Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation

Oleg Balabanov; Anthony Nouy

doi:10.1007/s10444-019-09725-6

Advances in Computational Mathematics

pp 1–51 | Cite as

Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation

Authors
Authors and affiliations

Oleg Balabanov
Anthony Nouy

Article

First Online: 03 December 2019

13 Downloads
2 Citations

Abstract

We propose a probabilistic way for reducing the cost of classical projection-based model order reduction methods for parameter-dependent linear equations. A reduced order model is here approximated from its random sketch, which is a set of low-dimensional random projections of the reduced approximation space and the spaces of associated residuals. This approach exploits the fact that the residuals associated with approximations in low-dimensional spaces are also contained in low-dimensional spaces. We provide conditions on the dimension of the random sketch for the resulting reduced order model to be quasi-optimal with high probability. Our approach can be used for reducing both complexity and memory requirements. The provided algorithms are well suited for any modern computational environment. Major operations, except solving linear systems of equations, are embarrassingly parallel. Our version of proper orthogonal decomposition can be computed on multiple workstations with a communication cost independent of the dimension of the full order model. The reduced order model can even be constructed in a so-called streaming environment, i.e., under extreme memory constraints. In addition, we provide an efficient way for estimating the error of the reduced order model, which is not only more efficient than the classical approach but is also less sensitive to round-off errors. Finally, the methodology is validated on benchmark problems.

Keywords

Model reduction Reduced basis Proper orthogonal decomposition Random sketching Subspace embedding

Mathematics Subject Classification (2010)

15B52 35B30 65F99 65N15

This is a preview of subscription content, to check access.

Preview

Unable to display preview. Download preview PDF.

Notes

Supplementary material

10444_2019_9725_MOESM1_ESM.tex (22 kb)

(TEX 21.9 KB)

Appendix

Here we list the proofs of propositions and theorems from the paper.

Proof of Proposition 2.2 (modified Cea’s lemma)

For all x ∈ U_r, it holds

$$ \begin{array}{@{}rcl@{}} \alpha_{r}(\mu) \| \mathbf{u}_{r}(\mu)- \mathbf{x} \|_{U} &\leq& \| \mathbf{r}(\mathbf{u}_{r}(\mu);\mu)- \mathbf{r}(\mathbf{x};\mu)\|_{U_{r}^{\prime}} \leq \| \mathbf{r}(\mathbf{u}_{r}(\mu);\mu) \|_{U_{r}^{\prime}}+ \| \mathbf{r}(\mathbf{x}; \mu) \|_{U_{r}^{\prime}}\\ &=& \| \mathbf{r}(\mathbf{x}; \mu) \|_{U_{r}^{\prime}} \leq \beta_{r}(\mu) \| \mathbf{u}(\mu)-\mathbf{x} \|_{U}, \end{array} $$

where the first and last inequalities directly follow from the definitions of α_r(μ) and β_r(μ), respectively. Now,

$$ \| \mathbf{u}(\mu)-\mathbf{u}_{r}(\mu)\|_{U} \!\leq\! \| \mathbf{u}(\mu)- \mathbf{x} \|_{U}+ \| \mathbf{u}_{r}(\mu)- \mathbf{x} \|_{U} \!\leq\! \| \mathbf{u}(\mu)- \mathbf{x} \|_{U}+ \frac{\beta_{r}(\mu)}{\alpha_{r}(\mu)} \| \mathbf{u}(\mu)- \mathbf{x} \|_{U}, $$

which completes the proof. □

Proof of Proposition 2.3

For all $\mathbf {a} \in \mathbb {K}^{r}$ and x := U_ra, it holds

$$ \begin{array}{ll} \frac{\| \mathbf{A}_{r}(\mu)\mathbf{a} \|}{\| \mathbf{a} \|}&= \underset{\mathbf{z} \in \mathbb{K}^{r} \backslash \{ \mathbf{0} \}} \max \frac{|\langle \mathbf{z}, \mathbf{A}_{r}(\mu)\mathbf{a} \rangle|}{\| \mathbf{z} \| \| \mathbf{a} \|} = \underset{ \mathbf{z} \in \mathbb{K}^{r} \backslash \{ \mathbf{0} \}} \max \frac{|\mathbf{z}^{\mathrm{H}}\mathbf{U}_{r}^{\mathrm{H}}\mathbf{A}(\mu)\mathbf{U}_{r}\mathbf{a}|}{\| \mathbf{z} \| \| \mathbf{a} \|} \\ &= \underset{\mathbf{y} \in U_{r} \backslash \{ \mathbf{0} \}} \max \frac{|\mathbf{y}^{\mathrm{H}}\mathbf{A}(\mu)\mathbf{x}|}{\| \mathbf{y} \|_{U} \| \mathbf{x} \|_{U}} = \frac{\| \mathbf{A}(\mu)\mathbf{x} \|_{U^{\prime}_{r}}}{ \| \mathbf{x} \|_{U}}. \end{array} $$

Then the proposition follows directly from definitions of α_r(μ) and β_r(μ). □

Proof of Proposition 2.4

We have

$$ \begin{array}{@{}rcl@{}} |s(\mu)- s_{r}^{\text{pd}}(\mu)| &=& |s(\mu) - s_{r}(\mu)+ \langle \mathbf{u}_{r}^{\text{du}}(\mu), \mathbf{r}(\mathbf{u}_{r}(\mu); \mu) \rangle| \\ &=& |\langle \mathbf{l}(\mu), \mathbf{u}(\mu)- \mathbf{u}_{r}(\mu) \rangle+ \langle \mathbf{A} (\mu)^{\mathrm{H}} \mathbf{u}_{r}^{\text{du}}(\mu), \mathbf{u}(\mu)- \mathbf{u}_{r}(\mu) \rangle| \\ &=& |\langle \mathbf{r}^{\text{du}} (\mathbf{u}_{r}^{\text{du}}(\mu); \mu), \mathbf{u}(\mu)- \mathbf{u}_{r}(\mu) \rangle |\\ &\leq& \| \mathbf{r}^{\text{du}} (\mathbf{u}_{r}^{\text{du}}(\mu); \mu) \|_{U^{\prime}} \| \mathbf{u}(\mu) - \mathbf{u}_{r}(\mu) \|_{U}, \end{array} $$

and the result follows from definition (2.10). □

Proof of Proposition 2.5

To prove the first inequality, we notice that $\mathbf {Q}\mathbf {P}_{U_{r}}\mathbf {U}_{m}$ has rank at most r. Consequently,

$$ \| \mathbf{Q}\mathbf{U}_{m} - {\mathbf{B}^{*}_{r}}\|^{2}_{F}\leq \| \mathbf{Q}\mathbf{U}_{m}- \mathbf{Q}\mathbf{P}_{{U_{r}}}\mathbf{U}_{m} \|^{2}_{F}= \sum\limits^{m}_{i=1} \| \mathbf{u} (\mu^{i}) - \mathbf{P}_{{U_{r}}} \mathbf{u} (\mu^{i})\|^{2}_{U}. $$

