Global stability Giovanni Alessandrini for coupled physics inverse - PowerPoint PPT Presentation

STABILITY FOR COUPLED PHYSICS IPs Global stability Giovanni Alessandrini for coupled physics inverse problems. Introduction A case study An example A priori assumptions Main Theorem Giovanni Alessandrini Stability for |u| Quantitative UCP Concluding remarks Università degli Studi di Trieste End Problémes Inverses et Imagerie 12-13/02/2014 Institut Henri Poincaré

STABILITY FOR Introduction COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example Since the ’80s, a dominant theme in Inverse Problems has A priori been: assumptions Main Theorem To image the interior of an object from measurements taken Stability for |u| in its exterior Quantitative • overdetermined boundary data, UCP Concluding • scattering data. remarks End With coupled physics IPs there is a shift to data associated to interior information.

STABILITY FOR Introduction COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example Since the ’80s, a dominant theme in Inverse Problems has A priori been: assumptions Main Theorem To image the interior of an object from measurements taken Stability for |u| in its exterior Quantitative • overdetermined boundary data, UCP Concluding • scattering data. remarks End With coupled physics IPs there is a shift to data associated to interior information.

STABILITY FOR Introduction COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example Interior data may provide much better stability than inverse A priori boundary problems, or inverse scattering problems. assumptions Available results require nondegeneracy conditions on the Main Theorem solutions of the involved direct problems: Stability for |u| Quantitative • Nonvanishing of solution. UCP • Nonvanishing of gradients. Concluding remarks • Nonvanishing of Jacobians. End • Nonvanishing of augmented Jacobians.

STABILITY FOR Introduction COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example Interior data may provide much better stability than inverse A priori boundary problems, or inverse scattering problems. assumptions Available results require nondegeneracy conditions on the Main Theorem solutions of the involved direct problems: Stability for |u| Quantitative • Nonvanishing of solution. UCP • Nonvanishing of gradients. Concluding remarks • Nonvanishing of Jacobians. End • Nonvanishing of augmented Jacobians.

STABILITY FOR Introduction COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example Interior data may provide much better stability than inverse A priori boundary problems, or inverse scattering problems. assumptions Available results require nondegeneracy conditions on the Main Theorem solutions of the involved direct problems: Stability for |u| Quantitative • Nonvanishing of solution. UCP • Nonvanishing of gradients. Concluding remarks • Nonvanishing of Jacobians. End • Nonvanishing of augmented Jacobians.

STABILITY FOR Introduction COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example Interior data may provide much better stability than inverse A priori boundary problems, or inverse scattering problems. assumptions Available results require nondegeneracy conditions on the Main Theorem solutions of the involved direct problems: Stability for |u| Quantitative • Nonvanishing of solution. UCP • Nonvanishing of gradients. Concluding remarks • Nonvanishing of Jacobians. End • Nonvanishing of augmented Jacobians.

STABILITY FOR Introduction COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example Interior data may provide much better stability than inverse A priori boundary problems, or inverse scattering problems. assumptions Available results require nondegeneracy conditions on the Main Theorem solutions of the involved direct problems: Stability for |u| Quantitative • Nonvanishing of solution. UCP • Nonvanishing of gradients. Concluding remarks • Nonvanishing of Jacobians. End • Nonvanishing of augmented Jacobians.

STABILITY FOR Global stability COUPLED PHYSICS IPs Giovanni Alessandrini Introduction Question . Is it possible to obtain global stability from An example measurements arising from arbitrary (nontrivial) solutions of A priori assumptions the direct problem? Main Theorem Stability for |u| The model problem A problem arising in microwave Quantitative imaging coupled with ultrasound, Triki (2010). UCP Concluding ∆ u + qu = 0 in Ω remarks End Find q ≥ constant > 0 given the local energy qu 2 and the boundary data u | ∂ Ω .

