Inmagneticresonanceelastography,oneseekstoreconstructtheshearmodulusfrommeasurementsofthedisplacementfieldinthewholebody.Inthispaper,wepresentanoptimizationapproachwhichsolvestheproblembysimplyminimizingadiscrepancyfunctional.Inordertorecoveracomplexanomalyinahomogenousmedium,wefirstobservethattheinformationcontainedinthewavefieldshouldbedecomposedintotwoparts,a'near-field'partintheregionaroundtheanomalyanda'far-field'partintheregionawayfromtheanomaly.Aswillbejustifiedboththeoreticallyandnumerically,separatingthesescalesprovidesalocalandprecisereconstruction.
