Reconstruction is the process of determining the parameters of a model that best match a set of experimental measurements. A standard approach to reconstruction employs gradient-guided optimization to find the best parameters. Although many complex physical situations involving numerous parameters would seem to be almost impossible to reconstruct because they can only be modeled through lengthy large-scale simulations, we will discuss recently-developed techniques that facilitate such reconstructions. These techniques can also help solve design optimization problems. Some of the basic techniques will be illustrated through the example of the simulation of the time-dependent diffusion of infrared light through biological tissue, for which we have demonstrated reconstruction of an unknown field of diffusion coefficients. Examples of tomographic reconstruction from radiographs will also be used for illustration. Potential application areas include modeling of the ocean, atmosphere, fluid flow, shock-wave phenomena, industrial processes, and engineering systems.
Key to facilitating optimization of complex calculations with respect to numerous variables is the efficient calculation of the gradient of an objective function, phi, provided by the Adjoint Differentiation In Code Technique (ADICT). ADICT involves applying the chain rule of differentiation to the computational simulation, with the two unique features that the computer code for the adjoint calculation proceeds in the reverse (adjoint) direction and explicitly "differentiates" the forward calculation. Aside from aiding the optimization process, the gradients of phi can provide an understanding of the sensitivity of the parameters to any set of measurements and can guide the analyst in correcting the simulation models to better match the data.
Other enabling techniques to be discussed include: quasi-Newton algorithms for efficient optimization and multiresolution analysis to constrain and accelerate the optimization process. Also useful for modeling of objects (or scenes) to be reconstructed are multiscale object models and deformable geometric models.
Keywords: reconstruction, optimization, adjoint differentiation, simulation, nonlinear analysis
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