Multi-fidelity POD surrogate-assisted optimization: Concept and aero-design study

Abstract

We combine multiple sets of variable-precision full-field simulations within a single surrogate model. The approach is based on an original formulation of the “Constrained Proper Orthogonal Decomposition” (C-POD), interpolating precise, albeit costly, high-fidelity data and approximating abondant, yet less accurate, lower-fidelity data. We compute the optimal high-dimensional subspace spanning the sparse high-fidelity full-field solutions and refine the output subspace definition thanks to the orthogonal information contained in abondant low-fidelity full-field solutions. We then build “hierarchised” multi-fidelity surrogate models based on the previously refined subspace and giving a fast estimation of the high-fidelity full-field solution of any new location in the design space. The proposed model is illustrated by exploring the prediction of an analytical 2D functional space on the one hand, and demonstrated by studying the efficiency of a 1.5-stage low-pressure compressor on the other.

Appendices

Appendix A: T _H and T _L detailed derivations

By construction,

$$ {\boldsymbol{\Psi}\boldsymbol{\Psi}^{\top}}={\boldsymbol{\Phi}\boldsymbol{\Phi}^{\top}}+{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}. $$

(A.1)

We use the definition in (2.5) ($\bar {\mathbf {z}}=\bar {\mathbf {y}}+\mathbf {d}$) and develop (2.1) giving $\mathbf {y}_{i}-\bar {\mathbf {y}}\in \text {Im}\boldsymbol {\Phi }\subset \text {Im} \boldsymbol {\Psi }{\textrm {I\kern -0.21em}}\Rightarrow {\left (\mathbf {I}-{\boldsymbol {\Psi }\boldsymbol {\Psi }^{\top }}\right )}\left (\mathbf {y}_{i}-\bar {\mathbf {y}}\right )=0$. Considering in addition the property given in (2.9) (Φ ^⊤ d = 0) yields the decomposition

$$\begin{array}{@{}rcl@{}} {\left(\mathbf{I}\!-{\!\boldsymbol{\Psi}\boldsymbol{\Psi}^{\top}}\right)\left(\mathbf{y}_{i}\,-\,\bar{\mathbf{z}}\right)} &\,=\,& {\left(\mathbf{I}\,-\,{\boldsymbol{\Psi}\boldsymbol{\Psi}^{\top}}\right)}\left(\mathbf{y}_{i}\!-\bar{\mathbf{\!y}}\right) \,-\, {\left(\mathbf{I}-{\boldsymbol{\Psi}\boldsymbol{\Psi}^{\top}}\right)}\mathbf{d} \end{array} $$

$$\begin{array}{@{}rcl@{}} &=& - {\left(\mathbf{I}-{\boldsymbol{\Psi}\boldsymbol{\Psi}^{\top}}\right)}\mathbf{d} \end{array} $$

$$\begin{array}{@{}rcl@{}} &=& - {\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)}\mathbf{d} - {\boldsymbol{\Phi}\boldsymbol{\Phi}^{\top}}\mathbf{d} \end{array} $$

$$\begin{array}{@{}rcl@{}} &=& - {\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)}\mathbf{d} \end{array} $$

(A.2)

Introducing this result into (2.4) reduces its first term to

$$ \mathbf{T}_{H} = \frac{M_{H}}{2}\mathbf{d}^{\top} {\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)}\mathbf{d}, $$

(A.3)

as introduced in (2.10).

Considering ${\boldsymbol {\Phi }^{\top } \boldsymbol {\Phi }}=\mathbf {I}_{M_{H}}$, we can further develop (I −Ψ Ψ ^⊤)(I −Φ Φ ^⊤) with (A.1) and (2.7) ($\boldsymbol {\Phi }^{\top }\boldsymbol {\Xi }=\mathbf {0}_{(M_{H}\times M_{L})}$) as follows :

$$\begin{array}{@{}rcl@{}} {\left(\mathbf{I}\!-{\boldsymbol{\!{\Psi}}\boldsymbol{\Psi}^{\top}}\right)}{\left(\mathbf{I}\,-\,{\boldsymbol{\Phi}\boldsymbol{\Phi}^{\top}}\right)} &\,=\,& {\left(\mathbf{I}\,-\,{\boldsymbol{\Phi}\boldsymbol{\Phi}^{\top}}\right)}{\left(\mathbf{I}-{\boldsymbol{\Phi}\boldsymbol{\Phi}^{\top}}\right)}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}{\left(\mathbf{I}-{\boldsymbol{\Phi}\boldsymbol{\Phi}^{\top}}\right)}\\ &=& \mathbf{I}-2{\boldsymbol{\Phi}\boldsymbol{\Phi}^{\top}}+{\boldsymbol{\Phi}\boldsymbol{\Phi}^{\top}}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}{\left(\mathbf{I}-{\boldsymbol{\Phi}\boldsymbol{\Phi}^{\top}}\right)}\\ &=& {\left(\mathbf{I}-{\boldsymbol{\Psi}\boldsymbol{\Psi}^{\top}}\right)} \end{array} $$

(A.4)

