A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Refined instability estimates for some inverse problems (2022)


Kow, P.-Z., & Wang, J.-N. (2022). Refined instability estimates for some inverse problems. Inverse Problems and Imaging, 16(6), 1619-1642. https://doi.org/10.3934/ipi.2022017


JYU-tekijät tai -toimittajat


Julkaisun tiedot

Julkaisun kaikki tekijät tai toimittajatKow, Pu-Zhao; Wang, Jenn-Nan

Lehti tai sarjaInverse Problems and Imaging

ISSN1930-8337

eISSN1930-8345

Julkaisuvuosi2022

Volyymi16

Lehden numero6

Artikkelin sivunumerot1619-1642

KustantajaAmerican Institute of Mathematical Sciences (AIMS)

JulkaisumaaYhdysvallat (USA)

Julkaisun kielienglanti

DOIhttps://doi.org/10.3934/ipi.2022017

Julkaisun avoin saatavuusEi avoin

Julkaisukanavan avoin saatavuus

Julkaisu on rinnakkaistallennettu (JYX)https://jyx.jyu.fi/handle/123456789/83337


Tiivistelmä

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 instability estimates for two inverse problems, including the inverse inclusion problem and the inverse scattering problem. Our aim is to derive explicitly the dependence of the instability estimates on key parameters.

The first result of this work is to show how the instability depends on the depth of the hidden inclusion and the conductivity of the background medium. This work can be regarded as a counterpart of the depth-dependent and conductivity-dependent stability estimate proved by Li, Wang, and Wang [28], or pure dependent stability estimate proved by Nagayasu, Uhlmann, and Wang [31]. We rigorously justify the intuition that the exponential instability becomes worse as the inclusion is hidden deeper inside a conductor or the conductivity is larger.

The second result is to justify the optimality of increasing stability in determining the near-field of a radiating solution of the Helmholtz equation from the far-field pattern. Isakov [16] showed that the stability of this inverse problem increases as the frequency increases in the sense that the stability estimate changes from a logarithmic type to a Hölder type. We prove in this work that the instability changes from an exponential type to a Hölder type as the frequency increases. This result is inspired by our recent work [25].


YSO-asiasanatosittaisdifferentiaaliyhtälötinversio-ongelmatkuvantaminenimpedanssitomografiasironta

Vapaat asiasanatinverse problems; instability; Calderón's problem; electrical impedance tomography; depth-dependent instability of exponential-type; Helmholtz equation; scattering theory; Rellich lemma; increasing stability phenomena; 35J15; 35R25; 35R30


Liittyvät organisaatiot


Hankkeet, joissa julkaisu on tehty


OKM-raportointiKyllä

Raportointivuosi2022

JUFO-taso2


Viimeisin päivitys 2024-26-03 klo 20:56