Transport along stochatic flows in fluid dynamics

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https://hal.inria.fr/hal-01377723

Submitted on 9 Nov 2016

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Transport along stochatic flows in fluid dynamics

Valentin Resseguier, Etienne Mémin, Bertrand Chapron

To cite this version:

Valentin Resseguier, Etienne Mémin, Bertrand Chapron. Transport along stochatic flows in fluid dynamics. Workshop - Statistical methods for dynamical stochastic models - DYNSTOCH 2016, Jun 2016, Rennes, France. �hal-01377723�

Transport along stochastic flows in fluid dynamics

Valentin Resseguier, Etienne Mémin, Bertrand Chapron

Fluids are very complex

with small vortices interacting with large vortices and

currents

Gula, Jonathan, M. Jeroen Molemaker, and James C.

McWilliams

"Gulf Stream dynamics along the southeastern US seaboard.”

Journal of Physical Oceanography 45.3 (2015): 690-715.

Fluids are very complex

with small vortices interacting with large vortices and

currents

Gula, Jonathan, M. Jeroen Molemaker, and James C.

McWilliams

"Gulf Stream dynamics along the southeastern US seaboard.”

Journal of Physical Oceanography 45.3 (2015): 690-715.

Why a random fluid dynamics?

• Take into account unresolved processes (small scales)

• Physical justification of empirical models

• Predicting possible distinct scenarios, extreme-events, …

• Quantification of modeling errors for data assimilation:

ensemble forecasts

• Randomized dynamics

• Simulation of the SQG under Moderate Uncertainty

• Stochastic reduced order model

Contents

Randomized

dynamics

• Stochastic flow:

Advection of tracer Θ

dX_t = w(X_t, t)dt + (X_t, t)dB_t

• Stochastic flow:

Advection of tracer Θ

dX_t = w(X_t, t)dt + (X_t, t)dB_t

(•, t)dB_t =⁴ Z

⌦

dz ˘ (•, z, t)dB_t(z) with infinite-dimensional Brownian motion

• Stochastic flow:

Advection of tracer Θ

dX_t = w(X_t, t)dt + (X_t, t)dB_t

(•, t)dB_t =⁴ Z

⌦

dz ˘ (•, z, t)dB_t(z) with infinite-dimensional Brownian motion

• A tracer is a function conserved along the flow:

D

⇥(t, X

) = d [⇥(t,

X

)] = 0

Ito-Wentzel formula

(Kunita 1997)

If both and are semimartingales (w.r.t. time) and is twice differentiable w.r.t. space

Then

d [⇥(t, X(t, y))] = d_t⇥ + (r⇥)^T dX + 1

2 tr(H_⇥d < X, X^T >) + d_t < (r⇥)^T , X >

Θ

Θ

Advection of tracer Θ

D _t ⇥ = 0

Advection of tracer Θ

Advection of tracer Θ

d

⇥ + w

· r ⇥dt + dB

· r ⇥ = r ·

✓ 1

2 a r ⇥

◆

dt

Advection

Advection of tracer Θ

d

⇥ + w

· r ⇥dt + dB

· r ⇥ = r ·

✓ 1

2 a r ⇥

◆

dt

Advection

Diffusion

Advection of tracer Θ

d

⇥ + w

· r ⇥dt + dB

· r ⇥ = r ·

✓ 1

2 a r ⇥

◆

dt

Advection

Diffusion

Advection of tracer Θ

d

⇥ + w

· r ⇥dt + dB

· r ⇥ = r ·

✓ 1

2 a r ⇥

◆

dt

a = ^T w^⇤ = w 1

2 (r · a)^T

Advection

Diffusion

Advection of tracer Θ

Drift

correction

d

⇥ + w

· r ⇥dt + dB

· r ⇥ = r ·

✓ 1

2 a r ⇥

◆

dt

a = ^T w^⇤ = w 1

2 (r · a)^T

Advection

Diffusion

Advection of tracer Θ

Drift

correction

Multiplicative random forcing

d

⇥ + w

· r ⇥dt + dB

· r ⇥ = r ·

✓ 1

2 a r ⇥

◆

dt

a = ^T w^⇤ = w 1

2 (r · a)^T

Advection

Diffusion

Advection of tracer Θ

Drift

correction

Multiplicative random forcing

d

⇥ + w

· r ⇥dt + dB

· r ⇥ = r ·

✓ 1

2 a r ⇥

◆

dt

a = ^T w^⇤ = w 1

2 (r · a)^T

Advection

Diffusion

Advection of tracer Θ

Drift

correction

Multiplicative random forcing

Balanced energy exchanges

d

⇥ + w

· r ⇥dt + dB

· r ⇥ = r ·

✓ 1

2 a r ⇥

◆

dt

a = ^T w^⇤ = w 1

2 (r · a)^T

(•, t)dB_t =⁴ Z

⌦

dz ˘ (•, z, t)dB_t(z)

=?

