We study the inverse source problem for the Helmholtz equation from boundary Cauchy data with multiple wave numbers. The main goal of this paper is to study the uniqueness and increasing stability when the (pseudo)convexity or non-trapping conditions for the related hyperbolic problem are not satisfied. We consider general elliptic equations of the second order and arbitrary observation sites. To show the uniqueness we use the analytic continuation, the Fourier transform with respect to the wave numbers and uniqueness in the lateral Cauchy problem for hyperbolic equations. Numerical examples in 2 spatial dimension support the analysis and indicate the increasing stability for large intervals of the wave numbers, while analytic proofs of the increasing stability are not available.

Periodic parameters are common and important in stochastic differential equations (SDEs) arising in many contemporary scientific and engineering fields involving dynamical processes. These parameters include the damping coefficient, the volatility or diffusion coefficient and possibly an external force. Identification of these periodic parameters allows a better understanding of the dynamical processes and their hidden intermittent instability. Conventional approaches usually assume that one of the parameters is known and focus on the recovery of rest parameters. By introducing the decorrelation time and calculating the standard Gaussian statistics (mean, variance) explicitly for the scalar Langevin equations with periodic parameters, we propose a parameter identification approach to simultaneously recovering all these parameters by observing a single trajectory of SDEs. Such an approach is summarized in form of regularization schemes with noisy operators and noisy right-hand sides and is further extended to parameter identification of SDEs which are indirectly observed by other random processes. Numerical examples show that our approach performs well in stable and weakly unstable regimes but may fail in strongly unstable regime which is induced by the strong intermittent instability itself.