A mixed formulation of the quasi-reversibility method J er emi - - PowerPoint PPT Presentation

A mixed formulation of the quasi-reversibility method

J´ er´ emi Dard´ e, Antti Hannukainen, Nuutti Hyv¨

• nen

Aalto University - Helsinki

Tuesday 3rd April 2012

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 1 / 27

Outline

1

Introduction Elliptic Cauchy problem Quasi-reversibility method - Standard formulation

2

Mixed formulation of quasi-reversibility First formulation Second formulation

3

Numerical results Data completion problem Inverse obstacle problem

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 2 / 27

Outline

1

Introduction Elliptic Cauchy problem Quasi-reversibility method - Standard formulation

2

Mixed formulation of quasi-reversibility First formulation Second formulation

3

Numerical results Data completion problem Inverse obstacle problem

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 3 / 27

Elliptic Cauchy problem

• Ω bounded open set of Rd, d ≥ 2, Γ ∪ Γc = ∂Ω, |Γ| = 0, |Γc| = 0.

Cauchy problem: for (f , gD, gN) ∈ L2(Ω) × H1/2(Γ) × H−1/2(Γ), find u ∈ H1(Ω) s.t.    ∆u = f in Ω u = gD

• n Γ

∇u.ν = gN

• n Γ

[Hadamard] Severely ill-posed problem: Cauchy problem has at most one solution u ∈ H1(Ω, ∆) :=

• v ∈ H1(Ω), ∆v ∈ L2(Ω)
• , which does not depend

continuously on the data (f , gD, gN) regularization method.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 4 / 27

Elliptic Cauchy problem

• Ω bounded open set of Rd, d ≥ 2, Γ ∪ Γc = ∂Ω, |Γ| = 0, |Γc| = 0.

Cauchy problem: for (f , gD, gN) ∈ L2(Ω) × H1/2(Γ) × H−1/2(Γ), find u ∈ H1(Ω) s.t.    ∆u = f in Ω u = gD

• n Γ

∇u.ν = gN

• n Γ

[Hadamard] Severely ill-posed problem: Cauchy problem has at most one solution u ∈ H1(Ω, ∆) :=

• v ∈ H1(Ω), ∆v ∈ L2(Ω)
• , which does not depend

continuously on the data (f , gD, gN) regularization method.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 4 / 27

Elliptic Cauchy problem

• Ω bounded open set of Rd, d ≥ 2, Γ ∪ Γc = ∂Ω, |Γ| = 0, |Γc| = 0.

Cauchy problem: for (f , gD, gN) ∈ L2(Ω) × H1/2(Γ) × H−1/2(Γ), find u ∈ H1(Ω) s.t.    ∇.A∇u = f in Ω u = gD

• n Γ

A∇u.ν = gN

• n Γ

[Hadamard] Severely ill-posed problem: Cauchy problem has at most one solution u ∈ H1(Ω, ∆) :=

• v ∈ H1(Ω), ∆v ∈ L2(Ω)
• , which does not depend

continuously on the data (f , gD, gN) regularization method.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 4 / 27

Elliptic Cauchy problem

Cauchy problem arises in many fields:

• plasma physics
• corrosion non-destructive evaluation
• electrocardiography
• ...
• inverse obstacle problems

Several methods of regularization:

• Optimization methods (e.g. minimization of Kohn-Vogelius cost function)
• Conformal mappings 2d case.
• Integral equations
• ...

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 5 / 27

Quasi-reversibility method - Standard formulation

• Introduced by Robert Latt`

es and Jacques-Louis Lions in The method of quasi-reversibility: applications to partial differential equations - 1969.

• Method based on the resolution of the

QR-problem: for ε > 0, find uε ∈ H2(Ω) s.t. uε = gD and ∇uε.ν = gN on Γ, and ∀v ∈ H2(Ω), v = ∇v.ν = 0 on Γ, (∆uε, ∆v)L2(Ω) + ε(uε, v)H2(Ω) = (f , ∆v)L2(Ω).

Theorem

The QR-problem admits a unique solution uε. Furthermore, if the Cauchy problem admits a (unique) solution u ∈ H2(Ω), uε

ε→0

− − − →

H2

u, ∆uε − ∆uL2(Ω) ≤ √εuH2(Ω), ε → uε − uH2(Ω) is an increasing function.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 6 / 27

Quasi-reversibility method - Standard formulation

• Introduced by Robert Latt`

es and Jacques-Louis Lions in The method of quasi-reversibility: applications to partial differential equations - 1969.

• Method based on the resolution of the

QR-problem: for ε > 0, find uε ∈ H2(Ω) s.t. uε = gD and ∇uε.ν = gN on Γ, and ∀v ∈ H2(Ω), v = ∇v.ν = 0 on Γ, (∆uε, ∆v)L2(Ω) + ε(uε, v)H2(Ω) = (f , ∆v)L2(Ω).

Theorem

The QR-problem admits a unique solution uε. Furthermore, if the Cauchy problem admits a (unique) solution u ∈ H2(Ω), uε

ε→0

− − − →

H2

u, ∆uε − ∆uL2(Ω) ≤ √εuH2(Ω), ε → uε − uH2(Ω) is an increasing function.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 6 / 27

Quasi-reversibility method - Standard formulation

• Introduced by Robert Latt`

es and Jacques-Louis Lions in The method of quasi-reversibility: applications to partial differential equations - 1969.