For the second inequality, let us denote the i-th column vector of B_r by b_i. Since $\mathbf {Q}\mathbf {R}_{U}^{-1}\mathbf {Q}^{\mathrm {H}}= \mathbf {Q}\mathbf {Q}^{\dagger }$, with Q^‡ the pseudo-inverse of Q, is the orthogonal projection onto range(Q), we have

$$ \begin{array}{ll} \| \mathbf{Q}\mathbf{U}_{m}- {\mathbf{B}_{r}}\|^{2}_{F} &\geq \|\mathbf{Q} \mathbf{R}_{U}^{-1}\mathbf{Q}^{\mathrm{H}} (\mathbf{Q}\mathbf{U}_{m}- {\mathbf{B}_{r}})\|^{2}_{F}= \sum\limits_{i=1}^{m} \|\mathbf{u} (\mu^{i}) - \mathbf{R}_{U}^{-1}\mathbf{Q}^{\mathrm{H}}{\mathbf{b}_{i}} \|^{2}_{U} \\ &\geq \sum\limits_{i=1}^{m} \|\mathbf{u} (\mu^{i})- \mathbf{P}_{{U_{r}}}\mathbf{u} (\mu^{i}) \|^{2}_{U}. \end{array} $$

□

Proof of Proposition 3.3

It is clear that $\langle \cdot , \cdot \rangle ^{\boldsymbol {\Theta }}_{X}$ and $\langle \cdot , \cdot \rangle ^{\boldsymbol {\Theta }}_{X^{\prime }}$ satisfy (conjugate) symmetry, linearity, and positive semi-definiteness properties. The definitenesses of $\langle \cdot , \cdot \rangle ^{\boldsymbol {\Theta }}_{X}$ and $\langle \cdot , \cdot \rangle ^{\boldsymbol {\Theta }}_{X^{\prime }}$ on Y and $Y^{\prime }$, respectively, follow directly from Definition 3.1 and Corollary 3.2. □

Proof of Proposition 3.4

Using Definition 3.1, we have

$$ \begin{array}{@{}rcl@{}} \| \mathbf{y}^{\prime} \|^{\boldsymbol{\Theta}}_{Z^{\prime}} &=& \underset{\mathbf{x} \in Z \backslash \{ \mathbf{0} \}} \max \frac{|\langle \mathbf{R}_{X}^{-1}\mathbf{y}^{\prime}, \mathbf{x} \rangle^{\boldsymbol{\Theta}}_{X}|}{\| \mathbf{x} \|^{\boldsymbol{\Theta}}_{X}}\leq \underset{\mathbf{x} \in Z \backslash \{ \mathbf{0} \}} \max \frac{|\langle \mathbf{R}_{X}^{-1}\mathbf{y}^{\prime}, \mathbf{x} \rangle_{X}|+ \varepsilon \| \mathbf{y}^{\prime} \|_{X^{\prime}} \| \mathbf{x} \|_{X}} {\| \mathbf{x} \|^{\boldsymbol{\Theta}}_{X}} \\ &\leq& \underset{\mathbf{x} \in Z \backslash \{ \mathbf{0} \}} \max \frac{|\langle \mathbf{R}_{X}^{-1}\mathbf{y}^{\prime}, \mathbf{x} \rangle_{X}|+ \varepsilon \| \mathbf{y}^{\prime} \|_{X^{\prime}} \| \mathbf{x} \|_{X}} {\sqrt{1-\varepsilon}\| \mathbf{x} \|_{X}} \\ &\leq& \frac{1}{\sqrt{1-\varepsilon}} \left( \underset{\mathbf{x} \in Z \backslash \{ \mathbf{0} \}} \max \frac{|\langle \mathbf{y}^{\prime}, \mathbf{x} \rangle|} {\| \mathbf{x} \|_{X}} + \varepsilon\| \mathbf{y}^{\prime} \|_{X^{\prime}} \right), \end{array} $$

which yields the right inequality. To prove the left inequality, we assume that $ \| \mathbf {y}^{\prime } \|_{Z^{\prime }}- \varepsilon \| \mathbf {y}^{\prime } \|_{X^{\prime }}\geq 0$. Otherwise, the relation is obvious because $\| \mathbf {y}^{\prime } \|^{\boldsymbol {\Theta }}_{Z^{\prime }}\geq 0$. By Definition 3.1, we have

$$ \begin{array}{ll} \| \mathbf{y}^{\prime} \|^{\boldsymbol{\Theta}}_{Z^{\prime}} &=\underset{\mathbf{x} \in Z \backslash \{ \mathbf{0} \}} \max \frac{|\langle \mathbf{R}_{X}^{-1}\mathbf{y}^{\prime}, \mathbf{x} \rangle^{\boldsymbol{\Theta}}_{X}|} {\| \mathbf{x} \|^{\boldsymbol{\Theta}}_{X}}\geq \underset{\mathbf{x} \in Z \backslash \{ \mathbf{0} \}} \max \frac{|\langle \mathbf{R}_{X}^{-1}\mathbf{y}^{\prime}, \mathbf{x} \rangle_{X}|- \varepsilon \| \mathbf{y}^{\prime} \|_{X^{\prime}} \| \mathbf{x} \|_{X} } {\| \mathbf{x} \|^{\boldsymbol{\Theta}}_{X}} \\ &\geq \underset{\mathbf{x} \in Z \backslash \{ \mathbf{0} \}} \max \frac{|\langle \mathbf{R}_{X}^{-1}\mathbf{y}^{\prime}, \mathbf{x} \rangle_{X}|- \varepsilon \| \mathbf{y}^{\prime} \|_{X^{\prime}} \| \mathbf{x} \|_{X}} {\sqrt{1+\varepsilon}\| \mathbf{x} \|_{X}} \\ &\geq \frac{1}{\sqrt{1+\varepsilon}} \left( \underset{\mathbf{x} \in Z \backslash \{ \mathbf{0} \}} \max \frac{|\langle \mathbf{y}^{\prime}, \mathbf{x} \rangle|} {\| \mathbf{x} \|_{X}}- \varepsilon \| \mathbf{y}^{\prime} \|_{X^{\prime}} \right), \end{array} $$

