We solve an overspecified as well as underspecified inverse Cauchy problems of the modified Helmholtz equation in an arbitrary 3D-bounded domain with a smooth boundary using a multiple/scale/direction Trefftz method (MSDTM), of which the directions are uniformly distributed on ${\mathbb{S}}^1$, and the scales are determined a priori by the collocation points on boundary to satisfy the specified boundary conditions. The three-dimensional numerical examples confirm the efficiency of the MSDTM. Although under a large noise up to 30%, the solutions of inverse Cauchy problems are quite accurate. More importantly, the present method converges very fast and saves much CPU time.

The inverse problem endowing with multiple unknown functions gradually becomes an important topic in the field of numerical heat transfer, and one fundamental problem is how to use limited minimal data to solve the inverse problem. With this in mind, in the present article we search the solution of a general inverse heat conduction problem when two boundary data on the space-time boundary are missing and recover two unknown temperature functions with the help of a few extra measurements of temperature data polluted by random noise. This twofold ill-posed inverse heat conduction problem is more difficult than the backward heat conduction problem and the sideways heat conduction problem, both with one unknown function to be recovered. Based on a stable adjoint Trefftz method, we develop a global boundary integral equation method, which together with the compatibility conditions and some measured data can be used to retrieve two unknown temperature functions. Several numerical examples demonstrate that the present method is effective and stable, even for those of strongly ill-posed ones under quite large noises.