• Method based on the resolution of the

QR-problem: for ε > 0, find uε ∈ H2(Ω) s.t. uε = gD and ∇uε.ν = gN on Γ, and ∀v ∈ H2(Ω), v = ∇v.ν = 0 on Γ, (∆uε, ∆v)L2(Ω) + ε(uε, v)H2(Ω) = (f , ∆v)L2(Ω).

Theorem

The QR-problem admits a unique solution uε. Furthermore, if the Cauchy problem admits a (unique) solution u ∈ H2(Ω), uε

ε→0

− − − →

H2

u, ∆uε − ∆uL2(Ω) ≤ √εuH2(Ω), ε → uε − uH2(Ω) is an increasing function.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 6 / 27

Quasi-reversibility method - Standard formulation

(∆uε, ∆v)L2(Ω) + ε(uε, v)H2(Ω) = (f , ∆v)L2(Ω) Advantages:

• variational form F.E.M.
• in case of noisy data: method1 to set the parameter of regularization ε functions
• f the amplitude of noise α, so that uε(α)

α→0

− − − → u. Drawbacks:

• convergence result if u ∈ H2(Ω) (and not u ∈ H1(Ω, ∆)).
• QR-problem is a fourth order problem.
• 1L. Bourgeois & J. Dard´

e, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 7 / 27

Discretization of the variational problem - Finite element methods

(∆uε, ∆v)L2(Ω) + ε(uε, v)H2(Ω) = (f , ∆v)L2(Ω) Different options:

• conforming finite element methods smooth (C 1) finite elements
• important number of degrees of freedom
• almost never available in numerical solvers
• non-conforming finite element methods Morley or Fraeijs de Veubeke
• Morley element exists for any dimension (not the case of F.V. element)
• seldom available in numerical solvers, especially for 3d problems
• convergence in mesh-dependent norm
• mixed formulation of the problem
• idea: to introduce a new unknown to deal with of high order derivatives
• new problem can be discretized using standard F.E.M.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 8 / 27

Discretization of the variational problem - Finite element methods

(∆uε, ∆v)L2(Ω) + ε(uε, v)H2(Ω) = (f , ∆v)L2(Ω) Different options:

• conforming finite element methods smooth (C 1) finite elements
• important number of degrees of freedom
• almost never available in numerical solvers
• non-conforming finite element methods Morley or Fraeijs de Veubeke
• Morley element exists for any dimension (not the case of F.V. element)
• seldom available in numerical solvers, especially for 3d problems
• convergence in mesh-dependent norm
• mixed formulation of the problem
• idea: to introduce a new unknown to deal with of high order derivatives
• new problem can be discretized using standard F.E.M.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 8 / 27

Discretization of the variational problem - Finite element methods

(∆uε, ∆v)L2(Ω) + ε(uε, v)H2(Ω) = (f , ∆v)L2(Ω) Different options:

• conforming finite element methods smooth (C 1) finite elements
• important number of degrees of freedom
• almost never available in numerical solvers
• non-conforming finite element methods Morley or Fraeijs de Veubeke
• Morley element exists for any dimension (not the case of F.V. element)
• seldom available in numerical solvers, especially for 3d problems
• convergence in mesh-dependent norm
• mixed formulation of the problem
• idea: to introduce a new unknown to deal with of high order derivatives
• new problem can be discretized using standard F.E.M.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 8 / 27

Discretization of the variational problem - Finite element methods

(∆uε, ∆v)L2(Ω) + ε(uε, v)H2(Ω) = (f , ∆v)L2(Ω) Different options:

• conforming finite element methods smooth (C 1) finite elements
• important number of degrees of freedom
• almost never available in numerical solvers
• non-conforming finite element methods Morley or Fraeijs de Veubeke
• Morley element exists for any dimension (not the case of F.V. element)
• seldom available in numerical solvers, especially for 3d problems
• convergence in mesh-dependent norm
• mixed formulation of the problem
• idea: to introduce a new unknown to deal with of high order derivatives
• new problem can be discretized using standard F.E.M.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 8 / 27

1

Introduction Elliptic Cauchy problem Quasi-reversibility method - Standard formulation

2

Mixed formulation of quasi-reversibility First formulation Second formulation

3

Numerical results Data completion problem Inverse obstacle problem

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 9 / 27

A first mixed formulation

• Introduced in A mixed formulation of quasi-reversibility to solve the Cauchy

problem for Laplace’s equation, L.Bourgeois, 2005. QR-problem: for ε > 0 ans δ > 0, find u ∈ H1(Ω), u|Γ = g0, and λ ∈ H1(Ω), λ|Γc = 0 s.t. ∀v ∈ H1(Ω), v|Γ = 0, ∀µ ∈ H1(Ω), µ|Γc = 0,    ε(u, v)H1(Ω) + (∇λ, ∇v)L2(Ω) = 0, (∇u, ∇µ)L2(Ω) − (λ, µ)L2(Ω) − δ(λ, µ)H1(Ω) = (f , µ)L2(Ω) + g1, µΓ.