which completes the proof. □

Proof of Proposition 3.7

Let us start with the case $\mathbb {K} = \mathbb {R}$. For the proof, we shall follow standard steps (see, e.g., [38, Section 2.1]). Given a d-dimensional subspace $V \subseteq \mathbb {R}^{n}$, let $\mathcal {S} = \{ \mathbf {x} \in V : \ \| \mathbf {x} \| = 1 \}$ be the unit sphere of V. According to [11, Lemma 2.4], for any γ > 0 there exists a γ-net $\mathcal {N}$ of $\mathcal {S}$¹ satisfying $\# \mathcal {N} \leq (1+2/\gamma )^{d}$. For η such that 0 < η < 1/2, let $\boldsymbol {\Theta } \in \mathbb {R}^{k \times n}$ be a rescaled Gaussian or Rademacher matrix with $k\geq 6\eta ^{-2} ({2} d \log (1+2/\gamma )+ \log (1/\delta ))$. By [1, Lemmas 4.1 and 5.1] and an union bound argument, we obtain for a fixed x ∈ V

$$ \mathbb{P}(|\|\mathbf{x}\|^{2} - \|\boldsymbol{\Theta} \mathbf{x}\|^{2}| \leq \eta \|\mathbf{x}\|^{2} ) \geq 1 -2\exp(-k\eta^{2}/6). $$

Consequently, using a union bound for the probability of success, we have

$$ \left\{\ | \| \mathbf{x} + \mathbf{y} \|^{2} - \|\boldsymbol{\Theta}(\mathbf{x} + \mathbf{y}) \|^{2} | \leq \eta \|\mathbf{x} + \mathbf{y} \|^{2}, \quad \forall \mathbf{x}, \mathbf{y} \in \mathcal{N}\right\}, $$

holds with probability at least 1 − δ. Then, we deduce that

$$ \left\{\ | \langle \mathbf{x} , \mathbf{y} \rangle - \langle \boldsymbol{\Theta} \mathbf{x} , \boldsymbol{\Theta} \mathbf{y} \rangle| \leq \eta, \quad \forall \mathbf{x}, \mathbf{y} \in \mathcal{N}\right\} $$

(7.1)

holds with probability at least 1 − δ. Now, let n be some vector in $\mathcal {S}$. Assuming γ < 1, it can be proven by induction that $ \mathbf {n} = {\sum }_{i\ge 0} \alpha _{i} \mathbf {n}_{i},$ where $\mathbf {n}_{i} \in \mathcal {N}$ and 0 ≤ α_i ≤ γⁱ.² If (7.1) is satisfied, then

$$ \begin{array}{@{}rcl@{}} \| \boldsymbol{\Theta} \mathbf{n} \|^{2} &=& \sum\limits_{i,j \geq 0} \langle \boldsymbol{\Theta} \mathbf{n}_{i}, \boldsymbol{\Theta}\mathbf{n}_{j} \rangle \alpha_{i}\alpha_{j} \\ &\leq& \sum\limits_{i,j \geq 0} ( \langle \mathbf{n}_{i}, \mathbf{n}_{j} \rangle \alpha_{i} \alpha_{j} + \eta \alpha_{i} \alpha_{j}) = 1 + \eta ( \sum\limits_{i \geq 0} \alpha_{i} )^{2} \le 1 + \frac{\eta}{(1-\gamma)^{2}}, \end{array} $$

and similarly $ \| \boldsymbol {\Theta } \mathbf {n} \|^{2} \geq 1-\frac {\eta }{(1-\gamma )^{2}}$. Therefore, if (7.1) is satisfied, we have

$$ | 1- \| \boldsymbol{\Theta} \mathbf{n} \|^{2}| \leq \eta / (1-\gamma)^{2}. $$

(7.2)

For a given ε ≤ 0.5/(1 − γ)², let η = (1 − γ)²ε. Since (7.2) holds for an arbitrary vector $\mathbf {n} \in \mathcal {S}$, using the parallelogram identity, we easily obtain that

$$ \ \left| \langle \mathbf{x}, \mathbf{y} \rangle - \langle \boldsymbol{\Theta} \mathbf{x}, \boldsymbol{\Theta} \mathbf{y} \rangle \right|\leq \varepsilon \| \mathbf{x} \| \| \mathbf{y} \| $$

(7.3)

holds for all x,y ∈ V if (7.1) is satisfied. We conclude that if $k\geq 6 \varepsilon ^{-2} (1-\gamma )^{-4} ({2} d \log (1+2/\gamma )+ \log (1/\delta ))$ then Θ is a ℓ₂ → ℓ₂ε-subspace embedding for V with probability at least 1 − δ. The lower bound for the number of rows of Θ is obtained by taking $\gamma = \arg \min \limits _{x \in (0,1)}(\log (1+2/x)/(1-x)^{4})\approx 0.0656$.