Using (2.1), (2.9), and (A.4) we modify the inner term in T _L ,

$${\left(\mathbf{I}-{\boldsymbol{\Psi}\boldsymbol{\Psi}^{\top}}\right)}\left(\mathbf{z}_{i}-\bar{\mathbf{z}}\right) = {\left(\mathbf{I}-{\boldsymbol{\Psi}\boldsymbol{\Psi}^{\top}}\right)}{\left(\mathbf{I}-{\boldsymbol{\Phi}\boldsymbol{\Phi}^{\top}}\right)}\left[\mathbf{z}_{i}-\mathbf{d}\right]. $$

We introduce (see (2.12))

$$ {\mathbf{z}_{i}}^{\perp} = {\left(\mathbf{I}-{\boldsymbol{\Phi}\boldsymbol{\Phi}^{\top}}\right)}{\mathbf{z}_{i}}, $$

(A.5)

into the previous equation,

$${\left(\mathbf{I}\,-\,{\boldsymbol{\Psi}\boldsymbol{\Psi}^{\top}}\right)}\left({\mathbf{z}_{i}}\,-\,\bar{\mathbf{z}}\right) \,=\, {\left(\mathbf{I}-{\boldsymbol{\Psi}\boldsymbol{\Psi}^{\top}}\right)}\left({\mathbf{z}_{i}}^{\perp}-{\left(\mathbf{I}-{\boldsymbol{\Psi}\boldsymbol{\Psi}^{\top}}\right)}\mathbf{d}\right), $$

and use (2.9) to obtain (2.11)

$$ \mathbf{T}_{L} = \frac{1}{2}\sum\limits_{i=1}^{M_{L}}\left\Vert{\left(\mathbf{I} -{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right)\right\Vert^{2}_{{\textrm{I\kern-0.21emR}}^{n}}. $$

(A.6)

Appendix B: Optimality condition of $\mathcal {J}\left (\mathbf {d},\boldsymbol {\Xi }\right )$

The optimality of $\mathcal {J}\left (\mathbf {d},\boldsymbol {\Xi }\right )$ w.r.t d satisfies

$$ \left\langle\frac{\partial \mathcal{J}}{\partial\mathbf{d}},\delta\mathbf{d}\right\rangle = 0~\forall~\delta\mathbf{d}\in Im~\mathbf{Q}_{2}, $$

(B.1)

with

$$\left\langle\frac{\partial \mathcal{J}}{\partial\mathbf{d}},\delta\mathbf{d}\right\rangle =-\delta\mathbf{d}^{\top}\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)\left[\mathbf{d}-\frac{M_{L}\cdot\boldsymbol{\mu}\left(\mathbf{Z}^{\perp}\right)}{\left(M_{H}+M_{L}\right)}\right] . $$

The optimal d has to satisfy the condition $\displaystyle \mathbf {d}_{\text {opt}}-\frac {M_{L}}{M_{H}+M_{L}}\boldsymbol {\mu }(\mathbf {Z}^{\perp })\in Im~\boldsymbol {\Xi }$, which is possible as both d and z _i ^⊥∈(I m Φ)^⊥ ∀ i ∈ [ [1,M _L ] ].

For the sake of simplicity, we impose

$$ \mathbf{d}_{\text{opt}}=\frac{M_{L}}{M_{H}+M_{L}}\boldsymbol{\mu}\left(\mathbf{Z}^{\perp}\right), $$

(B.2)

and introduce

$$ \mathbf{z}^{\perp}_{0} = \left(\sqrt{M_{H}}+1\right)\mathbf{d}_{\text{opt}}. $$

(B.3)

For the sake of readability, we refer to d _opt as d in the following derivations, as well as to $\mathcal {J}\left (\mathbf {d}_{\text {opt}},\boldsymbol {\Xi }\right )$ as $\mathcal {J}\left (\boldsymbol {\Xi }\right )$:

$$\begin{array}{@{}rcl@{}} \mathcal{J}\left(\boldsymbol{\Xi}\right)\, &=&\frac{M_{H}}{2}\mathbf{d}^{\top}\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)\mathbf{d}\\ &&+ \frac{1}{2}\sum\limits_{i=1}^{M_{L}}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right)^{\top}{\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right)\\ &=&+\frac{1}{2}\left(\sqrt{M_{H}}\mathbf{d}^{\top}\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)\sqrt{M_{H}}\mathbf{d}\right)\\ &&+ \frac{1}{2}\sum\limits_{i=1}^{M_{L}}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right)^{\top}{\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right)\\ &=&+\frac{1}{2}\left((\sqrt{M_{H}}+1\,-\,1)\mathbf{d}^{\top}\left(\mathbf{I}-\!{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)(\sqrt{M_{H}}\,+\,1\,-\,1)\mathbf{d}\right)\\ &&+ \frac{1}{2}\sum\limits_{i=1}^{M_{L}}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right)^{\top}{\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right)\\ &=&+\frac{1}{2}\sum\limits_{i=1}^{M_{L}}\left(\mathbf{z}^{\perp}_{0}-\mathbf{d}\right)^{\top}\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)\left(\mathbf{z}^{\perp}_{0}-\mathbf{d}\right)\\ &&+ \frac{1}{2}\sum\limits_{i=1}^{M_{L}}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right)^{\top}{\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right)\\ &=&+\frac{1}{2}\sum\limits_{i=0}^{M_{L}}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right)^{\top}{\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right). \end{array} $$

