Inverse Boundary Value Problem for the Helmholtz Equation with Multi-frequency Data

Abstract. We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-toNeumann map at selected frequencies as the data. We develop an explicit iterative reconstruction of the wavespeed using a multi-level nonlinear projected steepest descent iterative scheme in Banach spaces. We consider wavespeeds containing conormal singularities. A conditional Lipschitz estimate for the inverse problem holds for wavespeeds of the form of a linear combination of piecewise constant functions with an underlying domain partitioning, and gives a framework in which the scheme converges. The stability constant grows exponentially as the number of subdomains in the domain partitioning increases. To mitigate this growth of the stability constant, we introduce a hierarchy of compressive approximations of the solution to the inverse problem with piecewise constant functions. We establish an upper bound of the stability constant, which constrains the compression rate of the solution. Then, tracking the frequency dependencies through the approximation errors, we arrive at a procedure to select the frequencies such that convergence from level to level of our scheme is guaranteed.