STABILITY FOR Global stability COUPLED PHYSICS IPs Giovanni Alessandrini Introduction Question . Is it possible to obtain global stability from An example measurements arising from arbitrary (nontrivial) solutions of A priori assumptions the direct problem? Main Theorem Stability for |u| The model problem A problem arising in microwave Quantitative imaging coupled with ultrasound, Triki (2010). UCP Concluding ∆ u + qu = 0 in Ω remarks End Find q ≥ constant > 0 given the local energy qu 2 and the boundary data u | ∂ Ω .

STABILITY FOR Global stability COUPLED PHYSICS IPs Giovanni Alessandrini Introduction Question . Is it possible to obtain global stability from An example measurements arising from arbitrary (nontrivial) solutions of A priori assumptions the direct problem? Main Theorem Stability for |u| The model problem A problem arising in microwave Quantitative imaging coupled with ultrasound, Triki (2010). UCP Concluding ∆ u + qu = 0 in Ω remarks End Find q ≥ constant > 0 given the local energy qu 2 and the boundary data u | ∂ Ω .

STABILITY FOR Global stability COUPLED PHYSICS IPs Giovanni Alessandrini Introduction Question . Is it possible to obtain global stability from An example measurements arising from arbitrary (nontrivial) solutions of A priori assumptions the direct problem? Main Theorem Stability for |u| The model problem A problem arising in microwave Quantitative imaging coupled with ultrasound, Triki (2010). UCP Concluding ∆ u + qu = 0 in Ω remarks End Find q ≥ constant > 0 given the local energy qu 2 and the boundary data u | ∂ Ω .

STABILITY FOR The full problem COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example Ammari, Capdeboscq, De Gournay, Rozanova-Pierrat, Triki A priori (2011): assumptions div ( a ∇ u ) + k 2 qu = 0 in Ω Main Theorem Stability for |u| Find a , q ≥ constant > 0 given the local energies qu 2 , Quantitative a |∇ u | 2 (with several u ’s and k ’s!). UCP Concluding remarks Here: End u = electric field, q = electric permittivity, a − 1 = magnetic permeability.

STABILITY FOR The full problem COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example Ammari, Capdeboscq, De Gournay, Rozanova-Pierrat, Triki A priori (2011): assumptions div ( a ∇ u ) + k 2 qu = 0 in Ω Main Theorem Stability for |u| Find a , q ≥ constant > 0 given the local energies qu 2 , Quantitative a |∇ u | 2 (with several u ’s and k ’s!). UCP Concluding remarks Here: End u = electric field, q = electric permittivity, a − 1 = magnetic permeability.

STABILITY FOR Goals COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions • Stability of global type. Main Theorem Stability for |u| • Measurements for a single (nontrivial) solution u Quantitative possibly sign changing. UCP • No spectral assumptions on ∆ + q . Concluding remarks End

STABILITY FOR Goals COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions • Stability of global type. Main Theorem Stability for |u| • Measurements for a single (nontrivial) solution u Quantitative possibly sign changing. UCP • No spectral assumptions on ∆ + q . Concluding remarks End

STABILITY FOR Goals COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions • Stability of global type. Main Theorem Stability for |u| • Measurements for a single (nontrivial) solution u Quantitative possibly sign changing. UCP • No spectral assumptions on ∆ + q . Concluding remarks End

STABILITY FOR Goals COUPLED PHYSICS IPs Giovanni Alessandrini Introduction An example A priori assumptions • Stability of global type. Main Theorem Stability for |u| • Measurements for a single (nontrivial) solution u Quantitative possibly sign changing. UCP • No spectral assumptions on ∆ + q . Concluding remarks End

STABILITY FOR An example COUPLED PHYSICS IPs Giovanni Alessandrini In dimension n = 1, fix 0 < r < R and, for every Introduction k = 1 , 2 , . . . , set An example � A k A priori if | x | < r , assumptions q k ( x ) = 1 if r ≤ | x | ≤ R , Main Theorem Stability for |u| where Quantitative � π � 2 UCP r − 2 . A k = 2 + 2 k π Concluding remarks A solution to u xx + q k u = 0 in ( − R , R ) is End A k cos ( √ A k x ) � √ 1 if | x | < r , u k ( x ) = − sin ( | x | − r ) if r ≤ | x | ≤ R .