}{\Vert\xi^1\Vert}_X$

3. for $n=2$ to $N_\text{max}$ do:

• $z^n = \xi^n - \displaystyle\sum_{m=1}^{n-1} (\xi^n,\zeta^m)_X \zeta^m$

• $\zeta^n = \dfrac{z^n}{\Vert z^n\Vert}_X$

4. end for

 $(\zeta^n,\zeta^m)_X = \delta_{nm}, 1 \leq n,m \leq N_\text{max}$

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 $\mathbb{R}^{M\times M}$.

### 2.8. Formulation (Linear Compliant Case): a Galerkin method

#### 2.8.1. Galerkin Projection

Given $\mu \in \mathcal{D}^{\mu}$ evaluate

$s_N (\mu) = f(u_N (\mu);\mu)]$

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

$a (u_N (\mu), v; \mu) = f(v;\mu), \ \forall \: v \in X_N$

#### 2.8.2. Optimality

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

\begin{aligned} |||u(\mu) - u_N(\mu)|||_{\mu} &= \inf_{v_N \in X_N} |||u(\mu) - v_N(\mu)|||_\mu,\\ ||u(\mu) - u_N(\mu)||_X &\leq \sqrt{\dfrac{\gamma(\mu)}{\alpha(\mu)}} \inf_{v_N \in X_N} ||u(\mu) - v_N(\mu)||_X,\\ \end{aligned}

and

\begin{aligned} s(\mu)-s_N(\mu) &= |||u(\mu) - u_N(\mu)|||^2_\mu,\\ &= \inf_{v_N \in X_N} |||u(\mu) - v_N(\mu)|||^2_\mu, \end{aligned}

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

#### 2.8.3. Offline-online decomposition

Expand our RB approximations

$u_N(\mu)\ = \sum_{j=1}^N\ {u_N}_j(\mu)\ \zeta_j$

Express $s_N(\mu)$

$\displaystyle s_N(\mu) = \displaystyle\sum_{j=1}^N {u_N}_j(\mu)\ \left( \sum_{q=1}^{Q_f}\ \Theta^q_f(\mu)\ f^q(\zeta_j)\right)$

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

$\sum_{j=1}^N \left( \sum_{q=1}^{Q_a}\ \Theta^q_a(\mu)\ a^q( \zeta_i, \zeta_{j}) \right) {u_N}_j(\mu) = \sum_{q=1}^{Q_f}\ \Theta^q_f(\mu)\ f^q(\zeta_i)$

for $1 \leq i \leq N$

#### 2.8.4. Matrix form

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

where

\begin{aligned} (A_N)_{i \: j} (\mu) &= \sum_{q=1}^{Q_a}\ \Theta^q_a(\mu)\ a^q( \zeta_i, \zeta_{j}) & \quad 1 \leq i,j \leq N\\ F_{N \: i} &= \sum_{q=1}^{Q_f}\ \Theta^q_f(\mu) f^q (\zeta_i) & \quad 1 \leq i \leq N \end{aligned}

#### 2.8.5. Complexity analysis

Offline: independent of $\mu$

• Solve: $N$ FEM system depending on $\mathcal{N}$

• Form and store: $f^q (\zeta_i)$

• Form and store: $a^q( \zeta_i, \zeta_{j})$

Online: independent of $\mathcal{N}$

• Given a new $\mu \in \mathcal{D}^\mu$

• Form and solve $A_N(\mu)$ : $O(Q N^2)$ and $O(N^3)$

• Compute $s_N(\mu)$

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

#### 2.8.6. Condition number

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.