2018/12/10 by Wilkins Aquino, Aquino, Wilkins, Marc Bonnet +1
Earth and Planetary Sciences · Engineering · Mathematics · Medicine · #Analysis of PDEs (math.AP) #Classical Physics (physics.class-ph) #FOS: Mathematics #FOS: Physical sciences #Numerical methods in inverse problems #Photoacoustic and Ultrasonic Imaging #Seismic Imaging and Inversion Techniques #Ultrasonics and Acoustic Wave Propagation #Ultrasound Imaging and Elastography
paper · pdf · doi:10.48550/arxiv.1812.03653
openalex publication_date 2018/12/10 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28
We consider the identification of heterogeneous linear elastic moduli in the\ncontext of time-harmonic elastodynamics. This inverse problem is formulated as\nthe minimization of the modified error in constitutive equation (MECE), an\nenergy-based cost functional defined as an weighted additive combination\n\E+\κ\D of the error in constitutive equation (ECE)\n\E, expressed using an energy seminorm, and a quadratic error term\n\D incorporating the kinematical measurements. MECE-based\nidentification are known from existing computational evidence to enjoy\nattractive properties such as improved convexity, robustness to resonant\nfrequencies, and tolerance to incompletely specified boundary conditions (BCs).\nThe main goal of this work is to develop theoretical foundations, in a\ncontinuous setting, allowing to explain and justify some of the aforementioned\nbeneficial properties, in particular addressing the general case where BCs may\nbe underspecified. A specific feature of MECE formulations is that forward and\nadjoint solutions are governed by a fully coupled system, whose mathematical\nproperties play a fundamental role in the qualitative and computational aspects\nof MECE minimization. We prove that this system has a unique and stable\nsolution at any frequency, provided data is abundant enough (in a sense made\nprecise therein) to at least compensate for any missing information on BCs. As\na result, our formulation leads in such situations to a well-defined solution\neven though the relevant forward problem is not \a priori clearly\ndefined. This result has practical implications such as applicability of MECE\nto partial interior data (with important practical applications including\nultrasound elastography), convergence of finite element discretizations and\ndifferentiability of the reduced MECE functional. In addition, we establish\nthat usual least squares and pure ECE formulations are limiting cases of MECE\nformulations for small and large values of \κ, respectively. For the\nlatter case, which corresponds to exact enforcement of kinematic data, we\nfurthermore show that the reduced MECE Hessian is asymptotically positive for\nany parameter perturbation supported on the measurement region, thereby\ncorroborating existing computational evidence on convexity improvement brought\nby MECE functionals. Finally, numerical studies that support and illustrate our\ntheoretical findings, including a parameter reconstruction example using\ninterior data, are presented.\n