• Parametric or non-parametric estimation

• Observations can be:

Small-scale eulerian velocity

Small-scale Lagrangian velocity Tracer (solution of the SPDE)

• In the following simulations, very simple model (without estimations)

(x_i, t_j)dB_{tj ij}

(X_tj (x_i), t_j)dB_{tj ij} (⇥(x_i, t_j))_ij

dB_t = Z

⌦

dz r^? ˘(• z)dB_t(z) = r^? ˘ ⇤ dB_t b˘(k) = A 1_{{ <kkk< }}kkk ^↵

Simulation of the

SQG under Moderate Uncertainty

SQG MU

Code available online

SQG model:

Reference flow:

deterministic SQG 512 x 512

D_t⇥ = ⌫ ⁴⇥dt

w = ↵r^?( ) ¹² ⇥

SQG model:

Reference flow:

deterministic SQG 512 x 512

D_t⇥ = ⌫ ⁴⇥dt

w = ↵r^?( ) ¹² ⇥

One realization

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s

One realization

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s

Ensemble

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s

Ensemble

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s

Ensemble

x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s x ( m)

y(m)

t= 17 d ay s

0 2 4 6 8 10

x 10⁵ 0

2 4 6 8

x 10⁵

10⁻⁵ 10⁻⁴

10⁻⁶ 10⁻⁴ 10⁻²

|ˆb(κ)|2

κ! r a d . m⁻¹"

t= 17 d ay s

Ensemble

Random reduced order model from stochastic Navier-Stokes

equation

Why a reduced order model?

• Physical model (PDE) simplified using observations

• Very fast simulation of very complex system (e.g. for industrial application)

What is a reduced order

model?

PCA on data to reduce the state-space dimension

v (x, t) = v(x) +

X

i=0

b

(t)

(x) +

X

i=n+1

b

(t)

(x)

PCA on data to reduce the state-space dimension

v (x, t) = v(x) +

X

i=0

b

(t)

(x) +

X

i=n+1

b

(t)

(x)

v = w + B ˙

Resolved modes

PCA on data to reduce the state-space dimension

v (x, t) = v(x) +

X

i=0

b

(t)

(x) +

X

i=n+1

b

(t)

(x)

v = w + B ˙

Resolved modes Unresolved modes

⇡ 1

t dB

PCA on data to reduce the state-space dimension

v (x, t) = v(x) +

X

i=0

b

(t)

(x) +

X

i=n+1

b

(t)

(x)

v = w + B ˙

Resolved modes Unresolved modes

⇡ 1

t dB

Initial space Reduced subspace Solution

coordinates

Dimension d = 2 or 3

x M ~10⁷ n~10

(b_i(t))_i

(w_q(x_i, t))_qi

Assumptions of finite variations for and

Galerkin projection gives ODEs for resolved modes:

(stochastic Navier-Stokes)

Z

⌦ i

·

v = w + B ˙

db

= F

(b)dt

Assumptions of finite variations for and

Galerkin projection gives ODEs for resolved modes:

(stochastic Navier-Stokes)

Z

⌦ i

·

v = w + B ˙

db

= F

(b)dt

2^nd order polynomial:

coefficients given by physics,

j j and a(x, x, t) = 1

t < ( (x, t)B)_obs , ( (x, t)B)^T_obs >_t

Assumptions of finite variations for and

Galerkin projection gives ODEs for resolved modes:

(stochastic Navier-Stokes)

Z

⌦ i

·

v = w + B ˙

db

= F

(b)dt

2^nd order polynomial:

coefficients given by physics,

j j and a(x, x, t) = 1

t < ( (x, t)B)_obs , ( (x, t)B)^T_obs >_t

Model and estimation

(inspired by Genon-Catalot, Laredo, Picard 1992 )

a(x, x, t) = z₀(x) +

Xn

i=0

b_i(t)z_i(x)

z_i(x) = 1 t

Z t 0

b_i(t)d < (x, t)B, (x, t)B >

Galerkin projection gives SDEs for resolved modes:

(stochastic Navier-Stokes)

Z

⌦ i

·

db

= F

(b)dt + (↵

dB

)

b + (✓

dB

)

Galerkin projection gives SDEs for resolved modes:

(stochastic Navier-Stokes)

Z

⌦ i

·

n x M M x 1 n x 1 1 x M M x 1

db

= F

(b)dt + (↵

dB

)

b + (✓

dB

)

Galerkin projection gives SDEs for resolved modes:

(stochastic Navier-Stokes)