The statement of the proposition for the case $\mathbb {K} = \mathbb {C}$ can be deduced from the fact that if Θ is (ε,δ, 2d) oblivious ℓ₂ → ℓ₂ subspace embedding for $\mathbb {K} = \mathbb {R}$, then it is (ε,δ,d) oblivious ℓ₂ → ℓ₂ subspace embedding for $\mathbb {K} = \mathbb {C}$. A detailed proof of this fact is provided in the supplementary material. To show this, we first note that the real part and the imaginary part of any vector from a d-dimensional subspace $V^{*} \subseteq \mathbb {C}^{n}$ belong to a certain subspace $W \subseteq \mathbb {R}^{n}$ with $\dim (W) \leq 2d$. Further, one can show that if Θ is ℓ₂ → ℓ₂ε-subspace embedding for W, then it is ℓ₂ → ℓ₂ε-subspace embedding for V^∗. □

Proof of Proposition 3.9

Let $\boldsymbol {\Theta } \in \mathbb {R}^{k\times n}$ be a P-SRHT matrix, let V be an arbitrary d-dimensional subspace of $\mathbb {K}^{n}$, and let $\mathbf {V} \in \mathbb {K}^{n\times d}$ be a matrix whose columns form an orthonormal basis of V. Recall, Θ is equal to the first n columns of matrix $\boldsymbol {\Theta }^{*} = k^{-1/2} (\mathbf {R}\mathbf {H}_{s}\mathbf {D}) \in \mathbb {R}^{k \times s}$. Next we shall use the fact that for any orthonormal matrix $\mathbf{V} ^{*} \in \mathbb {K}^{s\times d}$, all singular values of a matrix Θ^∗V^∗ belong to interval $[\sqrt {1-\varepsilon }, \sqrt {1+\varepsilon }]$ with probability at least 1 − δ. This result is basically a restatement of [12, Lemma 4.1] and [35, Theorem 3.1] including the complex case and with improved constants. It can be shown to hold by mimicking the proof in [35] with a few additional algebraic operations. For a detailed proof of the statement, see the supplementary material.

By taking V^∗ with the first n × d block equal to V and zeros elsewhere, and using the fact that ΘV and Θ^∗V^∗ have the same singular values, we obtain that

$$ | \| \mathbf{V}{\mathbf{z}} \|^2 - \| \mathbf{Theta}\mathbf{V}{\mathbf{z}} \|^2 |= |{\mathbf{z}}^{\mathrm{H}} (\mathbf{I}- \mathbf{V}^{\mathrm{H}}\mathbf{Theta}^{\mathrm{H}}\mathbf{Theta}\mathbf{V}) {\mathbf{z}} | \leq \varepsilon \| {\mathbf{z}} \|^2= {\varepsilon} \| \mathbf{V}{\mathbf{z}} \|^2, \quad \forall {\mathbf{z}} \in \mathbb{K}^{d} $$

(4)

holds with probability at least 1 − δ. Using the parallelogram identity, it can be easily proven that relation (4) implies

$$ \ \left | \langle \mathbf{x}, \mathbf{y} \rangle - \langle \mathbf{Theta}\mathbf{x}, \mathbf{Theta}\mathbf{y} \rangle \right |\leq \varepsilon \| \mathbf{x} \| \| \mathbf{y} \|, \quad \forall \mathbf{x}, \mathbf{y} \in V.$$

We conclude that Θ is a (ε,δ,d) oblivious ℓ₂ → ℓ₂ subspace embedding. □

Proof of Proposition 3.11

Let V be any d-dimensional subspace of X and let V^∗ := {Qx : x ∈ V }. Since the following relations hold 〈⋅,⋅〉_U = 〈Q⋅,Q⋅〉 and $\langle \cdot , \cdot \rangle ^{\boldsymbol {\Theta }}_{U} = \langle \mathbf {Q} \cdot , \mathbf {Q} \cdot \rangle _{2}^{\boldsymbol {\Omega }}$, we have that sketching matrix Θ is an ε-embedding for V if and only if Ω is an ε-embedding for V^∗. It follows from the definition of Ω that this matrix is an ε-embedding for V^∗ with probability at least 1 − δ, which completes the proof. □

Proof of Proposition 4.1 (sketched Cea’s lemma)

The proof exactly follows the one of Proposition 2.2 with $\| \cdot \|_{U_{r}^{\prime }}$ replaced by $\| \cdot \|^{\boldsymbol {\Theta }}_{U_{r}^{\prime }}$. □

Proof of Proposition 4.2

According to Proposition 3.4, and by definition of a_r(μ), we have

$$ \begin{array}{ll} \alpha^{\boldsymbol{\Theta}}_{r}(\mu) &=\underset{\mathbf{x} \in U_{r} \backslash \{ \mathbf{0} \}} \min \frac{\| \mathbf{A}(\mu)\mathbf{x} \|^{\boldsymbol{\Theta}}_{U_{r}^{\prime}}}{\| \mathbf{x} \|_{U}}\geq \frac{1}{\sqrt{1+\varepsilon}}\underset{\mathbf{x} \in U_{r} \backslash \{ \mathbf{0} \}} \min \frac{(\| \mathbf{A}(\mu)\mathbf{x} \|_{U_{r}^{\prime}}- \varepsilon\| \mathbf{A}(\mu)\mathbf{x}\|_{U^{\prime}})}{\| \mathbf{x} \|_{U}} \\ &\geq \frac{1}{\sqrt{1+\varepsilon}}(1-\varepsilon a_{r}(\mu)) \underset{\mathbf{x} \in U_{r} \backslash \{ \mathbf{0} \}} \min \frac{\| \mathbf{A}(\mu)\mathbf{x} \|_{U_{r}^{\prime}}}{\| \mathbf{x} \|_{U}}. \end{array} $$