Theorem

This problem admits a unique solution (uε,δ, λε,δ). Furthermore, if limε→0 ε δ(ε) = 0, (uε, λε)

ε→0

− − − → (u, 0). Drawbacks:

• no procedure to set parameters of regularization in presence of noisy data
• additional unknown λ is actually known (λ ≈ ∆u − f = 0).

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 10 / 27

A first mixed formulation

• Introduced in A mixed formulation of quasi-reversibility to solve the Cauchy

problem for Laplace’s equation, L.Bourgeois, 2005. QR-problem: for ε > 0 ans δ > 0, find u ∈ H1(Ω), u|Γ = g0, and λ ∈ H1(Ω), λ|Γc = 0 s.t. ∀v ∈ H1(Ω), v|Γ = 0, ∀µ ∈ H1(Ω), µ|Γc = 0,    ε(u, v)H1(Ω) + (∇λ, ∇v)L2(Ω) = 0, (∇u, ∇µ)L2(Ω) − (λ, µ)L2(Ω) − δ(λ, µ)H1(Ω) = (f , µ)L2(Ω) + g1, µΓ.

Theorem

This problem admits a unique solution (uε,δ, λε,δ). Furthermore, if limε→0 ε δ(ε) = 0, (uε, λε)

ε→0

− − − → (u, 0). Drawbacks:

• no procedure to set parameters of regularization in presence of noisy data
• additional unknown λ is actually known (λ ≈ ∆u − f = 0).

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 10 / 27

A new mixed formulation: construction

• Cauchy problem: for (f , gD, gN) ∈ L2(Ω) × H1/2(Γ) × L2(Γ), find u s.t.

   ∆u = f in Ω u = gD

• n Γ

∇u.ν = gN

• n Γ
• p := ∇u, verifies ∇.p = ∇.∇u = f ∈ L2(Ω) in Ω and p.ν = ∇u.ν = gN on Γ.
• Cauchy problem: find (u, p) ∈ H1(Ω) × Hdiv(Ω) :=
• q ∈ L2(Ω)d, ∇.q ∈ L2(Ω)
• s.t.

       ∇.p = f in Ω ∇u = p in Ω u = gD

• n Γ

p.ν = gN

• n Γ

This problem admits at most one solution (u, p).

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 11 / 27

A new mixed formulation: construction

• Cauchy problem: for (f , gD, gN) ∈ L2(Ω) × H1/2(Γ) × L2(Γ), find u s.t.

   ∆u = f in Ω u = gD

• n Γ

∇u.ν = gN

• n Γ
• p := ∇u, verifies ∇.p = ∇.∇u = f ∈ L2(Ω) in Ω and p.ν = ∇u.ν = gN on Γ.
• Cauchy problem: find (u, p) ∈ H1(Ω) × Hdiv(Ω) :=
• q ∈ L2(Ω)d, ∇.q ∈ L2(Ω)
• s.t.

       ∇.p = f in Ω ∇u = p in Ω u = gD

• n Γ

p.ν = gN

• n Γ

This problem admits at most one solution (u, p).

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 11 / 27

A new mixed formulation: construction

• Cauchy problem: for (f , gD, gN) ∈ L2(Ω) × H1/2(Γ) × L2(Γ), find u s.t.

   ∆u = f in Ω u = gD

• n Γ

∇u.ν = gN

• n Γ
• p := ∇u, verifies ∇.p = ∇.∇u = f ∈ L2(Ω) in Ω and p.ν = ∇u.ν = gN on Γ.
• Cauchy problem: find (u, p) ∈ H1(Ω) × Hdiv(Ω) :=
• q ∈ L2(Ω)d, ∇.q ∈ L2(Ω)
• s.t.

       ∇.p = f in Ω ∇u = p in Ω u = gD

• n Γ

p.ν = gN

• n Γ

This problem admits at most one solution (u, p).

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 11 / 27

A new mixed formulation: construction

V :=

• v ∈ H1(Ω), v|Γ = gD
• ,

V0 :=

• v ∈ H1(Ω), v|Γ = 0
• ,

D :=

• q ∈ Hdiv(Ω), q.ν|Γ = gN
• ,

D0 :=

• q ∈ Hdiv(Ω), q.ν|Γ = 0
• .

V = ∅ and D = ∅.

• Mixed QR problem: for ε > 0, find uε ∈ V and pε ∈ D s.t.

       (∇uε − pε, ∇v)L2(Ω)d + ε(uε, v)H1(Ω) = 0, ∀v ∈ V0 − (∇uε − pε, q)L2(Ω)d + (∇.pε, ∇.q)L2(Ω) + ε(pε, q)Hdiv(Ω) = (f , ∇.q)L2(Ω), ∀q ∈ D0.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 12 / 27

A new mixed formulation: construction

V :=

• v ∈ H1(Ω), v|Γ = gD
• ,

V0 :=

• v ∈ H1(Ω), v|Γ = 0
• ,

D :=

• q ∈ Hdiv(Ω), q.ν|Γ = gN
• ,

D0 :=

• q ∈ Hdiv(Ω), q.ν|Γ = 0
• .

V = ∅ and D = ∅.

• Mixed QR problem: for ε > 0, find uε ∈ V and pε ∈ D s.t.

