Monotonicity and local uniqueness for the Helmholtz equation
Harrach, B., Pohjola, V., & Salo, M. (2019). Monotonicity and local uniqueness for the Helmholtz equation. Analysis and PDE, 12(7), 2019. https://doi.org/10.2140/apde.2019.12.1741
Julkaistu sarjassa
Analysis and PDEPäivämäärä
2019Tekijänoikeudet
© 2019 Mathematical Sciences Publishers
This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schrödinger) equation (1 + k2q)u = 0 in a bounded domain for fixed nonresonance frequency k > 0 and real-valued scattering coefficient function q. We show a monotonicity relation between the scattering coefficient q and the local Neumann-to-Dirichlet operator that holds up to finitely many eigenvalues. Combining this with the method of localized potentials, or Runge approximation, adapted to the case where finitely many constraints are present, we derive a constructive monotonicitybased characterization of scatterers from partial boundary data. We also obtain the local uniqueness result that two coefficient functions q1 and q2 can be distinguished by partial boundary data if there is a
neighborhood of the boundary part where q1 ≥ q2 and q1 6≡ q2.
Julkaisija
Mathematical Sciences PublishersISSN Hae Julkaisufoorumista
2157-5045Asiasanat
Julkaisu tutkimustietojärjestelmässä
https://converis.jyu.fi/converis/portal/detail/Publication/32197907
Metadata
Näytä kaikki kuvailutiedotKokoelmat
Rahoittaja(t)
Suomen Akatemia; Euroopan komissioRahoitusohjelmat(t)
Huippuyksikkörahoitus, SA; EU:n 7. puiteohjelma (FP7)
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Lisätietoja rahoituksesta
Pohjola and Salo were supported by the Academy of Finland (Finnish Centre of Excellence in Inverse Problems Research, grant number 284715) and by an ERC Starting Grant (grant number 307023).Lisenssi
Samankaltainen aineisto
Näytetään aineistoja, joilla on samankaltainen nimeke tai asiasanat.
-
Increasing stability in the linearized inverse Schrödinger potential problem with power type nonlinearities
Lu, Shuai; Salo, Mikko; Xu, Boxi (IOP Publishing, 2022)We consider increasing stability in the inverse Schrödinger potential problem with power type nonlinearities at a large wavenumber. Two linearization approaches, with respect to small boundary data and small potential ... -
Optimality of Increasing Stability for an Inverse Boundary Value Problem
Kow, Pu-Zhao; Uhlmann, Gunther; Wang, Jenn-Nan (Society for Industrial & Applied Mathematics (SIAM), 2021)In this work we study the optimality of increasing stability of the inverse boundary value problem (IBVP) for the Schrödinger equation. The rigorous justification of increasing stability for the IBVP for the Schrödinger ... -
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems
Covi, Giovanni; Mönkkönen, Keijo; Railo, Jesse (American Institute of Mathematical Sciences (AIMS), 2021)We prove a unique continuation property for the fractional Laplacian (−Δ)s when s∈(−n/2,∞)∖Z where n≥1. In addition, we study Poincaré-type inequalities for the operator (−Δ)s when s≥0. We apply the results to show that ... -
Refined instability estimates for some inverse problems
Kow, Pu-Zhao; Wang, Jenn-Nan (American Institute of Mathematical Sciences (AIMS), 2022)Many inverse problems are known to be ill-posed. The ill-posedness can be manifested by an instability estimate of exponential type, first derived by Mandache [29]. In this work, based on Mandache's idea, we refine the ... -
Dimension Bounds in Monotonicity Methods for the Helmholtz Equation
Harrach, Bastian; Pohjola, Valter; Salo, Mikko (Society for Industrial and Applied Mathematics, 2019)The article [B. Harrach, V. Pohjola, and M. Salo, Anal. PDE] established a monotonicity inequality for the Helmholtz equation and presented applications to shape detection and local uniqueness in inverse boundary problems. ...
Ellei toisin mainittu, julkisesti saatavilla olevia JYX-metatietoja (poislukien tiivistelmät) saa vapaasti uudelleenkäyttää CC0-lisenssillä.