Similarly,

$$ \begin{array}{ll} \beta^{\boldsymbol{\Theta}}_{r}(\mu) &= \underset{ \mathbf{x} \in \left( \text{span} \{ \mathbf{u}(\mu) \}+ U_{r} \right) \backslash \{ \mathbf{0} \}} \max \frac{ \| \mathbf{A}(\mu)\mathbf{x} \|^{\boldsymbol{\Theta}}_{U_{r}^{\prime}}}{\| \mathbf{x} \|_{U}} \\ &\leq \frac{1}{\sqrt{1-\varepsilon}} \underset{\mathbf{x} \in \left( \text{span} \{ \mathbf{u}(\mu) \} + U_{r} \right) \backslash \{ \mathbf{0} \}} \max \frac{\| \mathbf{A}(\mu)\mathbf{x} \|_{U_{r}^{\prime}}+ \varepsilon\| \mathbf{A}(\mu)\mathbf{x} \|_{U^{\prime}}}{\| \mathbf{x} \|_{U}} \\ &\leq \frac{1}{\sqrt{1-\varepsilon}} \left( \underset{\mathbf{x} \in \left( \text{span} \{ \mathbf{u}(\mu) \}+ U_{r} \right) \backslash \{ \mathbf{0} \}} \max \frac{\| \mathbf{A}(\mu)\mathbf{x} \|_{U_{r}^{\prime}}}{\| \mathbf{x} \|_{U}}+ \varepsilon \underset{\mathbf{x} \in U \backslash \{ \mathbf{0} \}} \max \frac{\| \mathbf{A}(\mu)\mathbf{x} \|_{U^{\prime}}}{\| \mathbf{x} \|_{U}} \right). \end{array} $$

□

Proof of Proposition 4.3

Let $\mathbf {a} \in \mathbb {K}^{r}$ and x := U_ra. Then

$$ \begin{array}{ll} \frac{\| \mathbf{A}_{r}(\mu)\mathbf{a} \|}{\| \mathbf{a} \|}&= \underset{\mathbf{z} \in \mathbb{K}^{r} \backslash \{ \mathbf{0} \}} \max \frac{|\langle \mathbf{z}, \mathbf{A}_{r}(\mu)\mathbf{a} \rangle|}{\| \mathbf{z} \| \| \mathbf{a} \|} = \underset{ \mathbf{z} \in \mathbb{K}^{r} \backslash \{ \mathbf{0} \}} \max \frac{|\mathbf{z}^{\mathrm{H}}\mathbf{U}_{r}^{\mathrm{H}}\boldsymbol{\Theta}^{\mathrm{H}}\boldsymbol{\Theta}\mathbf{R}_{U}^{-1}\mathbf{A}(\mu)\mathbf{U}_{r}\mathbf{a}|}{\| \mathbf{z} \| \| \mathbf{a} \|} \\ &= \underset{\mathbf{y} \in U_{r} \backslash \{ \mathbf{0} \}} \max \frac{|\mathbf{y}^{\mathrm{H}}\boldsymbol{\Theta}^{\mathrm{H}}\boldsymbol{\Theta}\mathbf{R}_{U}^{-1}\mathbf{A}(\mu)\mathbf{x}|}{\| \mathbf{y} \|^{\boldsymbol{\Theta}}_{U} \| \mathbf{x} \|^{\boldsymbol{\Theta}}_{U}}= \underset{\mathbf{y} \in U_{r} \backslash \{ \mathbf{0} \}} \max \frac{|\langle \mathbf{y}, \mathbf{R}_{U}^{-1}\mathbf{A}(\mu)\mathbf{x} \rangle^{\boldsymbol{\Theta}}_{U}|}{\| \mathbf{y} \|^{\boldsymbol{\Theta}}_{U} \| \mathbf{x} \|^{\boldsymbol{\Theta}}_{U}} \\ &= \frac{\| \mathbf{A}(\mu)\mathbf{x} \|^{\boldsymbol{\Theta}}_{U^{\prime}_{r}}}{ \| \mathbf{x} \|^{\boldsymbol{\Theta}}_{U}}. \end{array} $$

(7.4)

By definition,

$$ \sqrt{1-\varepsilon} \| \mathbf{x} \|_{U} \leq \| \mathbf{x} \|^{\boldsymbol{\Theta}}_{U} \leq \sqrt{1+\varepsilon} \| \mathbf{x} \|_{U}. $$

(7.5)

Combining (7.4) and (7.5) we conclude

$$ \frac{1}{\sqrt{1+\varepsilon}}\frac{\| \mathbf{A}(\mu)\mathbf{x} \|^{\boldsymbol{\Theta}}_{U^{\prime}_{r}}}{\| \mathbf{x} \|_{U}} \leq \frac{\|\mathbf{A}_{r}(\mu)\mathbf{a} \|}{\| \mathbf{a} \|}\leq \frac{1}{\sqrt{1-\varepsilon}}\frac{\| \mathbf{A}(\mu)\mathbf{x} \|^{\boldsymbol{\Theta}}_{U^{\prime}_{r}}}{\| \mathbf{x} \|_{U}}. $$

The statement of the proposition follows immediately from definitions of $\alpha ^{\boldsymbol {\Theta }}_{r}(\mu )$ and $\beta ^{\boldsymbol {\Theta }}_{r}(\mu )$. □

Proof of Proposition 4.4

The proposition directly follows from relations (2.10), (3.1), (3.2), and (4.7). □

Proof of Proposition 4.6

We have

$$ \begin{array}{ll} |s^{\text{pd}}(\mu)- s_{r}^{\text{spd}}(\mu)|&= | \langle \mathbf{u}_{r}^{\text{du}}(\mu), \mathbf{R}_{U}^{-1}\mathbf{r}(\mathbf{u}_{r}(\mu);\mu) \rangle_{U}-\langle \mathbf{u}_{r}^{\text{du}}(\mu), \mathbf{R}_{U}^{-1}\mathbf{r}(\mathbf{u}_{r}(\mu);\mu) \rangle^{\boldsymbol{\Theta}}_{U}| \\ &\leq \varepsilon \| \mathbf{r}(\mathbf{u}_{r}(\mu); \mu) ||_{U^{\prime}} \| \mathbf{u}_{r}^{\text{du}}(\mu) \|_{U} \\ &\leq \varepsilon \| \mathbf{r}(\mathbf{u}_{r}(\mu);\mu) ||_{U^{\prime}} \frac{\| \mathbf{A}(\mu)^{\mathrm{H}} \mathbf{u}_{r}^{\text{du}}(\mu) \|_{U^{\prime}}}{\eta(\mu)} \\ &\leq \varepsilon \| \mathbf{r}(\mathbf{u}_{r}(\mu);\mu) ||_{U^{\prime}} \frac{\| \mathbf{r}^{\text{du}}(\mathbf{u}_{r}^{\text{du}}(\mu);\mu)\|_{U^{\prime}}+ \| \mathbf{l}(\mu) \|_{U^{\prime}}}{\eta (\mu)}, \end{array} $$

