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

Outline Introduction 1 Elliptic Cauchy problem Quasi-reversibility method - Standard formulation Mixed formulation of quasi-reversibility 2 First formulation Second formulation Numerical results 3 Data completion problem Inverse obstacle problem J´ er´ emi Dard´ e (Aalto University) PICOF’12 Tuesday 3rd April 2012 2 / 27

Outline Introduction 1 Elliptic Cauchy problem Quasi-reversibility method - Standard formulation Mixed formulation of quasi-reversibility 2 First formulation Second formulation Numerical results 3 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 R d , d ≥ 2, Γ ∪ Γ c = ∂ Ω, | Γ | � = 0, | Γ c | � = 0. Cauchy problem: for ( f , g D , g N ) ∈ L 2 (Ω) × H 1 / 2 (Γ) × H − 1 / 2 (Γ), find u ∈ H 1 (Ω) s.t.  ∆ u = f in Ω  u = g D on Γ ∇ u .ν = g N on Γ  [Hadamard] Severely ill-posed problem: Cauchy problem has at most one solution u ∈ H 1 (Ω , ∆) := � v ∈ H 1 (Ω) , ∆ v ∈ L 2 (Ω) � , which does not depend continuously on the data ( f , g D , g N ) � 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 R d , d ≥ 2, Γ ∪ Γ c = ∂ Ω, | Γ | � = 0, | Γ c | � = 0. Cauchy problem: for ( f , g D , g N ) ∈ L 2 (Ω) × H 1 / 2 (Γ) × H − 1 / 2 (Γ), find u ∈ H 1 (Ω) s.t.  ∆ u = f in Ω  u = g D on Γ ∇ u .ν = g N on Γ  [Hadamard] Severely ill-posed problem: Cauchy problem has at most one solution u ∈ H 1 (Ω , ∆) := � v ∈ H 1 (Ω) , ∆ v ∈ L 2 (Ω) � , which does not depend continuously on the data ( f , g D , g N ) � 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 R d , d ≥ 2, Γ ∪ Γ c = ∂ Ω, | Γ | � = 0, | Γ c | � = 0. Cauchy problem: for ( f , g D , g N ) ∈ L 2 (Ω) × H 1 / 2 (Γ) × H − 1 / 2 (Γ), find u ∈ H 1 (Ω) s.t.  ∇ . A ∇ u = f in Ω  u = g D on Γ A ∇ u .ν = g N on Γ  [Hadamard] Severely ill-posed problem: Cauchy problem has at most one solution u ∈ H 1 (Ω , ∆) := � v ∈ H 1 (Ω) , ∆ v ∈ L 2 (Ω) � , which does not depend continuously on the data ( f , g D , g N ) � 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 u ε ∈ H 2 (Ω) s.t. u ε = g D and ∇ u ε .ν = g N on Γ, and ∀ v ∈ H 2 (Ω) , v = ∇ v .ν = 0 on Γ , (∆ u ε , ∆ v ) L 2 (Ω) + ε ( u ε , v ) H 2 (Ω) = ( f , ∆ v ) L 2 (Ω) . Theorem The QR-problem admits a unique solution u ε . Furthermore, if the Cauchy problem admits a (unique) solution u ∈ H 2 (Ω) , ε → 0 u ε − − − → u , H 2 � ∆ u ε − ∆ u � L 2 (Ω) ≤ √ ε � u � H 2 (Ω) , ε �→ � u ε − u � H 2 (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 u ε ∈ H 2 (Ω) s.t. u ε = g D and ∇ u ε .ν = g N on Γ, and ∀ v ∈ H 2 (Ω) , v = ∇ v .ν = 0 on Γ , (∆ u ε , ∆ v ) L 2 (Ω) + ε ( u ε , v ) H 2 (Ω) = ( f , ∆ v ) L 2 (Ω) . Theorem The QR-problem admits a unique solution u ε . Furthermore, if the Cauchy problem admits a (unique) solution u ∈ H 2 (Ω) , ε → 0 u ε − − − → u , H 2 � ∆ u ε − ∆ u � L 2 (Ω) ≤ √ ε � u � H 2 (Ω) , ε �→ � u ε − u � H 2 (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 u ε ∈ H 2 (Ω) s.t. u ε = g D and ∇ u ε .ν = g N on Γ, and ∀ v ∈ H 2 (Ω) , v = ∇ v .ν = 0 on Γ , (∆ u ε , ∆ v ) L 2 (Ω) + ε ( u ε , v ) H 2 (Ω) = ( f , ∆ v ) L 2 (Ω) . Theorem The QR-problem admits a unique solution u ε . Furthermore, if the Cauchy problem admits a (unique) solution u ∈ H 2 (Ω) , ε → 0 u ε − − − → u , H 2 � ∆ u ε − ∆ u � L 2 (Ω) ≤ √ ε � u � H 2 (Ω) , ε �→ � u ε − u � H 2 (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 (∆ u ε , ∆ v ) L 2 (Ω) + ε ( u ε , v ) H 2 (Ω) = ( f , ∆ v ) L 2 (Ω) Advantages: • variational form � F.E.M. • in case of noisy data: method 1 to set the parameter of regularization ε functions α → 0 of the amplitude of noise α , so that u ε ( α ) − − − → u . Drawbacks: • convergence result if u ∈ H 2 (Ω) (and not u ∈ H 1 (Ω , ∆)). • QR-problem is a fourth order problem . 1 L. 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 (∆ u ε , ∆ v ) L 2 (Ω) + ε ( u ε , v ) H 2 (Ω) = ( f , ∆ v ) L 2 (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 (∆ u ε , ∆ v ) L 2 (Ω) + ε ( u ε , v ) H 2 (Ω) = ( f , ∆ v ) L 2 (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 (∆ u ε , ∆ v ) L 2 (Ω) + ε ( u ε , v ) H 2 (Ω) = ( f , ∆ v ) L 2 (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