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# Efficient data assimilation of unsteady 3D turbulent flows using intrusive generative models

Barbet P., Talla M., Valls A., Petit M.,

Regnault F., Salminieri N., Wallian L.,

Resseguier V., Stabile G.

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Context

Observer for wind turbine application

*Application:* Real-time estimation and prediction of 3D turbulent flow using strongly-limited computational resources & few sensors

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## Context

### Observer for wind turbine application

*Application:* Real-time estimation and prediction of 3D turbulent flow using strongly-limited computational resources & few sensors

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## Context

### Observer for wind turbine application

*Application:* Real-time estimation and prediction of 3D turbulent flow using strongly-limited computational resources & few sensors

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# Context

## Observer for wind turbine application

*Application:* Real-time estimation and prediction of 3D turbulent flow using strongly-limited computational resources & few sensors

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# Context

## Observer for wind turbine application

*Application:* Real-time estimation and prediction of 3D turbulent flow using strongly-limited computational resources & few sensors

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# Context

## Observer for wind turbine application

*Application:* Real-time estimation and prediction of 3D turbulent flow using strongly-limited computational resources & few sensors

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Which simple model?

How to combine model & measurements?

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# Context

## Observer for wind turbine application

*Application:* Real-time estimation and prediction of 3D turbulent flow using strongly-limited computational resources & few sensors

<figure><img src="image_21.png" /></figure>

Which simple model?

How to combine model & measurements?

**Scientific problem :**

**Simulation & data assimilation** under severe dimensional reduction

typically, $10^7 \rightarrow O(10)$ degrees of freedom

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> Content

I. State of the art

> Intrusive reduced order model (ROM)

> Large Eddy Simulation (LES) & hyperreduction

> Data assimilation

II. Truncation-adapted stochastic closure

> Location uncertainty models (LUM)

> Generative Galerkin-ROM for turbulent flows

III. Numerical results

> Test cases

> Data assimilation does the job!

<figure><img src="image_23.png" /></figure>
I. State of the art

Intrusive reduced order model (ROM)

LES and hyperreduction

Data assimilation
Intrusive reduced order model (ROM)
Combine physical models and learning approaches

• Principal Component Analysis (PCA) on a *dataset* to reduce the dimensionality:

<figure><img src="image_24.png" /></figure>

• Approximation:
$$v(x, t) \approx \sum_{i=0}^{n} b_i(t) \phi_i(x)$$
Intrusive reduced order model (ROM)
Combine physical models and learning approaches

Principal Component Analysis (PCA) on a dataset to reduce the dimensionality:

<figure><img src="image_25.png" /></figure>

Approximation:

$$v(x, t) \approx \sum_{i=0}^{n} b_i(t) \phi_i(x)$$
Intrusive reduced order model (ROM)
Combine physical models and learning approaches

* Principal Component Analysis (PCA) on a *dataset* to reduce the dimensionality:

<figure><img src="image_26.png" /></figure>

* Approximation:

$$v(x, t) \approx \sum_{i=0}^{n} \left( b_i(t) \right) \phi_i(x)$$
Intrusive reduced order model (ROM)
Combine physical models and learning approaches

* Principal Component Analysis (PCA) on a *dataset* to reduce the dimensionality:

<figure><img src="image_27.png" /></figure>

* Approximation:

$$v(x, t) \approx \sum_{i=0}^{n} \left( b_i(t) \Phi_i(x) \right)$$

* Projection of the “physics” onto the spatial modes (POD-Galerkin)

$$\int_{\Omega} dx \Phi_i(x) \cdot ( \textcolor{red}{\text{Physical equation}} \text{ (e.g. Navier-Stokes)})$$
Intrusive reduced order model (ROM)
Combine physical models and learning approaches

* Principal Component Analysis (PCA) on a *dataset* to reduce the dimensionality:

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>
# Intrusive reduced order model (ROM)
## Combine physical models and learning approaches

* Principal Component Analysis (PCA) on a *dataset* to reduce the dimensionality:

<figure><img src="image_30.png" /></figure>
Intrusive reduced order model (ROM)
Combine physical models and learning approaches