(7.6)

and (4.10) follows by combining (7.6) with (2.15). □

Proof of Proposition 5.1

In total, there are at most $\binom {m}{r}$r-dimensional subspaces that could be spanned from m snapshots. Therefore, by using the definition of Θ, the fact that $\dim (Y_{r}(\mu ))\leq 2 r+1$ and a union bound for the probability of success, we deduce that Θ is a U → ℓ₂ε-subspace embedding for Y_r(μ), for fixed $\mu \in \mathcal {P}_{\text {train}}$, with probability at least 1 − m^− 1δ. The proposition then follows from another union bound. □

Proof of Proposition 5.4

We have

$$ {\varDelta}^{\text{POD}}(V) = \frac{1}{m} \| \mathbf{U}_{m}^{\boldsymbol{\Theta}} - \boldsymbol{\Theta} \mathbf{P}^{\boldsymbol{\Theta}}_{V} \mathbf{U}_{m}\|_{F}. $$

Moreover, matrix $\boldsymbol {\Theta } \mathbf {P}^{\boldsymbol {\Theta }}_{U_{r}} \mathbf {U}_{m}$ is the rank-r truncated SVD approximation of $\mathbf {U}_{m}^{\boldsymbol {\Theta }}$. The statements of the proposition can be then derived from the standard properties of the SVD. □

Proof of Theorem 5.5

Clearly, if Θ is a U → ℓ₂ε-subspace embedding for Y, then $\text {rank}(\mathbf {U}^{\boldsymbol {\Theta }}_{m})\geq r$. Therefore U_r is well-defined. Let $\{( \lambda _{i}, \mathbf {t}_{i}) \}_{i=1}^{l}$ and T_r be given by Definition 5.3. In general, $\mathbf {P}^{\boldsymbol {\Theta }}_{U_{r}}$ defined by (5.3) may not be unique. Let us further assume that $\mathbf {P}^{\boldsymbol {\Theta }}_{U_{r}}$ is provided for x ∈ U_m by $ \mathbf {P}^{\boldsymbol {\Theta }}_{U_{r}}\mathbf {x}:= \mathbf {U}_{r}\mathbf {U}_{r}^{\mathrm {H}}\boldsymbol {\Theta }^{\mathrm {H}}\boldsymbol {\Theta }\mathbf {x}, $ where $\mathbf {U}_{r}= \mathbf {U}_{m}[\frac {1}{\sqrt {\lambda _{1}}}\mathbf {t}_{1}, ..., \frac {1}{\sqrt {\lambda _{r}}}\mathbf {t}_{r}]$. Observe that $ \mathbf {P}^{\boldsymbol {\Theta }}_{U_{r}} \mathbf {U}_{m} = \mathbf {U}_{m} \mathbf {T}_{r} \mathbf {T}_{r}^{\mathrm {H}}. $ For the first part of the theorem, we establish the following inequalities. Let $\mathbf {Q} \in \mathbb {K}^{n\times n}$ denote the adjoint of a Cholesky factor of R_U, then

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{m} \sum\limits_{i=1}^{m} \| (\mathbf{I} -\mathbf{P}_{Y})(\mathbf{u}(\mu^{i})- \mathbf{P}^{\boldsymbol{\Theta}}_{U_{r}}\mathbf{u}(\mu^{i})) \|_{U}^{2}= \frac{1}{m} \| \mathbf{Q}(\mathbf{I} -\mathbf{P}_{Y})\mathbf{U}_{m} (\mathbf{I} - \mathbf{T}_{r}\mathbf{T}_{r}^{\mathrm{H}}) \|_{F}^{2}\\ &\leq& \frac{1}{m} \| \mathbf{Q}(\mathbf{I}- \mathbf{P}_{Y})\mathbf{U}_{m}\|_{F}^{2} \| \mathbf{I}- \mathbf{T}_{r}\mathbf{T}_{r}^{\mathrm{H}} \|^{2}= {\varDelta}_{Y} \| \mathbf{I}- \mathbf{T}_{r}\mathbf{T}_{r}^{\mathrm{H}} \|^{2}\leq {\varDelta}_{Y}, \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{m} \sum\limits_{i=1}^{m} \left( \| (\mathbf{I}- \mathbf{P}_{Y})(\mathbf{u}(\mu^{i})- \mathbf{P}^{\boldsymbol{\Theta}}_{U_{r}}\mathbf{u}(\mu^{i})) \|^{\boldsymbol{\Theta}}_{U} \right)^{2}= \frac{1}{m} \| \boldsymbol{\Theta}(\mathbf{I}- \mathbf{P}_{Y})\mathbf{U}_{m}(\mathbf{I}- \mathbf{T}_{r}\mathbf{T}_{r}^{\mathrm{H}}) \|_{F}^{2} \\ &\leq& \frac{1}{m} \| \boldsymbol{\Theta}(\mathbf{I}- \mathbf{P}_{Y})\mathbf{U}_{m} \|_{F}^{2} \| \mathbf{I}- \mathbf{T}_{r}\mathbf{T}_{r}^{\mathrm{H}} \|^{2} \leq(1+\varepsilon) {\varDelta}_{Y} \| \mathbf{I}- \mathbf{T}_{r}\mathbf{T}_{r}^{\mathrm{H}} \|^{2}\leq (1+\varepsilon){\varDelta}_{Y}. \end{array} $$