Z

⌦ i

·

additive noise multiplicative noise

n x M M x 1 n x 1 1 x M M x 1

db

= F

(b)dt + (↵

dB

)

b + (✓

dB

)

Galerkin projection gives SDEs for resolved modes:

(stochastic Navier-Stokes)

Z

⌦ i

·

additive noise multiplicative noise

2^nd order polynomial:

coefficients given by physics,

j j and a(x, x) = 1

t < ( (x)B)_obs , ( (x)B)^T_obs >_t

n x M M x 1 n x 1 1 x M M x 1

db

= F

(b)dt + (↵

dB

)

b + (✓

dB

)

Galerkin projection gives SDEs for resolved modes:

(stochastic Navier-Stokes)

Z

⌦ i

·

additive noise multiplicative noise

Correlations to estimate

2^nd order polynomial:

coefficients given by physics,

j j and a(x, x) = 1

t < ( (x)B)_obs , ( (x)B)^T_obs >_t

n x M M x 1 n x 1 1 x M M x 1

db

= F

(b)dt + (↵

dB

)

b + (✓

dB

)

Classical estimate SDE

Estimate of the correlation

multiplicative noise

Problem:

(↵_pidB_t)_obs = Z

⌦

G_pi [( dB_t)_obs]

↵ _pi ↵

_qj

Known but complex linear operator

Too complex to be computed for each time step

1 x M M x 1 d x M

(as a function of space)

db_i = F_i(b)dt + (↵_•idB_t)^Tb + (✓_idB_t)

↵_pi↵^T_qj = 1

t h↵_piB, ↵_qjBi_t

Classical estimate SDE

Estimate of the correlation

multiplicative noise

Problem:

(↵_pidB_t)_obs = Z

⌦

G_pi [( dB_t)_obs]

↵ _pi ↵

_qj

Known but complex linear operator

Too complex to be computed for each time step

1 x M M x 1 d x M

(as a function of space)

a(x, y) = 1

t < ( (x)B)_obs , ( (y)B)^T_obs >_t ^{Cannot be even}

memorized

d x d x M x M

db_i = F_i(b)dt + (↵_•idB_t)^Tb + (✓_idB_t)

↵_pi↵^T_qj = 1

t h↵_piB, ↵_qjBi_t

SDE

Estimate of the correlation

multiplicative noise

Solution:

↵ _pi ↵

_qj

db_i = F_i(b)dt + (↵_•idB_t)^T b + (✓_idB_t)

(from the PCA)

1 t

⌧ b_i,

Z t 0

b_p (↵_qjdB_t)

t

= X

k

↵_ki↵^T_qj 1 t

Z t 0

b_kb_p

| {z }

= _{p kp}

+✓_i↵^T_qj 1 t

Z t 0

b_p

| {z }

=0

= _p↵_pi↵^T_qj

SDE

Estimate of the correlation

multiplicative noise

Solution:

↵ _pi ↵

_qj

db_i = F_i(b)dt + (↵_•idB_t)^T b + (✓_idB_t)

(from the PCA)

1

t

⌧ b_i,

Z t 0

b_p (↵_qjdB_t⁰)

t

= Z

⌦

G_qj

1 t

⌧

(b_i)_obs,

Z t 0

(b_p dB_t⁰)_obs

t

1 t

⌧ b_i,

Z t 0

b_p (↵_qjdB_t)

t

= X

k

↵_ki↵^T_qj 1 t

Z t 0

b_kb_p

| {z }

= _{p kp}

+✓_i↵^T_qj 1 t

Z t 0

b_p

| {z }

=0

= _p↵_pi↵^T_qj

SDE

Estimate of the correlation

multiplicative noise

Solution:

↵ _pi ↵

_qj

db_i = F_i(b)dt + (↵_•idB_t)^T b + (✓_idB_t)

(from the PCA)

1

t

⌧ b_i,

Z t 0

b_p (↵_qjdB_t⁰)

t

= Z

⌦

G_qj

1 t

⌧

(b_i)_obs,

Z t 0

(b_p dB_t⁰)_obs

t

1 t

⌧ b_i,

Z t 0

b_p (↵_qjdB_t)

t

= X

k

↵_ki↵^T_qj 1 t

Z t 0

b_kb_p

| {z }

= _{p kp}

+✓_i↵^T_qj 1 t

Z t 0

b_p

| {z }

=0

= _p↵_pi↵^T_qj

d x M

Conclusion

Conclusion

• Random transport applicable to any fluid dynamics models

• Better small scales

• Estimate position and amplitude of errors, extreme events, likely scenarios

• Possible applications for your previous and future estimation methods (e.g. MLE with )(⇥(x_i, t_j))_ij

Code SQG MU:

link from Fluminance website - V. Resseguier

Thank you for your attention