The degree of ill-posedness of the linear operator equation Ax = b can be measured by the decay rate of the singular values of A. If the inverse problem is nonlinear and A is the corresponding Jacobian matrix, the singular values measure the local ill-posedness of the problem. We consider the case where the operator A is compact and maps infinite dimensional Hilbert spaces. The degree of ill-posedness can be measured by μ, where the singular values σ_n(A) ≈ n^(−μ). The problem becomes more difficult to solve numerically with increasing μ.
When the singular values quickly decay to zero, it is common to truncate them or introduce a regularization term. However, if the problem is severely ill-posed, a large amount of regularization may be required to solve the problem, and this can introduce significant bias error in the solution estimate. Regularization can be viewed as adding information to the ill-posed problem, and hence we consider the regularized inverse problem as simultaneous joint inversion. Simultaneous joint inversion has recently become a common method to incorporate multiple types of data and physics in a single inversion.
We extend discrete techniques of stacking matrices in joint inversion, to combining Green’s function solutions of multiple differential equations representing different types of data. The singular values of the joint operators indicate the effectiveness of combining multiple types of physics. This knowledge provides mathematical justification for joint inversion, and can be determined before the complicated machinery of discretizing and solving the problem is implemented. We will give an example of two differential equations with known Green’s function solutions. The decay rate of the singular values of the individual operators are compared to the singular values of the joint operator, and the extent to which the ill-posedness was resolved is quantified.
This is joint work with James Ford.