* Principal Component Analysis (PCA) on a *dataset* to reduce the dimensionality:

<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /></figure>
Intrusive reduced order model (ROM)
Combine physical models and learning approaches

* Principal Component Analysis (PCA) on a *dataset* to reduce the dimensionality:

<figure><img src="image_33.png" /></figure>
Intrusive reduced order model (ROM)
Combine physical models and learning approaches

* Principal Component Analysis (PCA) on a *dataset* to reduce the dimensionality:

<figure><img src="image_34.png" /></figure>

* Approximation:

$$v(x, t) \approx \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

* Projection of the “physics” onto the spatial modes (POD-Galerkin)
## Large Eddy Simulation (LES) & hyperreduction
How to deal with turbulent unsteady flows ?

* DNS cannot generate large amount of 3D turbulent flow data

* Possible with turbulence models > Smagorinsky LES

$$ \begin{array}{c} \partial_t v + C(v, v) \\ \text{Advection} \end{array} = \begin{array}{c} -\nabla p \\ \text{Pressure} \end{array} + \begin{array}{c} D(v) \\ \text{Viscous term} \end{array} + \begin{array}{c} F(v) \\ \text{Smagorinsky term} \end{array} $$

with F a non-polynomial differential operator

* Issue for Galerkin projection / ROM

* Solution : Hyperreduction > Discrete Empirical Interpolation Method (DEIM)

$$ F(x, t) \approx \hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x) = \bar{F}(x) + \sum_{k=1}^{m} \left( \sum_{i=1}^{m} G_{ik} \psi_i(x) \right) F(x_k, t) $$
## Large Eddy Simulation (LES) & hyperreduction
How to deal with turbulent unsteady flows ?

* DNS cannot generate large amount of 3D turbulent flow data

* Possible with turbulence models > Smagorinsky LES

$$ \begin{array}{c} \partial_t v + C(v, v) \\ \text{Advection} \end{array} = \begin{array}{c} -\nabla p \\ \text{Pressure} \end{array} + \begin{array}{c} D(v) \\ \text{Viscous term} \end{array} + \begin{array}{c} F(v) \\ \text{Smagorinsky term} \end{array} $$

with F a non-polynomial differential operator

* Issue for Galerkin projection / ROM

* Solution : Hyperreduction > Discrete Empirical Interpolation Method (DEIM)

$$ F(x, t) \approx \hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x) $$

Hardly work
for advection-dominated problem
(slow Kolmogorov N-width decay)
<figure><img src="image_35.png" /><figcaption>## Data assimilation
= (Bayesian) inverse problem to couple simulations and measurements y</figcaption></figure>
<figure><img src="image_36.png" /><figcaption>> **Data assimilation** = (Bayesian) inverse problem to couple simulations and measurements y</figcaption></figure>
<figure><img src="image_37.png" /><figcaption>> **Data assimilation** = (Bayesian) inverse problem to couple simulations and measurements y</figcaption></figure>
<figure><img src="image_38.png" /><figcaption>## Data assimilation = (Bayesian) inverse problem to couple simulations and measurements y</figcaption></figure>
<figure><img src="image_39.png" /><figcaption>> **Data assimilation** = (Bayesian) inverse problem to couple simulations and measurements y</figcaption></figure>
<figure><img src="image_40.png" /><figcaption>## Data assimilation = (Bayesian) inverse problem to couple simulations and measurements y</figcaption></figure>
<figure><img src="image_41.png" /><figcaption>## Data assimilation = (Bayesian) inverse problem to couple simulations and measurements y</figcaption></figure>
II. Truncation-adapted stochastic closure

Location uncertainty models (LUM)

Generative Galerkin-ROM for turbulent flows
<figure><img src="image_42.png" /></figure>

# Location uncertainty models (LUM)

## Randomized DNS

$$v = w + v'$$

Resolved fields:

w

Unresolved fields:

$$v' = \sigma \dot{B}$$

(Gaussian, white wrt t)

