Numerical solution of some geometric inverse problems
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Nume ical solu ion o some geome ic in e se p oblems
Anna DOUBOVA
Dp o. E.D.A.N. - Uni . o Se illa
join wo k wi h
E. FERN ´
ANDEZ-CARA - Dp o. E.D.A.N. - Uni . o Se illa
Nume ical Resolu ion o In e se P oblems
BCAM, Bilbao, 8-9 Janua y 2015
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Ou line
1Mo i a ion: Elas og aphy
2Wa e equa ion
3Elas ici y sys ems
4Recons uc ion and nume ical algo i hms
5Nume ical esul s
2D wa e equa ion
2D Lam´
e sys em
3D wa e equa ion
3D Lam´
e sys em
6Wo k in p og ess
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Geome ic In e se P oblems go e ned by PDEs
Mo i a ion: Elas og aphy
We conside :
Geome ic in e se p oblems
Wa e equa ion and Lam´
e sys ems
Mo i a ion: Elas og aphy
A non-in asi e me hod o umo de ec ion: when a mechanical
comp ession o ib a ion is applied, he umo de o ms less han he
su ounding issue
A echnique o de ec elas ic p ope ies o issue om acous ic wa e
gene a o s (applica ions in Medicine)
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Mo i a ion
Elas og aphy
Classical de ec ion me hods in mammog aphy:
Figu e: Palpa ion Figu e: x- ays
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Mo i a ion
Elas og aphy
Elas og aphy (“imaging palpa ion”) is be e sui ed han palpa ion and x- ays
echniques:
— Tumo s can be a om he su ace
—o small
— o may ha e p ope ies indis inguishable h ough palpa ion o x- ays
Figu e: S i ness is ep esen ed by a colo spec um, anging
om da k ed ( e y s i ) h ough o ange, yellow, and g een, o
blue ( e y so ).
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Wa e equa ion
N-dimensional wa e equa ion (N=2 o 3)
(a) Di ec p oblem:
Da a: Ω,T>0, ϕ,Dand γ⊂∂Ω
Resul : he solu ion u
(1)
u −∆u=0 in (Ω D)×(0,T)
u=ϕon (∂Ω) ×(0,T)
u=0 on (∂D)×(0,T)
u(x,0) = u0,u (x,0) = u1in Ω
In o ma ion:
(2)α=∂u
∂non γ×(0,T)
(b) In e se p oblem:
(Pa ial) da a: Ω,T,ϕand γ⊂∂Ω
(Addi ional) in o ma ion: α
Goal: Find Dsuch ha he solu ion o (1)sa is ies (2)
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
N-dimensional wa e equa ion
Uniqueness
ui
−∆ui=0 in Ω Di×(0,T),i=0,1
ui=ϕin ∂Ω×(0,T)
ui=0 in ∂Di×(0,T)
ui(x,0) = 0,ui
(x,0) = 0 in Ω Di
Theo em
T>T∗(Ω, γ),D0,D1a e con ex, ϕ6=0
∂u0
∂n=∂u1
∂non γ×(0,T)
=⇒D0=D1
Fundamen al esul s: H¨
o mande , Lions
A en ion: Weake han he geome ic condi ion (Only uniqueness, no
obse abili y!)