       (∇uε − pε, ∇v)L2(Ω)d + ε(uε, v)H1(Ω) = 0, ∀v ∈ V0 − (∇uε − pε, q)L2(Ω)d + (∇.pε, ∇.q)L2(Ω) + ε(pε, q)Hdiv(Ω) = (f , ∇.q)L2(Ω), ∀q ∈ D0.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 12 / 27

A new mixed formulation: construction

V :=

• v ∈ H1(Ω), v|Γ = gD
• ,

V0 :=

• v ∈ H1(Ω), v|Γ = 0
• ,

D :=

• q ∈ Hdiv(Ω), q.ν|Γ = gN
• ,

D0 :=

• q ∈ Hdiv(Ω), q.ν|Γ = 0
• .

V = ∅ and D = ∅.

• Mixed QR problem: for ε > 0, find uε ∈ V and pε ∈ D s.t.

       (∇uε − pε, ∇v)L2(Ω)d + ε(uε, v)H1(Ω) = 0, ∀v ∈ V0 − (∇uε − pε, q)L2(Ω)d + (∇.pε, ∇.q)L2(Ω) + ε(pε, q)Hdiv(Ω) = (f , ∇.q)L2(Ω), ∀q ∈ D0.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 12 / 27

A new mixed formulation: construction

V :=

• v ∈ H1(Ω), v|Γ = gD
• ,

V0 :=

• v ∈ H1(Ω), v|Γ = 0
• ,

D :=

• q ∈ Hdiv(Ω), q.ν|Γ = gN
• ,

D0 :=

• q ∈ Hdiv(Ω), q.ν|Γ = 0
• .

V = ∅ and D = ∅.

• Mixed QR problem: for ε > 0, find uε ∈ V and pε ∈ D s.t.

       (∇uε − pε, ∇v)L2(Ω)d + ε(uε, v)H1(Ω) = 0, ∀v ∈ V0 − (∇uε − pε, q)L2(Ω)d + (∇.pε, ∇.q)L2(Ω) + ε(pε, q)Hdiv(Ω) = (f , ∇.q)L2(Ω), ∀q ∈ D0.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 12 / 27

A new mixed formulation: construction

V :=

• v ∈ H1(Ω), v|Γ = gD
• ,

V0 :=

• v ∈ H1(Ω), v|Γ = 0
• ,

D :=

• q ∈ Hdiv(Ω), q.ν|Γ = gN
• ,

D0 :=

• q ∈ Hdiv(Ω), q.ν|Γ = 0
• .

V = ∅ and D = ∅.

• Mixed QR problem: for ε > 0, find uε ∈ V and pε ∈ D s.t.

       (∇uε − pε, ∇v)L2(Ω)d + ε(uε, v)H1(Ω) = 0, ∀v ∈ V0 − (∇uε − pε, q)L2(Ω)d + (∇.pε, ∇.q)L2(Ω) + ε(pε, q)Hdiv(Ω) = (f , ∇.q)L2(Ω), ∀q ∈ D0.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 12 / 27

Mixed formulation

• Mixed QR problem: for ε > 0, find uε ∈ V and pε ∈ D s.t.

   (∇uε − pε, ∇v)L2(Ω)d + ε(uε, v)H1(Ω) = 0, ∀v ∈ V0 −(∇uε − pε, q)L2(Ω)d + (∇.pε, ∇.q)L2(Ω) + ε(pε, q)Hdiv(Ω) = (f , ∇.q)L2(Ω), ∀q ∈ D0.

Theorem

For all ε > 0, the mixed QR problem admits a unique solution (uε, pε). Furthermore, if the Cauchy problem admits a (unique) solution u ∈ H1(Ω, ∆), we have uε

H1

− − − →

ε→0 u,

pε

Hdiv

− − − →

ε→0 ∇u,

∇.pε − f L2(Ω) ≤ √εu, ∇uH1(Ω)×Hdiv(Ω), ∇uε − pεL2(Ω)d ≤ √εu, ∇uH1(Ω)×Hdiv(Ω).

• Remark. If ∂Ω is “sufficiently smooth”, we actually have an equivalence: if the

Cauchy problem does not have a solution, then uε, pεH1(Ω)×Hdiv(Ω)

ε→0

− − − → ∞.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 13 / 27

Mixed formulation

• Mixed QR problem: for ε > 0, find uε ∈ V and pε ∈ D s.t.

   (∇uε − pε, ∇v)L2(Ω)d + ε(uε, v)H1(Ω) = 0, ∀v ∈ V0 −(∇uε − pε, q)L2(Ω)d + (∇.pε, ∇.q)L2(Ω) + ε(pε, q)Hdiv(Ω) = (f , ∇.q)L2(Ω), ∀q ∈ D0.

Theorem

For all ε > 0, the mixed QR problem admits a unique solution (uε, pε). Furthermore, if the Cauchy problem admits a (unique) solution u ∈ H1(Ω, ∆), we have uε

H1

− − − →

ε→0 u,

pε

Hdiv

− − − →

ε→0 ∇u,

∇.pε − f L2(Ω) ≤ √εu, ∇uH1(Ω)×Hdiv(Ω), ∇uε − pεL2(Ω)d ≤ √εu, ∇uH1(Ω)×Hdiv(Ω).

