Sensitivity analysis for nonlinear hyperbolic equations · 2016-06-23 · Camilla Fiorini...

Sensitivity analysis for nonlinear hyperbolic equations

21/6/2016 - Junior Seminar

Camilla Fiorini

Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar 1

Where am I from?

Régis Duvigneau (INRIA Sophia Antipolis), Christophe Chalons (LMV UVSQ).

Advisors:

University: Université Paris Saclay - Université de Versailles Saint-Quentin-en-Yvelines.

Lab: Laboratoire de Mathématiques de Versailles

‣ Analysis and PDEs

‣ Probability and Statistics

‣ Algebra

‣ Cryptography

Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar 1

Where am I from?

Régis Duvigneau (INRIA Sophia Antipolis), Christophe Chalons (LMV UVSQ).

Advisors:

University: Université Paris Saclay - Université de Versailles Saint-Quentin-en-Yvelines.

Lab: Laboratoire de Mathématiques de Versailles

‣ Analysis and PDEs

‣ Probability and Statistics

‣ Algebra

‣ Cryptography

Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar

Sensitivity Analysis

Sensitivity Analysis: the study of how variations in the output of a model

can be attributed to different sources of uncertainty in the model input.

Model: system of PDEs

p

p+ �p u+ �u

u

Therefore, we want to study the derivative of u with respect to p:

up =@u

@p

2


Applications

‣ Propagation of uncertainty or error: sensitivity can be used to study how uncertainty in a measurement of a parameter can affect the solution.

u(p+ �p) ' u(p) + �pup(p)

minp2P

J(u(p))

rpJ =@J

@uup

‣ Estimate of close solutions: using a first order Taylor expansion it is possible to estimate solution for different parameters values.

‣ Optimisation: sensitivity can be useful to solve problems such as

for which it is necessary to compute the gradient of the cost functional:

3


State equations

We will consider hyperbolic equations:

(@

t

u+ @

x

f(u) = 0 x 2 R, t > 0

u(x, 0) = g(x;p) x 2 R.

Hyperbolic equations are also known as conservation laws:

‣ u is the conserved variable

‣ f(u) is the flux function

‣ g(x;p) is the initial condition

4


Sensitivity equations

Under hypothesis of regularity, on can differentiate the state equations with respect to the parameter:

(@p(@tu) + @p(@xf(u)) = 0 x 2 R, t > 0

@pu(x, 0) = @pg(x;p) x 2 R.

Exchanging the derivatives in space and time with the ones with respect to the parameter one has:

(@

t

up + @

x

(f 0(u)up) = 0 x 2 R, t > 0

up(x, 0) = gp(x;p) x 2 R.

5


Example: the Burger’s equation

f(u) =u2

2g(x;A, xc, L) =

(A sin

2(

⇡L (x� xc) +

⇡2 ) x 2 (xc � L

2 , xc +L2 )

0 otherwise.

p

6



f(u) =u2

2g(x;A, xc, L) =

(A sin

2(

⇡L (x� xc) +

⇡2 ) x 2 (xc � L

2 , xc +L2 )

0 otherwise.

p

6


Solution of the state equations

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

The Burger’s equation can be rewritten as: @t

u+ u @x

u = 0

speed at which the initial condition is transported

7


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

Shock

Characteristics

The Burger’s equation can be rewritten as: @t

u+ u @x

u = 0

speed at which the initial condition is transported

Solution of the state equations

7


Two kinds of derivatives

‣ Classical derivative: it is defined everywhere but in the discontinuity

‣Weak derivative: it is defined also in the discontinuity, where it is a Dirac’s distribution

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

8



f(u) =u2

2g(x;A, xc, L) =

(A sin

2(

⇡L (x� xc) +

⇡2 ) x 2 (xc � L

2 , xc +L2 )

0 otherwise.

p

9



f(u) =u2

2g(x;A, xc, L) =

(A sin

2(

⇡L (x� xc) +

⇡2 ) x 2 (xc � L

2 , xc +L2 )

0 otherwise.

p

9


Choice of the derivative

Classical derivative:

Weak derivative:

‣ it does not corrupt the solution in the regular zones

‣ it is possible to estimate close solutions

‣ no correction to numerical schemes needed

10


Rankine-Hugoniot conditions

(@

t

u+ @

x

f(u) = 0 x 2 R, t > 0

u(x, 0) = g(x;p) x 2 R.f(u+)� f(u�) = �(u+ � u�)

(@

t

up + @

x

(f 0(u)up) = 0 x 2 R, t > 0

up(x, 0) = gp(x;p) x 2 R.f 0(u+)u+

p � f 0(u�)u�p = �(u+

p � u�p )

f 0(u+)u+p � f 0(u�)u�

p = �(u+p � u�

p ) + @p�(u+ � u�)

Across the shock, the state is governed by the Rankine-Hugoniot conditions:

If we wrote the same conditions for the sensitivity, we would have:

However, differentiating with respect to p the conditions for the state we obtain:

Idea: add to the sensitivity equation a source term that balances it out.

@

t

up + @

x

(f 0(u)up) = s(u+,u�) x 2 R, t > 0

11


Shock detection

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

@p�(u+ � u�)The term is zero in the regular zones, however this is not true if we

consider a discretisation of the equations.

It is necessary to define a shock detector.

12


Numerical results

The Riemann problem for the Burger’s equation:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

t

Shock detector

Analytical shockDetected shock

g(x;uR, uL, xc) guL(x;uR, uL, xc)

uL

uR

xc 1 xc

1

1

In this case it is easy to define a good shock detector.

13


Numerical results

Sensitivity with source term: Sensitivity without source term:

14


Numerical results

The same shock detector in a less simple case does not work:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6

-4

-2

0

2

4

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

gp(x;p)g(x;p)

It leads to an overcorrection:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

15

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

8

10t = 0.085568

NumericalAnalyticalNumerical wo corr

0 0.2 0.4 0.6 0.8 1-10

0

10

20

30

40

50

60

70t = 0.17575

NumericalAnalyticalNumerical wo corr


We defined a new shock detector based on the second derivative and on the breaking time.

Numerical results

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t

Shock detector

Analytical shockDetected shock

16


Numerical results

Sensitivity with source term: Sensitivity without source term:

17


Conclusion and future developments

‣ A general method for sensitivity analysis in case of discontinuities has been developed ;

‣ Shock detectors are specific to each case;

‣ The method has been extended to systems (Euler 1D);

‣We plan to increase the space dimension (2D or 3D).

18

Thank you for your attention!

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Sensitivity analysis for nonlinear hyperbolic equations · 2016-06-23 · Camilla Fiorini...

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