Now, we have

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{m} \sum\limits_{i=1}^{m} \| \mathbf{u}(\mu^{i})- \mathbf{P}_{U_{r}}\mathbf{u}(\mu^{i}) \|_{U}^{2}\leq \frac{1}{m} \sum\limits_{i=1}^{m} \| \mathbf{u}(\mu^{i})- \mathbf{P}^{\boldsymbol{\Theta}}_{U_{r}}\mathbf{u}(\mu^{i}) \|_{U}^{2} \\ &=& \frac{1}{m} \sum\limits_{i=1}^{m} \left( \| \mathbf{P}_{Y}(\mathbf{u}(\mu^{i})- \mathbf{P}^{\boldsymbol{\Theta}}_{U_{r}}\mathbf{u}(\mu^{i})) \|_{U}^{2}+ \| (\mathbf{I}- \mathbf{P}_{Y})(\mathbf{u}(\mu^{i})- \mathbf{P}^{\boldsymbol{\Theta}}_{U_{r}}\mathbf{u}(\mu^{i})) \|_{U}^{2} \right) \\ & \leq& \frac{1}{m} \sum\limits_{i=1}^{m} \| \mathbf{P}_{Y}(\mathbf{u}(\mu^{i})- \mathbf{P}^{\boldsymbol{\Theta}}_{U_{r}}\mathbf{u}(\mu^{i})) \|_{U}^{2}+ {\varDelta}_{Y} \leq \frac{1}{m} \frac{1}{1-\varepsilon} \sum\limits_{i=1}^{m} \left( \| \mathbf{P}_{Y}(\mathbf{u}(\mu^{i})- \mathbf{P}^{\boldsymbol{\Theta}}_{U_{r}}\mathbf{u}(\mu^{i})) \|^{\boldsymbol{\Theta}}_{U} \right)^{2}+ {\varDelta}_{Y} \\ &\leq& \frac{1}{1-\varepsilon} \frac{1}{m} \sum\limits_{i=1}^{m} 2 \left( \left( \| \mathbf{u}(\mu^{i})- \mathbf{P}^{\boldsymbol{\Theta}}_{U_{r}}\mathbf{u}(\mu^{i}) \|^{\boldsymbol{\Theta}}_{U} \right)^{2} + \left( \| (\mathbf{I}- \mathbf{P}_{Y})(\mathbf{u}(\mu^{i})- \mathbf{P}^{\boldsymbol{\Theta}}_{U_{r}}\mathbf{u}(\mu^{i})) \|^{\boldsymbol{\Theta}}_{U} \right)^{2} \right)+ {\varDelta}_{Y} \\ &\leq& \frac{1}{1-\varepsilon} \frac{1}{m} \sum\limits_{i=1}^{m} 2 \left( \| \mathbf{u}(\mu^{i})- \mathbf{P}_{U^{*}_{r}}\mathbf{u}(\mu^{i}) \|^{\boldsymbol{\Theta}}_{U} \right)^{2}+ (\frac{2(1+\varepsilon)}{1-\varepsilon}+1){\varDelta}_{Y} \\ &\leq& \frac{2(1+\varepsilon)}{1-\varepsilon} \frac{1}{m} \sum\limits_{i=1}^{m} \| \mathbf{u}(\mu^{i})- \mathbf{P}_{U^{*}_{r}}\mathbf{u}(\mu^{i}) \|_{U}^{2}+ (\frac{2(1+\varepsilon)}{1-\varepsilon}+1){\varDelta}_{Y}, \end{array} $$

which is equivalent to (5.7).

The second part of the theorem can be proved as follows. Assume that Θ is U → ℓ₂ε-subspace embedding for U_m, then

$$ \begin{array}{@{}rcl@{}} &&\frac{1}{m} \sum\limits_{i=1}^{m} \| \mathbf{u}(\mu^{i})- \mathbf{P}_{U_{r}}\mathbf{u}(\mu^{i}) \|_{U}^{2} \leq \frac{1}{m} \sum\limits_{i=1}^{m} \| \mathbf{u}(\mu^{i})- \mathbf{P}^{\boldsymbol{\Theta}}_{U_{r}}\mathbf{u}(\mu^{i}) \|_{U}^{2}\\ &\leq& \frac{1}{m} \frac{1}{1-\varepsilon}\sum\limits_{i=1}^{m} \left( \| \mathbf{u}(\mu^{i})- \mathbf{P}^{\boldsymbol{\Theta}}_{U_{r}}\mathbf{u}(\mu^{i}) \|^{\boldsymbol{\Theta}}_{U} \right)^{2} \leq \frac{1}{m} \frac{1}{1-\varepsilon} \sum\limits_{i=1}^{m} \left( \| \mathbf{u}(\mu^{i})- \mathbf{P}_{U^{*}_{r}}\mathbf{u}(\mu^{i}) \|^{\boldsymbol{\Theta}}_{U} \right)^{2}\\ &\leq& \frac{1}{m} \frac{1+\varepsilon}{1-\varepsilon} \sum\limits_{i=1}^{m} \| \mathbf{u}(\mu^{i})- \mathbf{P}_{U^{*}_{r}}\mathbf{u}(\mu^{i}) \|_{U}^{2}, \end{array} $$