From Ito-Wentzell
formula (Kunita 1990)
with Ito notations

Momentum conservation

$$\frac{d}{dt} (w(t, X_t)) = F \quad (\text{Forces})$$

Positions of fluid parcels $X_t$ :

$$\frac{d}{dt} X_t = w(t, X_t) + \sigma(t, X_t) \underbrace{\frac{dB_t}{dt}}_{\substack{\text{Gaussian} \\ \text{process} \\ \text{white in time}}}$$

<figure><img src="image_43.png" /></figure>
## Location uncertainty models (LUM)
Randomized DNS

$$v = w + v'$$

Resolved fields:

w

Unresolved fields:

$v' = \sigma \dot{B}$

(Gaussian, white wrt t)

From Ito-Wentzell
formula (Kunita 1990)
with Ito notations

<figure><img src="image_44.png" /></figure>
<figure><img src="image_45.png" /></figure>

# Location uncertainty models (LUM)
## Randomized LES

$$v = w + v'$$

$$F = \hat{F} + F'$$

Resolved fields:

w

$\hat{F}$

Unresolved fields:

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

<figure><img src="image_46.png" /></figure>
## Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

$$\tilde{b} = \binom{1}{b} : n + 1$$

$$\Pi_{\phi} g = \sum_{j=1}^{m} \langle g, \phi_i \rangle \phi_i$$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$
## Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

$$v = w + v'$$

$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

$$\tilde{b} = \binom{1}{b} : n + 1$$

$$\Pi_{\phi} g = \sum_{j=1}^{m} \langle g, \phi_i \rangle \phi_i$$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$
> Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

Full order : $M \sim 10^7$
Reduced order : $n, m \sim 10$

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

$$\tilde{b} = \binom{1}{b} : n + 1$$
$$\Pi_{\phi} g = \sum_{j=1}^{m} \langle g, \phi_i \rangle \phi_i$$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$
## Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

Full order : $M \sim 10^7$
Reduced order : $n, m \sim 10$

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

<figure><img src="image_47.png" /></figure>

$$\tilde{b} = \binom{1}{b} : n + 1$$
$$\Pi_{\phi} g = \sum_{j=1}^{m} \langle g, \phi_i \rangle \phi_i$$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$
> Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

Full order : $M \sim 10^7$
Reduced order : $n, m \sim 10$

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

<figure><img src="image_48.png" /></figure>

2nd order polynomial

$$\tilde{b} = \binom{1}{b} : n + 1$$
$$\Pi_{\phi} g = \sum_{j=1}^{m} \langle g, \phi_i \rangle \phi_i$$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$
## Generative Galerkin-ROM for turbulent flows (Red LUM)
### POD-Galerkin gives SDEs for resolved modes

Full order : $M \sim 10^7$
Reduced order : $n, m \sim 10$

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

<figure><img src="image_49.png" /></figure>

2nd order polynomial

Projected additive &
multiplicative noises

$$\tilde{b} = \binom{1}{b} : n + 1$$

$$\Pi_{\phi} g = \sum_{j=1}^{m} \langle g, \phi_i \rangle \phi_i$$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$
## Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

$$v = w + v' \\ F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K(\sigma \dot{B}, \sigma_F \dot{B}) \tilde{b}(t)$$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$
## Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K(\sigma \dot{B}, \sigma_F \dot{B}) \tilde{b}(t)$$

<figure><img src="image_50.png" /></figure>

2nd order polynomial

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t \nu'(\nu')^T$

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \begin{pmatrix} 1 \\ b \end{pmatrix}$$

$$a(x, x) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q}\xi^F)$$
## Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K(\sigma \dot{B}, \sigma_F \dot{B}) \tilde{b}(t)$$

<figure><img src="image_51.png" /></figure>

2nd order polynomial

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t \nu'(\nu')^T$

Randomized Navier-Stokes

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \begin{pmatrix} 1 \\ b \end{pmatrix}$$

$$a(x, x) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q}\xi^F)$$
## Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K(\sigma \dot{B}, \sigma_F \dot{B}) \tilde{b}(t)$$

<figure><img src="image_52.png" /></figure>

2nd order polynomial

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t \nu'(\nu')^T$

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \begin{pmatrix} 1 \\ b \end{pmatrix}$$

$$a(x, x) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$

Randomized Navier-Stokes

PCA modes

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q}\xi^F)$$
## Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K(\sigma \dot{B}, \sigma_F \dot{B}) \tilde{b}(t)$$

