NUMERICAL ASPECTS OF PDE-CONSTRAINED OPTIMIZATION · Navid Allahverdi Numerics in Optimization....

NUMERICAL ASPECTS OF PDE-CONSTRAINED OPTIMIZATION

Navid AllahverdiJoint work with A. Pozo (BCAM) and E. Zuazua (BCAM&Ikerbasque)

BCAM – Basque Center for Applied Mathematics

NUMERIWAVES Working Group MeetingMARCH 31, 2014

IntroductionNumerical Experiments Formulation

PDE-constrained Optimization

Design Optimization

minp

J(u, p)

s.t. F (u, p) = 0,h(p) = 0,g(p) ≤ 0,

J(u, p): objective function, cost functionp: design variables such as geometry,u: state variables,F (u, p): system of PDEs governing uh(p) and g(p): constraints on p

Navid Allahverdi Numerics in Optimization


Optimal Control

Continuous SystemGiven u∗ and T find u0 such that minimizes:

J(u0) =12

∫R

(u(x ,T )− u∗(x))2 dx , (1)

where u is the solution of the viscous Burgers equation:∂tu + ∂x

(u2

2

)= ν∂xx u, x ∈ R, t > 0,

u(x , 0) = u0(x), x ∈ R.(2)

Ex. data assimilation, sonic boom propagation.solving min. problem = solving inverse (backward) problem (T −→ 0)backward problem for heat equation is ill-posed.backward problem for hyperbolic equation may have many solutions.



Optimal Control

Discretized systemIn the discretized system, we minimize:

J∆(u0j ) =

∆x2

∑Z

(uNj − u∗j )2 (3)

where unj is the solution of the discretized viscous Burgers equation:

un+1j = un

j −∆t∆x (gn

j+1/2 − gnj−1/2) + ν∆t

∆x2 (unj−1 − 2un

j + unj+1), n = 0, . . . ,N − 1,

u0j = 1

∆x

∫ xj+1/2xj−1/2

u0(x)dx .

(4)

Engquist-Osher (EO): gEO(v ,w) =v(v + |v |)

4+

w(w − |w |)4

, (5)

Lax-Friedrichs modified (LFM): gLFM(v ,w) =v2 + w2

4−

∆x∆t

(w − v4

). (6)


IntroductionNumerical Experiments

Problem Description

Target functionLet’s choose T = 50, and following target function:

u∗(x) =

3

2000

(− e−(5

√20+x)2 + e−(2

√20+x)2

+√π x(

erf(5√

20− x) + erf(2√

20 + x))), |x − 5| ≤ 25,

0, elsewhere.

(7)

Discretization

x ∈ (a, b)

spatial discretization: ∆x =(b − a)

M, M = 400, 600, 800, 1000, finite volume cells

temporal discretization to satisfy CFL:

∆t∆x

maxj|u0

j |+ ν∆t

∆x2 ≤12

(8)

physical viscosity is chosen as ν = 0.0001.



Engquist-Osher

−30 −20 −10 0 10 20 30 40 50

−0.05

0.00

0.05

0.10

0.15

0.20Engquist Osher Scheme, T=50, N=400 Cells

u0

uT

u ∗

−30 −20 −10 0 10 20 30 40 50

−0.05

0.00

0.05

0.10

0.15


u0

uT

u ∗

−30 −20 −10 0 10 20 30 40 50

−0.05

0.00

0.05

0.10

0.15


u0

uT

u ∗

−30 −20 −10 0 10 20 30 40 50

−0.05

0.00

0.05

0.10

0.15


u0

uT

u ∗



Engquist-Osher Flux, 800 Cells



Modified Lax-Friedrichs

−30 −20 −10 0 10 20 30 40 50

−0.05

0.00

0.05

0.10

0.15

0.20Modified Lax-Friedrichs, T=50, N=400 Cells

u0

uT

u ∗

−30 −20 −10 0 10 20 30 40 50

−0.05

0.00

0.05

0.10

0.15


u0

uT

u ∗

−30 −20 −10 0 10 20 30 40 50

−0.05

0.00

0.05

0.10

0.15


u0

uT

u ∗

−30 −20 −10 0 10 20 30 40 50

−0.05

0.00

0.05

0.10

0.15


u0

uT

u ∗



Modified Lax Friedrichs Flux, 400 Cells



Modified Lax Friedrichs Flux, 800 Cells



Backward and forward solvers

solve the optimization (backward) problem with MLF and obtain u0

solve the forward problem with EO from u0



Functional landscape of J∆(u0)

visualize the descretized functional landscapes

parameterize u0 for two degrees of freedom (p, q)

u0(p, q ; x) ={

u0N + p sin(qx), near steep fronts

u0N , elsewhere

(9)

u0N (x) = h x

(erf(m(x − a))− erf(m(x − b)

)for pairs of (p, q) →u0(p, q ; x) →J∆(u0) → plot(p, q, J∆).

−30 −20 −10 0 10 20 30 40 50−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

0.10Initial N-wave with superimposed oscillations

N−waveN−wave+oscillation

−30 −20 −10 0 10 20 30 40 50

−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

0.10Initial datum and target function

u0

uT



Functional Landscape

Functional landscape (p, q, J); Engquist-Osher (left) and modified Lax-Friedrichs(right) fluxes are used for discretizing the Burgers equation. The landscape of LFM isless sensitive to the variation of parameters p, q due the dissipative nature of LFM.

p

-0.03-0.02

-0.010.00

0.010.02

0.03

q

01

23

45

6

J(p, q)

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

p

-0.03-0.02

-0.010.00

0.010.02

0.03

q

01

23

45

6

J(p, q)

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040



Conclusion

The performance of Modified Lax-Friedrichs is not satisfactory in large timehorizon. The initial data obtained are quite oscillatory in the vicinity of sharpslopes for large cell sizes. The spurious oscillations are due to the numericalviscosity inherent in the LFM method and they do not have any physicalsignificance in relation with the dynamics of the Burgers equation.The performance of Engquist-Osher is satisfactory. The obtained initial data isless sensitive to the choice of the cell size ∆x and domain discretization.The discretization scheme can alter the underlying continuous dynamics of theequation as observed from functional landscapes. For example, the oscillatoryoptimal initial datum obtained with LFM is not a best fit for the functional if theinitial datum is evolved in time with EO scheme in forward mode.The numerical viscosity present in LFM is proportional to the square of the cellsize (∆x)2. The numerical viscosity is adjusted by the size of cells and hence thesolution is dependent on the domain discretization.


NUMERICAL ASPECTS OF PDE-CONSTRAINED OPTIMIZATION · Navid Allahverdi Numerics in Optimization....

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