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Elas ici y sys ems
Iso opic case
u − ∇ · (µ(x)(∇u+∇u ) + λ(x)(∇ · u)Id.) = 0 in Ω D×(0,T)
u=ϕon ∂Ω×(0,T)
u=0 on ∂D×(0,T)
u(0) = u0,u (0) = u1in Ω D
Obse a ion: σ(u)·n:= µ(x)(∇u+∇u ) + λ(x)(∇ · u)Id.·non γ×(0,T)
Explana ions:
u= (u1,u2,u3)is he displacemen ec o
Small displacemen s. Hence, linea elas ici y
Iso opy assump ions. The issue is desc ibed by λand µ
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
N-dimensional elas ici y sys em
Uniqueness
ui
− ∇ · (µ(x)(∇ui+∇)(ui) ) + λ(x)(∇ · ui)Id.) = 0 in Ω Di×(0,T)
ui=ϕon ∂Ω×(0,T)
ui=0 on ∂Di×(0,T)
ui(0) = 0,ui
(0) = 0 in Ω Di
Theo em (Cons an coe icien s)
T>T∗(Ω, γ),D0,D1a e con ex, ϕ6=0
σ(u0)·n=σ(u1)·n on γ×(0,T))=⇒D0=D1
Fo uniqueness, he key poin is Unique con inua ion p ope y
(Imanu ilo –Yamamo o, 2008, complex condi ions on µ, λ)
AD, E. Fe n´
andez-Ca a, wo k in p og ess
(∃O he unique con inua ion esul s o s a iona y p oblems:
Lin–Wang, 2005; Escau iaza, 2005; Alessand ini–Mo asi, 2001;
Nakamu a–Wang, 2006; Imanu ilo –Yamamo o, 2012)
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 2-D wa e equa ion I
Case o a ball
Tes 1:T=5, u0=10x,u1=0, ϕ=10x
x0des = -3,y0des = 0, des = 0.4
x0ini = 0,y0ini = 0, ini = 0.6
NLop (AUGLAG + CRS2), NoI e = 1007, F eeFem++:
x0cal = -2.998645439,y0cal = 0.000425214708
cal = 0.4001667063
Figu e: Ini ial mesh: iangles 992, e ices 526 Figu e: The desi ed cen e and adius o he ball
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 2-D wa e equa ion II
Case o a ball
Figu e: Compu ed cen e and adius: AUGALG + CRS2
x0cal= -2.998645439,y0cal = 0.000425214708, cal=0.4001667063
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 2-D wa e equa ion III
Case o a ball
0 50 100 150 200 250 300 350 400 450 500
0
500
1000
1500
2000
2500
3000
3500
I e a ions
Cu en Func ion Values
Func ion alue
Figu e: E olu ion o Jdu ing he i s 500 i e a ions o CRS2
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 2-D wa e equa ion IV
Case o a ball
0.2 0.3 0.4 0.5 0.6 0.7 0.8
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
J(x0, )
Figu e: The unc ional Jwi h espec o he a iable
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 2-D wa e equa ion V
Case o a ball
−5
−4
−3
−2
−1
−2
−1
0
1
2
0
20
40
60
80
100
120
Figu e: The unc ional Jwi h espec o he a iables x0and y0
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 2-D wa e equa ion VI
Case o a ball
Tes 2:T=5, u0=10x,u1=0, ϕ=10x
x0des = -3,y0des = 0, des = 0.4
x0ini = 0,y0ini = 0, ini = 0.6
NLop (AUGLAG + DIRECT), NoI e = 1001, F eeFem++:
x0cal = -2.962962963
y0cal = -0.01219326322
cal = 0.4220164609
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 2-D wa e equa ion VII
Case o a ball
20 ANNA DOUBOVA AND ENRIQUE FERN ´
ANDEZ-CARA
Figu e 14. Some expe iences wi h AUGLAG and DIRECTNoScal as
he subsidia y op imiza ion algo i hm.
Rema k 3. As in he one-dimensional case, i is possible o in oduce ano he
me hod o he pa ial iden i ica ion on B(x0; ) elying on di↵e en ia ion wi h
espec o he domain. Thus, o each su icien ly small m2R2⇥R, le us assume
ha ↵=@u
@n|⇥(0,T )and ↵m=@um
@n|⇥(0,T )a e known, wi h
m=(d, s),D+m=B(x0+d; +s).
He e, u( eps. um) is he solu ion o (35) co esponding o B=B(x0; )( esp.
