TY - JOUR

T1 - NUMERICAL METHODS FOR MULTISCALE INVERSE PROBLEMS

AU - Frederick, Christina

AU - Engquist, Björn

N1 - Funding Information:
This work has benefited from valuable discussions with Assyr Abdulle, Kui Ren, Pingbing Ming, and Fenyang Tang. This research was supported in part by NSF grant DMS-1217203, the Texas Consortium for Computational Seismology, and Institut Mittag-Leffler. CF was also supported in part by NSF grant DMS-1317015.
Funding Information:
Acknowledgements. This work has benefited from valuable discussions with As-syr Abdulle, Kui Ren, Pingbing Ming, and Fenyang Tang. This research was supported in part by NSF grant DMS-1217203, the Texas Consortium for Computational Seismology, and Institut Mittag-Leﬄer. CF was also supported in part by NSF grant DMS-1317015.
Publisher Copyright:
© 2017. International Press.

PY - 2017

Y1 - 2017

N2 - We consider the inverse problem of determining the highly oscillatory coefficient aϱ in partial differential equations of the form −∇ · (aϱ∇uϱ)+buϱ = f from given measurements of the solutions. Here, ϱ indicates the smallest characteristic wavelength in the problem (0 < ϱ « 1). In addition to the general difficulty of finding an inverse is the challenge of multiscale modeling, which is hard even for forward computations. The inverse problem in its full generality is typically ill-posed, and one common approach is to reduce the dimension by seeking effective parameters. We will here include microscale features directly in the inverse problem and avoid ill-posedness by assuming that the microscale can be accurately represented by a low-dimensional parametrization. The basis for our inversion will be a coupling of the parametrization to analytic homogenization or a coupling to efficient multiscale numerical methods when analytic homogenization is not available. We will analyze the reduced problem, b = 0, by proving uniqueness of the inverse in certain problem classes and by numerical examples and also include numerical model examples for medical imaging, b > 0, and exploration seismology, b < 0.

AB - We consider the inverse problem of determining the highly oscillatory coefficient aϱ in partial differential equations of the form −∇ · (aϱ∇uϱ)+buϱ = f from given measurements of the solutions. Here, ϱ indicates the smallest characteristic wavelength in the problem (0 < ϱ « 1). In addition to the general difficulty of finding an inverse is the challenge of multiscale modeling, which is hard even for forward computations. The inverse problem in its full generality is typically ill-posed, and one common approach is to reduce the dimension by seeking effective parameters. We will here include microscale features directly in the inverse problem and avoid ill-posedness by assuming that the microscale can be accurately represented by a low-dimensional parametrization. The basis for our inversion will be a coupling of the parametrization to analytic homogenization or a coupling to efficient multiscale numerical methods when analytic homogenization is not available. We will analyze the reduced problem, b = 0, by proving uniqueness of the inverse in certain problem classes and by numerical examples and also include numerical model examples for medical imaging, b > 0, and exploration seismology, b < 0.

KW - 35B27

KW - 35R25

KW - 65N21

KW - 65N30

KW - Inverse problems

KW - heterogeneous multiscale method

KW - periodic homogenization

KW - stability

UR - http://www.scopus.com/inward/record.url?scp=85067492034&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85067492034&partnerID=8YFLogxK

U2 - 10.4310/CMS.2017.V15.N2.A2

DO - 10.4310/CMS.2017.V15.N2.A2

M3 - Article

AN - SCOPUS:85067492034

SN - 1539-6746

VL - 15

SP - 305

EP - 328

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

IS - 2

ER -