<figure><img src="image_53.png" /></figure>

2nd order polynomial

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t \nu' (\nu')^T$

<figure><img src="image_54.png" /></figure>

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \begin{pmatrix} 1 \\ b \end{pmatrix}$$

$$a(x, x) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$
> Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K(\sigma \dot{B}, \sigma_F \dot{B}) \tilde{b}(t)$$

<figure><img src="image_55.png" /></figure>

Projected additive &
multiplicative noises

2nd order polynomial

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t \overline{(v' (v')^T)}$

<figure><img src="image_56.png" /></figure>

Randomized Navier-Stokes

<figure><img src="image_57.png" /></figure>

PCA modes

<figure><img src="image_58.png" /></figure>

PCA residual $v'$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \begin{pmatrix} 1 \\ b \end{pmatrix}$$

$$a(x, x) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$

variant of Resseguier et al. (2021). SIAM-ASA J Uncertain. hal-03169957
## Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot (-\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B}) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K(\sigma \dot{B}, \sigma_F \dot{B}) \tilde{b}(t)$$

<figure><img src="image_59.png" /></figure>

Projected additive &
multiplicative noises

2nd order polynomial

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t \overline{(v' (v')^T)}$

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \begin{pmatrix} 1 \\ b \end{pmatrix}$$

$$a(x, x) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$

Randomized Navier-Stokes

PCA modes

PCA residual $v'$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$
## Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot \left( -\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B} \right) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K \left( \sigma \dot{B}, \sigma_F \dot{B} \right) \tilde{b}(t)$$

2nd order polynomial

Projected additive & multiplicative noises

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t \, v'(v')^T$

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \begin{pmatrix} 1 \\ b \end{pmatrix}$$

$$a(x, x) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$

Randomized Navier-Stokes

PCA modes

PCA residual $v'$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q}\xi^F)$$
## Generative Galerkin-ROM for turbulent flows (Red LUM)
### POD-Galerkin gives SDEs for resolved modes

Full order : $M \sim 10^7$

Reduced order : $n, m \sim 10$

$$v = w + v'$$

$$F = \hat{F} + F'$$

**Resolved fields :**

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

**Unresolved fields :**

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot \left( -\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B} \right) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K \left( \sigma \dot{B}, \sigma_F \dot{B} \right) \tilde{b}(t)$$

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t \, v'(v')^T$

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \left( \frac{1}{b} \right)$$

$$a(x, t) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$

variant of Resseguier et al. (2021). SIAM-ASA J Uncertain. hal-03169957

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q}\xi^F)$$
## Generative Galerkin-ROM for turbulent flows (Red LUM)
### POD-Galerkin gives SDEs for resolved modes

Full order : $M \sim 10^7$

Reduced order : $n, m \sim 10$

$$v = w + v'$$

$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot \left( -\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B} \right) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K \left( \sigma \dot{B}, \sigma_F \dot{B} \right) \tilde{b}(t)$$

2nd order polynomial

Projected additive &
multiplicative noises

**Covariance to estimate**

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t \, v' (v')^T$

Randomized Navier-Stokes

PCA modes

PCA residual $v'$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \begin{pmatrix} 1 \\ b \end{pmatrix}$$

$$a(x, x) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$

variant of Resseguier et al. (2021). SIAM-ASA J Uncertain. hal-03169957
> Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

Full order : $M \sim 10^7$
Reduced order : $n, m \sim 10$

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot \left( -\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B} \right) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K \left( \underbrace{\sigma \dot{B}, \sigma_F \dot{B}}_{\text{2M} \times 1} \right) \tilde{b}(t)$$

<figure><img src="image_60.png" /></figure>

2nd order polynomial

Projected additive &
multiplicative noises

**Covariance to estimate**

$$\mathbb{E}(K_{jq}(\dots) K_{ip}(\dots)) / dt \approx \Delta t K_{jq} \left[ \frac{b_p}{b_p^2} \frac{\Delta b_i}{\Delta t} (v', F') \right]$$

