Automatic diﬀerentiation applied to an inverse … EuroAd Workshop - Didier Auroux... ·...

Didier Auroux, Thomas Migliore

Laboratoire J. A. Dieudonne,

Universite de Nice Sophia Antipolis

in collaboration with Laurent Hascoet (INRIA), Jacques Blum (Univ. Nice), Laurent Loth

and Daniel Coelho (ANDRA)

Automatic differentiation applied to

an inverse problem in geology

Ninth Euro Automatic Differentiation Workshop,

INRIA Sophia Antipolis. November 26-27, 2009

Motivations

Nuclear waste storage : storage in deep level is considered by ANDRA

as the reference strategy for wastes with high or medium activity and long-lived.

Need for simulations of the radionucleide transport in underground impact

of a possible propagation of radioelements.

The modelling of the flow in porous media around the storage requires to

know the physical parameters of the different geological layers.

Those parameters (porosity and diffusion) are not directly accessible by mea-

surements, hence we have to solve an inverse problem to recover them.

9th Euro AD Workshop, INRIA, November 27 2009 1/26

Mathematical model

The propagation of concentration in porous medium is described by a

convection-diffusion equation, i.e. a transport equation, mass balance, for

each radionucleide. The equation has the following form

∂t(ωR C) −∇.(D.∇C − q C) + λ ωR C − Q = 0

where

∗ C(x, t) is the concentration of the radioelement (measured) ;

∗ ω(x) is the porosity of the medium (unknown) ;

∗ λ is the decay coefficient (λ = log(2)4,951152.1014 s−1) ;

∗ q(x, t) is the Darcy velocity (q = 0 m/s) ;

∗ R is the constant delay factor (R = 1) ;

∗ D(x) is the diffusion tensor (unknown) ;

∗ Q(x, t) denotes a source term (Q = 0 mol/m3/s).


Direct problem

We use the zonation method : the media is considered as a set of homogeneous

zones. The parameters are constant by zones : D = Di and ω = ωi in Ωi.

D =

10−11 m2/s on Ω1

10−6 m2/s on Ω2

10−10 m2/s on Ω3

ω =

0, 2 on Ω1

10 on Ω2

0, 3 on Ω3

C(t = 0) =

0 mol/m3 on Ω1 ∪ Ω3

1 mol/m3 on Ω2

D∇C(t, x).n = 0 on ∂Ω


Parameter estimation

Concentration values : the concentration is measured at final time on the

three lines γi, and on the subdomain Ω2 at every time.

Inverse problem : given Cobs, find porosity ω and diffusion D.

Cost function : quadratic misfit between the observations and the correspon-

ding quantities calculated by the model :

J(C(ω, D)) = α

∫ T

0

∫

Ω2

[

C − Cobs

]2dx dt + β

3∑

j=1

∫

γj

[

C(T ) − Cobs(T )]2

dx.

The weights α and β have been chosen to give comparable influence to the

two terms.

Goal : minimize J in order to find the optimal values of porosity and diffusion.


Twin experiments

Use of synthetic data :

• Choose values for the diffusion coefficient and the porosity ;

• Compute the resulting concentration Cobs in solving the forward problem ;

• Identify the diffusion and porosity from the observed concentration values.

Advantage : we can compare the porosity and the diffusion field calculated

by the inverse method to the initial porosity and diffusion field.


Minimization

We use gradient descent algorithms to minimize the criterion J we need to

compute the gradient of J with respect to ω and D.

The numerical gradient has been obtained by Automatic Differentation of the

direct code TRACES with the software TAPENADE developed by the Tropics

Team at INRIA Sophia-Antipolis generation of the adjoint code of TRACES.

We use the minimization software N2QN1 (quasi-Newton BFGS), which is a

bound constrained optimizer from Gilbert & Lemarechal (INRIA).


Automatic Differentiation

Direct code : 20.000 lines of mixed fortran77/fortran90 in 40 routines, not

supposed to be differentiated.

Some of the issues we met, while trying to make two successive program runs :

• I/O files in ascii mode, access from several routines, files not closed at the

end of the program

• Matrix allocations everywhere, almost no deallocation

• Control variables defined in an ascii file (difficult to change between two runs)

• Memory issues if we consider a reasonable number of time steps (1 trajectory

= 1 Go, 1 run = several Go of memory, increasing memory over runs)

• Allocatable arrays used uninitialized NaN in the adjoint



Main modifications :

• Store the trajectory (for generating the observations in twin eperiments)

• Define the cost function

• Change I/O ASCII files to binary files (with direct access)

• Rewrite initialization, allocations, deallocations, ...

• Rewrite the main time loop routine (do call onetimestep enddo)

• Remove the prints to standard output

and then play again with Tapenade !