which completes the proof. □

References

1.
Achlioptas, D.: Database-friendly random projections: Johnson-Lindenstrauss with binary coins. J. Comput. Syst. Sci. 66(4), 671–687 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
2.
Ailon, N., Liberty, E.: Fast dimension reduction using Rademacher series on dual bch codes. Discret. Comput. Geom. 42(4), 615 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
3.
Alla, A., Kutz, J.N.: Randomized model order reduction. tech. report, arXiv:http://arxiv.org/abs/1611.02316 (2016)
4.
Baker, C.G., Gallivan, K.A., Dooren, P.V.: Low-rank incremental methods for computing dominant singular subspaces. Linear Algebra Appl. 436(8), 2866–2888 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
5.
Balabanov, O., Nouy, A.: Randomized linear algebra for model reduction. Part ii: minimal residual methods and dictionary-based approximation. arXiv:http://arxiv.org/abs/1910.14378 (2019)
6.
Bebendorf, M.: Why finite element discretizations can be factored by triangular hierarchical matrices. SIAM J. Numer. Anal. 45(4), 1472–1494 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
7.
Bebendorf, M.: Hierarchical Matrices. Springer, Berlin (2008)zbMATHGoogle Scholar
8.
Benner, P., Cohen, A., Ohlberger, M., Willcox, K. (eds.): Model reduction and approximation: theory and algorithms. SIAM, Philadelphia, PA (2017)Google Scholar
9.
Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
10.
Boman, E.G., Hendrickson, B., Vavasis, S.: Solving elliptic finite element systems in near-linear time with support preconditioners. SIAM J. Numer. Anal. 46 (6), 3264–3284 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
11.
Bourgain, J., Lindenstrauss, J., Milman, V.: Approximation of zonoids by zonotopes. Acta Math. 162(1), 73–141 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
12.
Boutsidis, C., Gittens, A.: Improved matrix algorithms via the subsampled randomized Hadamard transform. SIAM J. Matrix Anal. Appl. 34(3), 1301–1340 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
13.
Braconnier, T., Ferrier, M., Jouhaud, J.-C., Montagnac, M., Sagaut, P.: Towards an adaptive POD/SVD surrogate model for aeronautic design. Comput. Fluids 40(1), 195–209 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
14.
Buhr, A., Engwer, C., Ohlberger, M., Rave, S.: A numerically stable a posteriori error estimator for reduced basis approximations of elliptic equations. In: Onate, X.O.E., Huerta, A. (eds.) Proceedings of the 11th World Congress on Computational Mechanics. CIMNE, pp 4094–4102, Barcelona (2014)Google Scholar
15.
Buhr, A., Smetana, K.: Randomized local model order reduction. SIAM J. Sci. Comput. 40(4), A2120–A2151 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
16.
Casenave, F., Ern, A., Lelièvre, T.: Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method. ESAIM: Mathematical Modelling and Numerical Analysis 48(1), 207–229 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
17.
Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Numer. Math. Sci. Comput. (2014)Google Scholar
18.
Engquist, B., Ying, L.: Sweeping preconditioner for the Helmholtz equation: hierarchical matrix representation. Commun. Pure Appl. Math. 64(5), 697–735 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
19.
Gross, D., Nesme, V.: Note on sampling without replacing from a finite collection of matrices. arXiv:http://arxiv.org/abs/1001.2738 (2010)
20.
Haasdonk, B.: Reduced basis methods for parametrized PDEs – a tutorial introduction for stationary and instationary problems. In: Benner, P., Cohen, A., Ohlberger, M., Willcox, K. (eds.) Model Reduction and Approximation, pp 65–136. SIAM, Philadelphia (2017)CrossRefGoogle Scholar
21.
Hackbusch, W.: Hierarchical matrices: algorithms and analysis, vol. 49. Springer, Berlin (2015)zbMATHCrossRefGoogle Scholar
22.
Halko, N., Martinsson, P.-G., Tropp, J.A.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217–288 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
23.
Hesthaven, J.S., Rozza, G., Stamm, B.: Certified reduced basis methods for parametrized partial differential equations, 1st edn. Springer Briefs in Mathematics, Switzerland (2015)zbMATHGoogle Scholar
24.
Himpe, C., Leibner, T., Rave, S.: Hierarchical approximate proper orthogonal decomposition. SIAM J. Sci. Comput. 40(5), A3267–A3292 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
25.
Hochman, A., Villena, J.F., Polimeridis, A.G., Silveira, L.M., White, J.K., Daniel, L.: Reduced-order models for electromagnetic scattering problems. IEEE Trans. Antennas Propag. 62(6), 3150–3162 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
26.
Knezevic, D.J., Peterson, J.W.: A high-performance parallel implementation of the certified reduced basis method. Comput. Methods Appl. Mech. Eng. 200(13), 1455–1466 (2011)zbMATHCrossRefGoogle Scholar
27.
Lee, Y.T., Sidford, A.: Efficient accelerated coordinate descent methods and faster algorithms for solving linear systems. In: 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), vol. 00, pp 147–156 (2014)Google Scholar
28.
Maday, Y., Nguyen, N.C., Patera, A.T., Pau, S.H.: A general multipurpose interpolation procedure: the magic points. Communications on Pure & Applied Analysis 8(1), 383 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
29.
Martinsson, P.-G.: A fast direct solver for a class of elliptic partial differential equations. J. Sci. Comput. 38(3), 316–330 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
30.
Oxberry, G.M., Kostova-Vassilevska, T., Arrighi, W., Chand, K.: Limited-memory adaptive snapshot selection for proper orthogonal decomposition. Int. J. Numer. Methods Eng. 109(2), 198–217 (2017)MathSciNetCrossRefGoogle Scholar
31.
Quarteroni, A., Manzoni, A., Negri, F.: Reduced basis methods for partial differential equations: an introduction, vol. 92. Springer, Berlin (2015)zbMATHGoogle Scholar
32.
Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Meth. Eng. 15(3), 229 (2008)zbMATHCrossRefGoogle Scholar
33.
Sarlos, T.: Improved approximation algorithms for large matrices via random projections. In: 2006 IEEE 47th annual symposium on foundations of computer science (FOCS), pp 143–152. IEEE (2006)Google Scholar
34.
Sirovich, L.: Turbulence and the dynamics of coherent structures. I. coherent structures. Q. Appl. Math. 45(3), 561–571 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
35.
Tropp, J.A.: Improved analysis of the subsampled randomized Hadamard transform. Adv. Adapt. Data Anal. 3(01n02), 115–126 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
36.
Tropp, J.A.: User-friendly tail bounds for sums of random matrices. Found. Comput. Math. 12(4), 389–434 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
37.
Tropp, J.A., et al.: An introduction to matrix concentration inequalities. Foundations and Trends®;in Machine Learning 8(1-2), 1–230 (2015)zbMATHCrossRefGoogle Scholar
38.
Woodruff, D.P., et al.: Sketching as a tool for numerical linear algebra. Foundations and Trends®;in Theoretical Computer Science 10(1–2), 1–157 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
39.
Xia, J., Chandrasekaran, S., Gu, M., Li, X.S.: Superfast multifrontal method for large structured linear systems of equations. SIAM J. Matrix Anal. Appl. 31(3), 1382–1411 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
40.
Zahm, O., Nouy, A.: Interpolation of inverse operators for preconditioning parameter-dependent equations. SIAM J. Sci. Comput. 38(2), A1044–A1074 (2016)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

Authors and Affiliations

Oleg Balabanov
- 1
- 2
Anthony Nouy
- 1
Email author

1.Centrale NantesLMJL, UMR CNRS 6629NantesFrance
2.Polytechnic University of CataloniaLaCànBarcelonaSpain

Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation

Abstract

Keywords

Mathematics Subject Classification (2010)

Preview

Notes

Supplementary material

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article