B=B(x0+d; +s)). Using domain a ia ion echniques, we ob ain he ollowing
o mula:
↵m↵=L(d, s)+1
2Q((d, s),(d, s)) + o(|d|2+|s)|2),
whe e
L(d, s)= @z
@n⇥(0,T ),Q((d, s),(d, s)) = @w
@n⇥(0,T ),
Figu e: Some expe iences wi h AUGLAG and DIRECT
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 2-D wa e equa ion VIII
Case o a ball
Figu e: Desi ed and compu ed adius and cen e s o he ball
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 2-D wa e equa ion I
Case o an ellipse
Op imiza ion p oblem: case o an ellipse
Gi en: eα=eα(x, ).
Find x0,y0and θand a,bsuch ha (x0,y0, θ, a,b)∈Xeand
J(x0,y0, θ, a,b)≤J(x0
0,y0
0, θ0,a0,b0)∀(x0
0,y0
0, θ0,a0,b0)∈Xe,(2)
he unc ion J:Xe7→ Ris de ined by
J(x0,y0, θ, a,b) := 1
2ZZγ×(0,T)
|α[x0,y0, θ, a,b]−eα|2ds d
wi h
α[x0,y0, θ, a,b] = ∂u
∂non γ×(0,T)
Xe:= {(x0,y0, θ, a,b)∈R5:E(x0,y0, θ, a,b)⊂Ω}
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 2-D wa e equa ion II
Case o an ellipse
Resolu ion o an op imiza ion p oblem. Now, J=J(x0,y0, θ, a,b)
Tes 3:T=5, u0=10x,u1=0, ϕ=10x
x0des=-3,y0des=-3,sin( he ades)=0,ades=0.8,bdes=0.4
x0ini=-1,y0ini=-1,sin( he aini)=0,aini=0.5,bini=0.5
NLop (AUGLAG + DIRECTNoScal), NoI e = 2001, F eeFem++
x0cal = -2.963301665
y0cal = -3.035106437
sin( he acal) = 0.112178021
acal = 0.8446502058
bcal = 0.4166666667
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 2-D Lam´
e sys em III
Case o an ellipse
Figu e: Compu ed solu ion a he inal ime
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 3-D wa e equa ion I
Case o a sphe e
Tes 6:T=5, u0=10x,u1=0, ϕ=10x
x0des = -2,y0des = -2,z0des = -2, des = 1
x0ini = 0,y0ini = 0,z0des = 0, ini = 0.6
NLop (AUGLAG + DIRECTNoScal), NoI e = 438, F eeFem++
x0cal = -1.975308642
y0cal = -2.232383275
z0cal = -2.305542854
cal = 1.05
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 3-D wa e equa ion II
Case o a sphe e
Figu e: Ini ial mesh. Poin s: 829, e ahed a: 4023, aces: 8406, edges: 5210,
bounda y aces: 720, bounda y edges: 1080
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 3-D wa e equa ion III
Case o a sphe e
Figu e: Desi ed con igu a ion
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 3-D wa e equa ion IV
Case o a sphe e
Figu e: Compu ed obse a ion and mesh
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 3-D wa e equa ion V
Case o a sphe e
Figu e: Solu ion o he wa e equa ion co esponding o compu ed da a
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 3-D Lam´
e sys em I
Case o a sphe e
Tes 7:T=5,
u01 =10x,u02 =10y,u03 =10z
u11 =0, u12 =0, u13 =0
ϕ1=10x,ϕ2=10y,ϕ3=10z
x0des = -2,y0des = -2,z0des = -2, des = 1
x0ini = 0,y0ini = 0,z0des = 0, ini = 0.6
NLop (AUGLAG + DIRECTNoScal), NoI e = 444 , F eeFem++:
x0cal = -1.981405274
y0cal = -2.225232904
z0cal = -2.148084171
cal = 0.9504115226
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 3-D Lam´
e sys em II
Case o a sphe e
Figu e: Ini ial mesh. Poin s: 829, e ahed a: 4023, aces: 8406, edges: 5210,
bounda y aces: 720, bounda y edges: 1080
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 3-D Lam´
e sys em III
Case o a sphe e
Figu e: Desi ed con igu a ion
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm
Nume ical esul s: 3-D Lam´
e sys em IV
Case o a sphe e
A. Doubo a Nume ical solu ion o some geome ic in e se p oblemsm