• Remark. If ∂Ω is “sufficiently smooth”, we actually have an equivalence: if the

Cauchy problem does not have a solution, then uε, pεH1(Ω)×Hdiv(Ω)

ε→0

− − − → ∞.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 13 / 27

Sketch of proof

• Existence and uniqueness of (uε, pε) Lax-Milgram theorem.
• uε − u ∈ V0, pε − ∇u ∈ D0
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv
• uε − u, pε − ∇uH1×Hdiv ≤ u, ∇uH1×Hdiv
• ∇uε − pεL2(Ω)d, ∇.pε − f L2(Ω) ≤ √εu, ∇uH1×Hdiv.
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv ⇒ εn

n→∞

− − − → 0 s.t. (uεn, pεn) ⇀

H1×Hdiv

(w, r)        uεn|Γ = gD pεn.ν|Γ = gN ∇uεn − pεnL2(Ω)d ≤ C√εn ∇.pεn − f L2(Ω) ≤ C√εn = ⇒        w|Γ = gD r.ν|Γ = gN ∇w − r = 0 ∇.r − f = 0 = ⇒ (w, r) = (u, ∇u).

• J´

er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 14 / 27

Sketch of proof

• Existence and uniqueness of (uε, pε) Lax-Milgram theorem.
• uε − u ∈ V0, pε − ∇u ∈ D0
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv
• uε − u, pε − ∇uH1×Hdiv ≤ u, ∇uH1×Hdiv
• ∇uε − pεL2(Ω)d, ∇.pε − f L2(Ω) ≤ √εu, ∇uH1×Hdiv.
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv ⇒ εn

n→∞

− − − → 0 s.t. (uεn, pεn) ⇀

H1×Hdiv

(w, r)        uεn|Γ = gD pεn.ν|Γ = gN ∇uεn − pεnL2(Ω)d ≤ C√εn ∇.pεn − f L2(Ω) ≤ C√εn = ⇒        w|Γ = gD r.ν|Γ = gN ∇w − r = 0 ∇.r − f = 0 = ⇒ (w, r) = (u, ∇u).

• J´

er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 14 / 27

Sketch of proof

• Existence and uniqueness of (uε, pε) Lax-Milgram theorem.
• uε − u ∈ V0, pε − ∇u ∈ D0
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv
• uε − u, pε − ∇uH1×Hdiv ≤ u, ∇uH1×Hdiv
• ∇uε − pεL2(Ω)d, ∇.pε − f L2(Ω) ≤ √εu, ∇uH1×Hdiv.
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv ⇒ εn

n→∞

− − − → 0 s.t. (uεn, pεn) ⇀

H1×Hdiv

(w, r)        uεn|Γ = gD pεn.ν|Γ = gN ∇uεn − pεnL2(Ω)d ≤ C√εn ∇.pεn − f L2(Ω) ≤ C√εn = ⇒        w|Γ = gD r.ν|Γ = gN ∇w − r = 0 ∇.r − f = 0 = ⇒ (w, r) = (u, ∇u).

• J´

er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 14 / 27

Sketch of proof

• Existence and uniqueness of (uε, pε) Lax-Milgram theorem.
• uε − u ∈ V0, pε − ∇u ∈ D0
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv
• uε − u, pε − ∇uH1×Hdiv ≤ u, ∇uH1×Hdiv
• ∇uε − pεL2(Ω)d, ∇.pε − f L2(Ω) ≤ √εu, ∇uH1×Hdiv.
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv ⇒ εn

n→∞

− − − → 0 s.t. (uεn, pεn) ⇀

H1×Hdiv

(w, r)        uεn|Γ = gD pεn.ν|Γ = gN ∇uεn − pεnL2(Ω)d ≤ C√εn ∇.pεn − f L2(Ω) ≤ C√εn = ⇒        w|Γ = gD r.ν|Γ = gN ∇w − r = 0 ∇.r − f = 0 = ⇒ (w, r) = (u, ∇u).

• J´

er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 14 / 27

Sketch of proof

• Existence and uniqueness of (uε, pε) Lax-Milgram theorem.
• uε − u ∈ V0, pε − ∇u ∈ D0
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv
• uε − u, pε − ∇uH1×Hdiv ≤ u, ∇uH1×Hdiv
• ∇uε − pεL2(Ω)d, ∇.pε − f L2(Ω) ≤ √εu, ∇uH1×Hdiv.
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv ⇒ εn

n→∞

− − − → 0 s.t. (uεn, pεn) ⇀

H1×Hdiv

(w, r)        uεn|Γ = gD pεn.ν|Γ = gN ∇uεn − pεnL2(Ω)d ≤ C√εn ∇.pεn − f L2(Ω) ≤ C√εn = ⇒        w|Γ = gD r.ν|Γ = gN ∇w − r = 0 ∇.r − f = 0 = ⇒ (w, r) = (u, ∇u).

• J´

er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 14 / 27

Sketch of proof

• Existence and uniqueness of (uε, pε) Lax-Milgram theorem.
• uε − u ∈ V0, pε − ∇u ∈ D0
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv
• uε − u, pε − ∇uH1×Hdiv ≤ u, ∇uH1×Hdiv
• ∇uε − pεL2(Ω)d, ∇.pε − f L2(Ω) ≤ √εu, ∇uH1×Hdiv.
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv ⇒ εn

n→∞

− − − → 0 s.t. (uεn, pεn) ⇀

H1×Hdiv

(w, r)        uεn|Γ = gD pεn.ν|Γ = gN ∇uεn − pεnL2(Ω)d ≤ C√εn ∇.pεn − f L2(Ω) ≤ C√εn = ⇒        w|Γ = gD r.ν|Γ = gN ∇w − r = 0 ∇.r − f = 0 = ⇒ (w, r) = (u, ∇u).

