An Inverse Solution for Determining Arbitrary Boundary-Conditions using a Least-Squares Approach

Despite the versatility of numerical approaches to the inverse problem (Beck et al. (1) and Xue et al. (2)), there is still a strong need for analytical solutions. In fact, many numerical simulations require a starting point and must be verified and bounded to help ensure the validity of the solution. In addition to bounding the problem, there is always a need for closed-form solutions or first-order approximations that can be quickly used to highlight the significance of various parameters and their often complicated interrelationships. Even with this enduring importance, significant limitations remain including a reliance on higher-order derivatives that magnify data errors, restrictions to small time frames, or the inability to handle arbitrary boundary-conditions. Fortunately, many of these limitations can be avoided and the inverse-solution found for a variety of geometries by using a generalized direct-solution combined with a least-squares approach.