On the topography of the cost functional in linear and nonlinear inverse problems

摘要

We analyze, through linear algebra, the topography of the cost functional in linear and nonlinear inverse problems with the aim of illuminating general characteristics. To a first-order approximation, the local data misfit function in any inverse problem is valley-shaped and elongated in the direc-tions of the null space of the Jacobian and/or in the direc-tions of the smallest singular values. In nonlinear inverse problems, valleys persist; however, local minima might also coexist in the misfit space and might be related to nonlinear effects ignored by the Gauss-Newton approximation to the Hessian, the regularization term designed to provide convex-ity to the misfit function, or to noise in the data. Further-more, noise perturbs the size of the equivalence region making location of solutions easier but finding a global minimum harder (in the case of existence). Understanding the behavior of the cost functional is an important step in the developing techniques to appraise inverse solutions and estimate uncertainties caused by noise, incomplete sam-pling, regularization, and more fundamentally, simplified physical models.