• J´

er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 14 / 27

Sketch of proof

• Existence and uniqueness of (uε, pε) Lax-Milgram theorem.
• uε − u ∈ V0, pε − ∇u ∈ D0
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv
• uε − u, pε − ∇uH1×Hdiv ≤ u, ∇uH1×Hdiv
• ∇uε − pεL2(Ω)d, ∇.pε − f L2(Ω) ≤ √εu, ∇uH1×Hdiv.
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv ⇒ εn

n→∞

− − − → 0 s.t. (uεn, pεn) ⇀

H1×Hdiv

(w, r)        uεn|Γ = gD pεn.ν|Γ = gN ∇uεn − pεnL2(Ω)d ≤ C√εn ∇.pεn − f L2(Ω) ≤ C√εn = ⇒        w|Γ = gD r.ν|Γ = gN ∇w − r = 0 ∇.r − f = 0 = ⇒ (w, r) = (u, ∇u).

• J´

er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 14 / 27

Sketch of proof

• Existence and uniqueness of (uε, pε) Lax-Milgram theorem.
• uε − u ∈ V0, pε − ∇u ∈ D0
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv
• uε − u, pε − ∇uH1×Hdiv ≤ u, ∇uH1×Hdiv
• ∇uε − pεL2(Ω)d, ∇.pε − f L2(Ω) ≤ √εu, ∇uH1×Hdiv.
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv ⇒ εn

n→∞

− − − → 0 s.t. (uεn, pεn) ⇀

H1×Hdiv

(w, r)        uεn|Γ = gD pεn.ν|Γ = gN ∇uεn − pεnL2(Ω)d ≤ C√εn ∇.pεn − f L2(Ω) ≤ C√εn = ⇒        w|Γ = gD r.ν|Γ = gN ∇w − r = 0 ∇.r − f = 0 = ⇒ (w, r) = (u, ∇u).

• J´

er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 14 / 27

Sketch of proof

• Existence and uniqueness of (uε, pε) Lax-Milgram theorem.
• uε − u ∈ V0, pε − ∇u ∈ D0
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv
• uε − u, pε − ∇uH1×Hdiv ≤ u, ∇uH1×Hdiv
• ∇uε − pεL2(Ω)d, ∇.pε − f L2(Ω) ≤ √εu, ∇uH1×Hdiv.
• uε, pεH1×Hdiv ≤ u, ∇uH1×Hdiv ⇒ εn

n→∞

− − − → 0 s.t. (uεn, pεn) ⇀

H1×Hdiv

(w, r)        uεn|Γ = gD pεn.ν|Γ = gN ∇uεn − pεnL2(Ω)d ≤ C√εn ∇.pεn − f L2(Ω) ≤ C√εn = ⇒        w|Γ = gD r.ν|Γ = gN ∇w − r = 0 ∇.r − f = 0 = ⇒ (w, r) = (u, ∇u).

• J´

er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 14 / 27

Noise and Discretization

• Method to set parameter of regularization in presence of noisy data for the

standard formulation works for the mixed formulation

• Mixed QR problem: for ε > 0, find (uε, pε) ∈ V × D s.t.

       (∇uε − pε, ∇v)L2(Ω)d + ε(uε, v)H1(Ω) = 0, ∀v ∈ V0 −(∇uε − pε, q)L2(Ω)d + (∇.pε, ∇.q)L2(Ω) + ε(pε, q)Hdiv(Ω) = (f , ∇.q)L2(Ω), ∀q ∈ D0.

• uε ∈ H1(Ω) Pm Lagrange F.E.
• pε ∈ Hdiv(Ω) RTm Raviart-Thomas F.E. or BDMm Brezzi-Douglas-Marini

F.E.

• Convergence result:

uε,h − uεH1(Ω), pε,h − pεHdiv(Ω) ≤ hk √1 + ε √ε C(u).

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 15 / 27

Noise and Discretization

• Method to set parameter of regularization in presence of noisy data for the

standard formulation works for the mixed formulation

• Mixed QR problem: for ε > 0, find (uε, pε) ∈ V × D s.t.

       (∇uε − pε, ∇v)L2(Ω)d + ε(uε, v)H1(Ω) = 0, ∀v ∈ V0 −(∇uε − pε, q)L2(Ω)d + (∇.pε, ∇.q)L2(Ω) + ε(pε, q)Hdiv(Ω) = (f , ∇.q)L2(Ω), ∀q ∈ D0.

• uε ∈ H1(Ω) Pm Lagrange F.E.
• pε ∈ Hdiv(Ω) RTm Raviart-Thomas F.E. or BDMm Brezzi-Douglas-Marini

F.E.

• Convergence result:

uε,h − uεH1(Ω), pε,h − pεHdiv(Ω) ≤ hk √1 + ε √ε C(u).

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 15 / 27

1

Introduction Elliptic Cauchy problem Quasi-reversibility method - Standard formulation

2

Mixed formulation of quasi-reversibility First formulation Second formulation

3

Numerical results Data completion problem Inverse obstacle problem

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 16 / 27

Scalar 2d case

• Ω := D((0, 0), 1) \ D((0, 0), 0.4), Γ = ∂D((0, 0), 1).
• Mesh characteristics: 50 pts on Γ, 25

pts on Γc, hmax ∼ 0.18.