Tapenade, adjoint code and validation :

1. Remove a few bugs in Tapenade (many thanks to Laurent Hascoet !)

2. Validation of the adjoint on a few time steps

48.000 lines of code, several months of work, and then :

Divided differences = 0.39085362941408853000E-06 (epsilon=0.20E-10)

AD Forward derivative = 0.39089480467175335000E-06

AD Reverse 〈xb|xd〉 = 0.39089480467178204000E-06

Slowdown factors : tangent 1.28 adjoint 4.31


Gradient validation

3. Validation of the adjoint with finite differences on a few time steps

r(ε) =J(K + ε δK) − J(K)

ε 〈∇J(K), δK〉−→ε→0

1


Gradient validation

4. and then validation on the “full” program

Diffusion parameters : 9.6E-7 0.00021 Porosity parameters : 0.0011 0.32

Cost function : f = 2.0258486318199945E-8

Adjoint gradient : g = -0.0018222876539832878 0.00008205308963214484

0.000001979478809560615 1.418571652819071E-7

Finite diff gradient : g = -0.0018222227804698817 0.00008205309952898589

0.00000197947976722247 1.418515488854936E-7

Slowdown factors : tangent 1.41 adjoint 5.44

Max Stack size : 10199 blocks of 16384 bytes => total :159.359375 Mbytes

C Traffic : 4945 Mb and 305450 millionths

F Traffic : 62 Mb and 73256 millionths

Divided differences = 0.30208125371924738381E-05 (epsilon=0.20E-10)

AD Forward derivative = 0.30210378956057821803E-05

AD Reverse <xb|xd> = 0.30210378956057436403E-05

Various evolutions of the code, modification of the location/number of obser-

vations, modification of the number of control variables, . . .


Numerical results - no noise

Parameters Initial Identified Reference

D1 5.10−4 0, 0999999999 0, 1

D3 1.10−2 0, 9999999999 1, 0

ω1 5.10−3 0, 1999999999 0, 2

ω3 2.10−3 0, 3000000000 0, 3

0 10 20 30 40 50 60 70 80 9010

−25

10−20

10−15

10−10

10−5

100

105

Iterations

Val

ues

total costcost 1cost 2

Evolution of the cost function

0 10 20 30 40 50 60 70 80 9010

−15

10−10

10−5

100

105

1010

Iterations

Val

ues

gradient (D1)

gradient (D3)

gradient (ω1)

gradient (ω3)

(absolute value)

Evolution of the gradient



0 10 20 30 40 50 60 70 80 900

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Iterations

Por

osity

ω1 identified

ω1 real

ω3 identified

ω3 real

Evolution of porosity

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

IterationsD

iffus

ion

D1 identified

D1 real

D3 identified

D3 real

Evolution of diffusion


Numerical results - noisy data

5% noise

on the data

Parameters Initial Identified Reference

D1 5.10−4 0, 1018767983 0, 1

D3 1.10−2 0, 9552432582 1, 0

ω1 5.10−3 0, 2049359853 0, 2

ω3 2.10−3 0, 1729395267 0, 3

0 10 20 30 40 50 60 70 80 9010

−2

10−1

100

101

102

103

104

Iterations

Val

ues



0 10 20 30 40 50 60 70 80 9010

−10

10−5

100

105

1010

Iterations

Val

ues

gradient (D1)

gradient (D3)

gradient (ω1)

gradient (ω3)

(absolute value)




0 10 20 30 40 50 60 70 80 900

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Iterations

Por

osity

ω1 identified

ω1 real

ω3 identified

ω3 real

Evolution of porosity

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

IterationsD

iffus

ion

D1 identified

D1 real

D3 identified

D3 real

Evolution of diffusion


Zone identification

Problem : find the different zones (Ω1 and Ω3)

⋆ The coefficients ω and D are still

fixed in Ω2.

⋆ We use data generated with the

previous 3 zones.

⋆ We start the identification with 440

zones (one zone for each cell) in the

vicinity of the interface between Ω1

and Ω3.

⋆ We try to recover the true areas by

identifying one diffusion and porosity

coefficient per zone.


Numerical results - N zones

Diffusion at

iteration 1

Diffusion at

iteration 947

True

diffusion



Porosity at

iteration 1

Porosity at

iteration 947

True

porosity



Evolution of the cost function and its gradient


Zone identification

⋆ We need to regularize the inverse

problem.

⋆ We add some a priori physical infor-

mation : the domain has a revolution

symmetry five concentric zones.

⋆ We start the identification with these

7 zones (five + Ω1 + Ω2) where the

coefficients are constant.

⋆ We try to recover the true areas by

identifying one diffusion and porosity

coefficient by zone.



True porosity

Porosite − iteration1

0

0.05

0.1

0.15

0.2

0.25

Porosity at iteration 1 Porosity at iteration 179



True diffusion Diffusion at iteration 1 Diffusion at iteration 179



0 20 40 60 80 100 120 140 160 18010

−25

10−20

10−15

10−10

10−5

100

105

Iterations

Val

ues



0 20 40 60 80 100 120 140 160 18010

−10

10−5

100

105

IterationsV

alue

s

norm of the gradient




5% noise

on the data

Porosity at iteration 299 Diffusion at iteration 299



0 50 100 150 200 250 30010

−2

10−1

100

101

102

103

Iterations

Val

ues



0 50 100 150 200 250 30010

−4

10−2

100

102

104

106

IterationsV

alue

s

norm of the gradient



Conclusions

• We now have the adjoint code of TRACES, thanks to TAPENADE

• The parameter estimation works in 2D

• The adjoint and optimization techniques have been validated with a large

number of zones

• The zone identification problem can be solved by using concentric zones

∗ Sensitivity studies (small sensitivity to the porosity in Ω3 ?)

∗ 3D code (2D axisymmetric)

∗ Compare with other methods (topological optimization, refinement indica-

tors, . . .)


Automatic diﬀerentiation applied to an inverse … EuroAd Workshop - Didier Auroux... ·...

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