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t v'(v')^T$

<figure><img src="image_61.png" /></figure>

<figure><img src="image_62.png" /></figure>

<figure><img src="image_63.png" /></figure>

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \left(\frac{1}{b}\right)$$

$$a(x, x) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$

variant of Resseguier et al. (2021). SIAM-ASA J Uncertain. hal-03169957
## Generative Galerkin-ROM for turbulent flows (Red LUM)
### POD-Galerkin gives SDEs for resolved modes

Full order : $M \sim 10^7$

Reduced order : $n, m \sim 10$

$$v = w + v'$$

$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot \left( -\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B} \right) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K \left( \sigma \dot{B}, \sigma_F \dot{B} \right) \tilde{b}(t)$$

<figure><img src="image_64.png" /></figure>

2nd order polynomial

Projected additive &
multiplicative noises

**Covariance to estimate**

$$\mathbb{E}(K_{jq}(\dots) K_{ip}(\dots)) / dt \approx \Delta t K_{jq} \left[ \frac{b_p}{b_p^2} \frac{\Delta b_i}{\Delta t} (v', F') \right]$$

<figure><img src="image_65.png" /></figure>

Randomized Navier-Stokes

<figure><img src="image_66.png" /></figure>

PCA modes

<figure><img src="image_67.png" /></figure>

PCA residual $v'$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t v' (v')^T$

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \left(\frac{1}{b}\right)$$

$$a(x, t) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$

variant of Resseguier et al. (2021). SIAM-ASA J Uncertain. hal-03169957
> Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

Full order : $M \sim 10^7$

Reduced order : $n, m \sim 10$

$v = w + v'$

$F = \hat{F} + F'$

Resolved fields :

$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$

$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$

Unresolved fields :

$v' = \sigma \dot{B}$

$F' = \sigma_F \dot{B}$

(Gaussian, white wrt t)

$$ \int_{\Omega} \phi_i(x) \cdot \left( -\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B} \right) dx $$

$$ M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K \left( \sigma \dot{B}, \sigma_F \dot{B} \right) \tilde{b}(t) $$

<figure><img src="image_68.png" /></figure>

2nd order polynomial

Projected additive &
multiplicative noises

**Covariance to estimate**

$$ \mathbb{E}(K_{jq}(\dots) K_{ip}(\dots)) / dt \approx \Delta t K_{jq} \left[ \frac{b_p}{b_p^2} \frac{\Delta b_i}{\Delta t} (v', F') \right] $$

<figure><img src="image_69.png" /></figure>

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t v'(v')^T$

$\bar{f} = \frac{1}{T} \int_{0}^{T} f$

$\tilde{b} = \left(\frac{1}{b}\right)$

$a(x, t) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$

$$ K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F) $$
> Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

Full order : $M \sim 10^7$
Reduced order : $n, m \sim 10$

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot \left( -\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B} \right) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K \left( \sigma \dot{B}, \sigma_F \dot{B} \right) \tilde{b}(t)$$

<figure><img src="image_70.png" /></figure>

2nd order polynomial

Projected additive &
multiplicative noises

**Covariance to estimate**

$$\mathbb{E}(K_{jq}(\dots) K_{ip}(\dots)) / dt \approx \Delta t K_{jq} \left[ \frac{b_p}{b_p^2} \frac{\Delta b_i}{\Delta t} (v', F') \right]$$

<figure><img src="image_71.png" /></figure>

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F) d\xi$$

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t v'(v')^T$

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \left(\frac{1}{b}\right)$$

$$a(x, t) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$

variant of Resseguier et al. (2021). SIAM-ASA J Uncertain. hal-03169957
> Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

Full order : $M \sim 10^7$

Reduced order : $n, m \sim 10$

$v = w + v'$

$F = \hat{F} + F'$

Resolved fields :

$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$

$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$

Unresolved fields :

$v' = \sigma \dot{B}$

$F' = \sigma_F \dot{B}$

(Gaussian, white wrt t)

$$\int_{\Omega} \phi_i(x) \cdot \left( -\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B} \right) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K \left( \sigma \dot{B}, \sigma_F \dot{B} \right) \tilde{b}(t)$$