• Problem parameters: ε = 10−4, P2 Lagrange and BDM1 F.E. spaces.
• Exact solution: u = y 3

3 − x2 y, u = sin(π x) cosh(π y)

• Computation done using Freefem++.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 17 / 27

Exact solution: u = y 3 3 − x2 y

IsoValue

• 0.367694
• 0.315166
• 0.280148
• 0.245129
• 0.210111
• 0.175092
• 0.140074
• 0.105055
• 0.070037
• 0.0350185

5.54637e-17 0.0350185 0.070037 0.105055 0.140074 0.175092 0.210111 0.245129 0.280148 0.367694 IsoValue

• 0.000293702
• 0.000255091
• 0.000229351
• 0.00020361
• 0.000177869
• 0.000152129
• 0.000126388
• 0.000100648
• 7.49069e-05
• 4.91663e-05
• 2.34256e-05

2.315e-06 2.80556e-05 5.37963e-05 7.95369e-05 0.000105278 0.000131018 0.000156759 0.000182499 0.000246851

u and uε − u in Ω.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 18 / 27

Exact solution: u = y 3 3 − x2 y

Vec Value 0.0529944 0.105989 0.158983 0.211978 0.264972 0.317966 0.370961 0.423955 0.47695 0.529944 0.582938 0.635933 0.688927 0.741922 0.794916 0.847911 0.900905 0.953899 1.00689 Vec Value 0.000363842 0.000727684 0.00109153 0.00145537 0.00181921 0.00218305 0.00254689 0.00291073 0.00327458 0.00363842 0.00400226 0.0043661 0.00472994 0.00509379 0.00545763 0.00582147 0.00618531 0.00654915 0.00691299

∇u and pε − ∇u in Ω.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 18 / 27

Exact solution: u = y 3 3 − x2 y

• 0.025
• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02 0.025

• 3
• 2
• 1

1 2 3 u uε u and uε on Γc.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 18 / 27

Exact solution: u = sin(π x) cosh(π y)

IsoValue

• 9.36283
• 8.02528
• 7.13359
• 6.24189
• 5.35019
• 4.45849
• 3.56679
• 2.67509
• 1.7834
• 0.891698
• 2.66399e-15

0.891698 1.7834 2.67509 3.56679 4.45849 5.35019 6.24189 7.13359 9.36283 IsoValue

• 0.0106538
• 0.00906162
• 0.00800014
• 0.00693866
• 0.00587717
• 0.00481569
• 0.00375421
• 0.00269273
• 0.00163125
• 0.000569765

0.000491717 0.0015532 0.00261468 0.00367616 0.00473764 0.00579913 0.00686061 0.00792209 0.00898357 0.0116373

u and uε − u in Ω.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 19 / 27

Exact solution: u = sin(π x) cosh(π y)

Vec Value 1.94244 3.88489 5.82733 7.76977 9.71221 11.6547 13.5971 15.5395 17.482 19.4244 21.3669 23.3093 25.2518 27.1942 29.1366 31.0791 33.0215 34.964 36.9064 Vec Value 0.0123351 0.0246702 0.0370054 0.0493405 0.0616756 0.0740107 0.0863458 0.098681 0.111016 0.123351 0.135686 0.148021 0.160357 0.172692 0.185027 0.197362 0.209697 0.222032 0.234367

∇u and pε − ∇u in Ω.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 19 / 27

Exact solution: u = sin(π x) cosh(π y)

• 1.5
• 1
• 0.5

0.5 1 1.5

• 3
• 2
• 1

1 2 3 u uε u and uε on Γc.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 19 / 27

Scalar 3d case

• Ω := D((0, 0), 1) × ]0, 2[ \ D((0, 0), 0.4) × ]0.7, 1.4[.
• Γ := ∂D((0, 0), 1) × ]0, 2[.
• Problem solved using P1 × RT0 F.E.
• Mesh created with GMSH, computed with Freefem++, visualization using

Medit and Matlab.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 20 / 27

Exact solution: u = x2 + y 2 − 2 z2

uε and uε − u in Ω.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 21 / 27

Exact solution: u = x2 + y 2 − 2 z2

−2 −1 1 2 3 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 −3.5 −3 −2.5 −2 −1.5 −1 −2 −1 1 2 3 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075

uε and uε − u on ∂D((0, 0), 0.4) × ]0.7, 1.4[.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 21 / 27

Exact solution: u = cosh(z)(sin(x/ √ 2) + cos(y/ √ 2))

uε and uε − u in Ω.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 22 / 27

Exact solution: u = cosh(z)(sin(x/ √ 2) + cos(y/ √ 2))

−2 −1 1 2 3 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1 1.5 2 2.5 −2 −1 1 2 3 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 −0.3 −0.25 −0.2 −0.15 −0.1

uε and uε − u on ∂D((0, 0), 0.4) × ]0.7, 1.4[.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 22 / 27

