Multiple Level Sets for Piecewise Constant Surface Reconstruction in Highly Ill-Posed Problems

This paper considers highly ill-posed surface recovery inverse problems, where the sought surface in 2D or 3D is piecewise constant with several possible level values. These levels may further be potentially unknown. Multiple level set functions are used when there are more than two such levels, and we extend the methods and theory of our previous works to handle such more complex situations. A rather efficient method is developed. Inverse potential problems in two and three space dimensions are solved numerically, demonstrating the method's capabilities for several both known and unknown level values.