Oberseminar Wissenschaftliches Rechnen (Martin Klakow)

Testing unknowns for certain features with statistical hypothesis testing without reconstructing the true solution was limited by two drawbacks in the past: Testing for ⟨φ, u†⟩ = ⟨Φ, Tu†⟩ ?=
0 , required φ to be in the range of the adjoint of the forward operator T∗, and computing Φ from T∗Φ = φ is generally ill-posed, which decreases the power of the test.

To overcome these, Kretschmann et al. have developed optimal regularized hypothesis testing for Gaussian distributed data.[1]
In our work, we adapt the regularized testing approach to Poisson distributed data. We discuss how to interpret this sort of data as a Hilbert space process, how to apply regularized testing, and how to estimate the noise variance, which depends on the measurement time. We also introduce a variant based on Tikhonov regularization, which can be applied, if only a single data set is
available, and discuss its properties. We investigate the performance in 2-dimensional numerical simulations and discuss the application to super-resolution fluorescence microscopy.
References
[1] R. Kretschmann, D. Wachsmuth, F. Werner, Optimal regularized hypothesis testing in
statistical inverse problems, Inverse Problems,40(1)(2024),015013