Inverse obstacle problem

• D bounded, connected open set in Rd, Γ ⊂ ∂D.
• Problem: for (gD, gN) = (0, 0), find O ⊂ D s.t. Ω(O) := D \ O is connected,

and u ∈ H1(Ω(O)) ∩ C 0(Ω(O)) s.t.        ∇.A∇u = in Ω(O) u = gD

• n Γ

A∇u.ν = gN

• n Γ

u =

• n ∂O.
• This problem has at most one solution (O, u).
• Example - control of plasma shape in Tokamak: in the vacuum chamber V of a

tokamak, in presence of a plasma P, the poloidal component of the magnetic field ψ verifies the following equations (in cylindrical coordinates):        ∇ 1

r ∇ψ

= in V \ P ψ = gD

• n ∂V

1 r ∇ψ.ν

= gN

• n ∂V

ψ = cte

• n ∂P.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 23 / 27

Inverse obstacle problem

• D bounded, connected open set in Rd, Γ ⊂ ∂D.
• Problem: for (gD, gN) = (0, 0), find O ⊂ D s.t. Ω(O) := D \ O is connected,

and u ∈ H1(Ω(O)) ∩ C 0(Ω(O)) s.t.        ∇.A∇u = in Ω(O) u = gD

• n Γ

A∇u.ν = gN

• n Γ

u =

• n ∂O.
• This problem has at most one solution (O, u).
• Example - control of plasma shape in Tokamak: in the vacuum chamber V of a

tokamak, in presence of a plasma P, the poloidal component of the magnetic field ψ verifies the following equations (in cylindrical coordinates):        ∇ 1

r ∇ψ

= in V \ P ψ = gD

• n ∂V

1 r ∇ψ.ν

= gN

• n ∂V

ψ = cte

• n ∂P.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 23 / 27

Exterior approach2

Exterior approach algorithm:

• Construct a sequence of open sets (ωm)m∈N such that

O ⊂ ωm+1 ⊂ ωm

• Every step: solve a Cauchy problem outside ωm to reconstruct u → mixed QR

formulation

• 2L. Bourgeois & J. Dard´

e, A quasi-reversibility approach to solve the inverse obstacle problem.

• J. Dard´

e, The ”exterior approach”: a new framework to solve inverse obstacle problems.

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 24 / 27

Control of plasma

• Control of plasma shape in Tokamak: in the vacuum chamber V of a tokamak,

in presence of a plasma P, the poloidal component of the magnetic field ψ verifies the following equations (in cylindrical coordinates):        ∇ 1

r ∇ψ

= in V \ P ψ = gD

• n ∂V

1 r ∇ψ.ν

= gN

• n ∂V

ψ = cte

• n ∂P.

1 2 3 4 −3 −2 −1 1 2 3 1.5 2 2.5 3 −1 −0.5 0.5 1 1.5

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 25 / 27

Control of plasma

• Control of plasma shape in Tokamak: in the vacuum chamber V of a tokamak,

in presence of a plasma P, the poloidal component of the magnetic field ψ verifies the following equations (in cylindrical coordinates):        ∇ 1

r ∇ψ

= in V \ P ψ = gD

• n ∂V

1 r ∇ψ.ν

= gN

• n ∂V

ψ = cte

• n ∂P.

1 2 3 4 −3 −2 −1 1 2 3 1.5 2 2.5 −0.5 0.5 1

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 25 / 27

3d inverse obstacle problem with Laplace equation

• Inverse obstacle problem: knowing (gD, gN) on Γ ⊂ ∂D, find (O, u) s.t.

       ∆u = in D \ O u = gD

• nΓ

∇u.ν = gN

• m Γ

u =

• n ∂O

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 26 / 27

Conclusion

We propose a new mixed formulation of the quasi-reversibility method:

• convergence obtained for u ∈ H1(Ω, ∆)
• discretization by standard finite elements
• method to set parameter of regularization functions of amplitude of noise
• scalar Cauchy problem (2d,3d)
• inverse obstacle problem, scalar case (2d, 3d)
• vectorial Cauchy problem (2d)

Future works:

• method to set parameter of regularization functions of amplitude of noise →

to be tested

• vectorial Cauchy problem (3d)
• inverse obstacle problem, vectorial case (2d, 3d).

Thank you!

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 27 / 27

Conclusion

We propose a new mixed formulation of the quasi-reversibility method:

• convergence obtained for u ∈ H1(Ω, ∆)
• discretization by standard finite elements
• method to set parameter of regularization functions of amplitude of noise
• scalar Cauchy problem (2d,3d)
• inverse obstacle problem, scalar case (2d, 3d)
• vectorial Cauchy problem (2d)

Future works:

• method to set parameter of regularization functions of amplitude of noise →

to be tested

• vectorial Cauchy problem (3d)
• inverse obstacle problem, vectorial case (2d, 3d).

Thank you!

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 27 / 27

Conclusion

We propose a new mixed formulation of the quasi-reversibility method:

• convergence obtained for u ∈ H1(Ω, ∆)
• discretization by standard finite elements
• method to set parameter of regularization functions of amplitude of noise
• scalar Cauchy problem (2d,3d)
• inverse obstacle problem, scalar case (2d, 3d)
• vectorial Cauchy problem (2d)

Future works:

• method to set parameter of regularization functions of amplitude of noise →

to be tested

• vectorial Cauchy problem (3d)
• inverse obstacle problem, vectorial case (2d, 3d).

Thank you!

J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 27 / 27