| Abstract: |
| The solution of coefficient inverse problems in partial differential equations (PDEs) remains a significant computational challenge, particularly in the context of linear elasticity where the vector-valued state variable and the high dimensionality of the parameter space result in substantial computational costs. This talk presents an exploratory framework for accelerating the estimation of the shear modulus ($\mu$) in nearly incompressible media by leveraging Reduced Order Modeling (ROM) techniques. While ROM is traditionally applied to accelerate forward simulations, its integration into the optimization loop for inverse problems requires careful alignment between the reduced basis and the trajectory of the optimizer. We discuss an initial approach using Proper Orthogonal Decomposition (POD) to construct a reduced basis for the forward and adjoint operators within a gradient-based optimization routine. Preliminary results comparing the efficiency and reconstruction accuracy of this reduced-order approach for scalar elliptic problems are presented. An ongoing extension of this framework to the nearly incompressible elasticity equations will be discussed. This work represents an initial attempt to move toward a more goal-oriented ROM strategy where the reduced basis is informed by the requirements of the inversion process. |
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