To complete the basis Ψ = [Φ ∣ Ξ], we now seek Ξ satisfying

$$\begin{array}{@{}rcl@{}} \underset{\boldsymbol{\Xi}}{\min}~\frac{1}{2}\sum\limits_{i=0}^{M_{L}}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right)^{\top}{\left(\mathbf{I}-{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)}\left({\mathbf{z}_{i}}^{\perp}-\mathbf{d}\right), \end{array} $$

$$\begin{array}{@{}rcl@{}} \text{s.t} \end{array} $$

$$\begin{array}{@{}rcl@{}} {\boldsymbol{\Xi}^{\top} \boldsymbol{\Xi}}=\mathbf{I}_{m_{L}} \text{ with } m_{L}\leq M_{L}, \end{array} $$

$$\begin{array}{@{}rcl@{}} \boldsymbol{\Phi}^{\top}\boldsymbol{\Xi}=\mathbf{0}. \end{array} $$

(B.4)

Equation (B.4) implies Ξ ∈ I m Q ₂ ⇒∃t, Ξ = Q ₂ t, where Q ₂ is the orthonormal basis given by the first QR-decomposition (2.3) spanning (I m Φ)^⊥, and t is a (n − M _H ) × m _L real-valued matrix. Keeping in mind the orthonormality of Q ₂, $\mathbf {Q}_{2}^{\top }\mathbf {Q}_{2}=\mathbf {I}_{n-M_{H}}$ yields $\mathbf {t}^{\top }\mathbf {t}=\mathbf {t}^{\top }\mathbf {Q}_{2}^{\top }\mathbf {Q}_{2}\mathbf {t}={\boldsymbol {\Xi }^{\top } \boldsymbol {\Xi }}=\mathbf {I}_{m_{L}}$.

One can notice that z _i ^⊥−d ∈ I m Q ₂ ∀ i ∈ [ [0,M _L ] ], and introduce $\mathbf {u}_{i}\in {\textrm {I\kern -0.21emR}}^{(n-M_{H})}$, such that z _i ^⊥−d = Q ₂ u _i .

By replacing Ξ and z _i ^⊥−d, respectively with Q ₂ t and Q ₂ u _i in (B.4), we obtain:

$$\begin{array}{@{}rcl@{}} {\left({\mathbf{z}_{i}}^{\perp}\!-\mathbf{d}\right)^{\top}{\left(\mathbf{I}\,-\,{\boldsymbol{\Xi}\boldsymbol{\Xi}^{\top}}\right)}\left({\mathbf{z}_{i}}^{\perp}\,-\,\mathbf{d}\right)} &=& \mathbf{u}_{i}^{\top}\mathbf{Q}_{2}^{\top}\left(\mathbf{I}-\mathbf{Q}_{2}\mathbf{t}\mathbf{t}^{\top}\mathbf{Q}_{2}^{\top}\right)\mathbf{Q}_{2}\mathbf{u}_{i}\\ &=&\mathbf{u}_{i}^{\top}\mathbf{Q}_{2}^{\top}\mathbf{Q}_{2}\mathbf{u}_{i}\\ &&-\mathbf{u}_{i}^{\top}\mathbf{Q}_{2}^{\top}\mathbf{Q}_{2}\mathbf{t}\mathbf{t}^{\top}\mathbf{Q}_{2}^{\top}\mathbf{Q}_{2}\mathbf{u}_{i}\\ &=&\mathbf{u}_{i}^{\top}\mathbf{u}_{i}\\ &&-\mathbf{u}_{i}^{\top}\mathbf{t}\mathbf{t}^{\top}\mathbf{u}_{i}\\ &=&\mathbf{u}_{i}^{\top}{\left(\mathbf{I}-{\boldsymbol{\mathrm{t}}\boldsymbol{\mathrm{t}}^{\top}}\right)}\mathbf{u}_{i}. \end{array} $$

(B.5)

The completion problem is therefore reduced to

$$\begin{array}{@{}rcl@{}} \underset{\mathbf{t}}{\min}~\frac{1}{2}\sum\limits_{i=0}^{M_{L}}\mathbf{u}_{i}^{\top}{\left(\mathbf{I}-{\boldsymbol{\mathrm{t}}\boldsymbol{\mathrm{t}}^{\top}}\right)}\mathbf{u}_{i}, \end{array} $$

$$\begin{array}{@{}rcl@{}} \text{s.t} \end{array} $$

$$\begin{array}{@{}rcl@{}} {\boldsymbol{\mathrm{t}}^{\top} \boldsymbol{\mathrm{t}}}=\mathbf{I}_{m_{L}} \text{ with } m_{L}\leq M_{L}, \end{array} $$

(B.6)

which defines a classical POD problem on t.