<figure><img src="image_72.png" /></figure>

2nd order polynomial

Projected additive &
multiplicative noises

**Covariance to estimate**

$$\mathbb{E}(K_{jq}(\dots) K_{ip}(\dots)) / dt \approx \Delta t K_{jq} \left[ \frac{b_p}{b_p^2} \frac{\Delta b_i}{\Delta t} (v', F') \right]$$

<figure><img src="image_73.png" /></figure>

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t v'(v')^T$

$\bar{f} = \frac{1}{T} \int_{0}^{T} f$

$\tilde{b} = \left(\frac{1}{b}\right)$

$a(x, t) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$

$$K_{jq}[\xi, \xi^F] = \int_{\Omega} \phi_j \cdot (-C(\xi, \phi_q) + \delta_{0q} \xi^F)$$

variant of Resseguier et al. (2021). SIAM-ASA J Uncertain. hal-03169957
## Generative Galerkin-ROM for turbulent flows (Red LUM)
POD-Galerkin gives SDEs for resolved modes

$$v = w + v'$$
$$F = \hat{F} + F'$$

Resolved fields :

$$w(x, t) = \sum_{i=0}^{n} b_i(t) \phi_i(x)$$

$$\hat{F}(x, t) = \sum_{i=0}^{m} r_i(t) \psi_i(x)$$

Unresolved fields :

$$v' = \sigma \dot{B}$$

$$F' = \sigma_F \dot{B}$$

(Gaussian, white wrt t)

Full order : $M \sim 10^7$
Reduced order : $n, m \sim 10$

$$\int_{\Omega} \phi_i(x) \cdot \left( -\partial_t w - C(w^*, w) - C(\sigma \dot{B}, w) + (D_{\sigma} + D)(w) + \hat{F}(w) + \sigma_F \dot{B} \right) dx$$

$$M \frac{d}{dt} b(t) = H(\tilde{b}(t)) + K \left( \sigma \dot{B}, \sigma_F \dot{B} \right) \tilde{b}(t) \quad (n+1) \times 1$$

<figure><img src="image_74.png" /></figure>

Projected additive &
multiplicative noises

**Covariance to estimate**

2nd order polynomial

$$\mathbb{E}(K_{jq}(\dots) K_{ip}(\dots)) / dt \approx \Delta t K_{jq} \left[ \frac{b_p}{b_p^2} \frac{\Delta b_i}{\Delta t} (v', F') \right]$$

Coefficients given by :

* Randomized Navier-Stokes

* $(\phi_j)_j, (\psi_j)_j$

* $a(x) \approx \Delta t v'(v')^T$

$$\bar{f} = \frac{1}{T} \int_{0}^{T} f$$

$$\tilde{b} = \left(\frac{1}{b}\right)$$

$$a(x, t) = \mathbb{E}\{(\sigma dB_t)(\sigma dB_t)^T\}/dt$$

### New estimator

* Truncation-adapted

* Consistency proven ($\Delta t \to 0$)

* Numerically efficient

* Physically-based
  → Robustness in
  extrapolation
# Summary
## Generative Galerkin-ROM + Data assimilation

<figure><img src="image_75.png" /></figure>

<figure><img src="image_76.png" /></figure>

<figure><img src="image_77.png" /></figure>

<figure><img src="image_78.png" /></figure>
# Summary
## Generative Galerkin-ROM + Data assimilation

<figure><img src="image_79.png" /></figure>
# Summary
## Generative Galerkin-ROM + Data assimilation

<figure><img src="image_80.png" /></figure>
## III. Numerical results

* Test cases

* Data assimilation does the job!

