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In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equations in the presence of small inclusions. As a byproduct, we derive a topological derivative based algorithm for the reconstruction of piecewise smooth functions. This algorithm can be used for edge detection in imaging, topological optimization, and for inverse problems, such as Quantitative Photoacoustic Tomography, for which we demonstrate the effectiveness of our asymptotic expansion method numerically.

Asymptotic Expansions for Higher Order Elliptic Equations with an Application to Quantitative Photoacoustic Tomography