Sensitivity analysis for nonlinear hyperbolic equations 21/6/2016 - Junior Seminar 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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
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Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
9
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
9
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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.
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Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
Numerical results
Sensitivity with source term: Sensitivity without source term:
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Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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
Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
Numerical results
Sensitivity with source term: Sensitivity without source term:
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Camilla Fiorini Sensitivity analysis for hyperbolic equations / 18Junior Seminar
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).
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Thank you for your attention!