<figure><img src="image_81.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Liberté
Égalité
Fraternité

INRAO
> Test cases

Wake flow past a cylinder

Open▼FOAM®

DNS

“true” simulation
(Re 300, 10⁷ degrees of freedom)

LES

“true” simulation
(Re 3900, 10⁷ degrees of freedom)
> Test cases

Wake flow past a cylinder

Open▼FOAM®

**DNS**
"true" simulation
(Re 300, 10⁷ degrees of freedom)

<figure><img src="image_82.png" /></figure>

**LES**
"true" simulation
(Re 3900, 10⁷ degrees of freedom)
> Test cases

Wake flow past a cylinder

Open▼FOAM®

**DNS**
"true" simulation
(Re 300, 10⁷ degrees of freedom)

<figure><img src="image_83.png" /></figure>

**LES**
"true" simulation
(Re 3900, 10⁷ degrees of freedom)

<figure><img src="image_84.png" /></figure>
> Test cases
Wake flow past a cylinder

For data assimilation:
Measurement point (blurred & noisy velocity)

Open▼FOAM®

<figure><img src="image_85.png" /></figure>

DNS
"true" simulation
(Re 300, 10⁷ degrees
of freedom)

Synthetic PIV
blurred & noisy
2D velocity

<figure><img src="image_86.png" /></figure>

LES
"true" simulation
(Re 3900, 10⁷ degrees
of freedom)
<figure><img src="image_87.png" /></figure>

# Data assimilation does the job!

DNS, Re 300, $n = 4$ modes

<figure><img src="image_88.png" /></figure>

<figure><img src="image_89.png" /></figure>

From $10^7$ to 4 degrees of freedom

Single measurement point (blurred & noisy velocity)

Synthetic PIV
blurred & noisy
2D velocity

**DNS**
"true" simulation
(Re 300, $10^7$ degrees of freedom)

**Reference :**
POD-projection of the
"true" simulation
(DNS)
($H^1$optimal from 4-
degrees of freedom
linear decomposition)

**Our method :**
Data assimilation
with
POD-Galerkin of
randomized DNS

(4 degrees of freedom)
<figure><img src="image_90.png" /></figure>

# Data assimilation does the job!

DNS, Re 300, $n = 4$ modes

<figure><img src="image_91.png" /></figure>

From $10^7$ to 4 degrees of freedom

Single measurement point (blurred & noisy velocity)

<figure><img src="image_92.png" /></figure>

Synthetic PIV
blurred & noisy
2D velocity

### DNS

“true” simulation
(Re 300, $10^7$ degrees of freedom)

### Reference :

POD-projection of the
“true” simulation
(DNS)
($H^1$optimal from 4-
degrees of freedom
linear decomposition)

<figure><img src="image_93.png" /></figure>

### Our method :

Data assimilation
with
POD-Galerkin of
randomized DNS

(4 degrees of freedom)
<figure><img src="image_94.png" /></figure>

# Data assimilation does the job!

LES, Re 3900, $n = 16$ modes

<figure><img src="image_95.png" /></figure>

From $10^7$ to 16 degrees of freedom
9 measurement points (blurred & noisy velocity)

**LES**
"true" simulation
(Re 3900, $10^7$ degrees of freedom)

<figure><img src="image_96.png" /></figure>

**Synthetic PIV**
blurred & noisy
2D velocity

**Reference :**
POD-projection of the
"true" simulation
(LES)
($H^1$optimal from 16-
degrees of freedom
linear decomposition)

<figure><img src="image_97.png" /></figure>

**Our method :**
Data assimilation
with
POD-Galerkin of
randomized LES

(16 degrees of
freedom)
> Conclusion

<figure><img src="image_98.png" /></figure>

RÉPUBLIQUE
FRANÇAISE

Liberté
Égalité
Fraternité

<figure><img src="image_99.png" /></figure>
# Conclusion

<figure><img src="image_100.png" /></figure>

* Intrusive ROM : for very fast and robust CFD ($O(10^7) \rightarrow O(10)$ degrees of freedom)

* Smagorinsky LES term handled by randomized DEIM

* Closure problems handled by LU-like models

* Dimension-adapted estimator for correlated additive & multiplicative noises

* Provide a dimension-adapted "generative model" for prior / Model error quantification

* Data assimilation (Bayesian inverse problem) :
  to correct the fast simulation on-line

* Unsteady 3D turbulent flow estimation/prediction in the whole spatial domain (large-scale structures)

* From sparse & noisy measurements

* Robust far outside the training set

# Next steps

Hiring postdoc !

* Better (deterministic & stochastic) hyperreduction

* Along-spanwise measurements

* Non-Markovian closure

* Real measurements