Open Questions

The nonlinear diffusion arises in various settings. In optics, this is relevant to the nonlinear optics, such as multi-photon microscopy, which includes two-photon excitation microscopy, three-photon excitation microscopy, second-order harmonic imaging, third-order harmonic imaging, etc. The models are derived from Maxwell's equation or the wave equation in a nonlinear turbid medium.

1. Theoretically, the rigorous derivation for the nonlinear model is not trivial. The linear model is derived with Dyson's expansion with careful estimates of each diagram. But the nonlinear model is only known formally.  [ It is now formally proved whether it is a multi-photon absorption model. ]

2. In the diffusion regime, the multi-photon microscopy is a nonlinear diffusion equation. A similar equation is encountered in both qPAT and UMOT (AOT) if one carefully deals with the Liouville transform. In some cases, the nonlinear diffusion equation does not give a unique physical solution, but it is possible to remedy this with multiple measurements. The theory is absent for all the non-linearity except for the linear qPAT.

3. Apart from the nonlinear diffusion, the transport equation also permits a nonlinear term in scattering. This is in general more difficult to deal with; however, in the UMOT, the nonlinear term appears to be in an easier shape due to the enhancement of singularity on the connection segment of two point sources. However, it is not the same case for optics. The uniqueness and stability estimates are mostly open for optics.

4. Numerically, the reconstruction algorithms are not trivial at all. The qPAT with linear diffusion has a straightforward imaging by taking a Gauge transform. For the nonlinear case, the optimization-based algorithms are expensive in general, direct reconstruction mostly depends on the maximum principle. For other cases, it is not clear whether or not there exist direct imaging techniques. 

5. In the AOT with fluorescence excitation [LYZ18], if the measurement is the coupled current from both excitation and emission particles, can we reconstruct the relevant coefficients from two observations? Furthermore, the general theory will be: given a nonlinear diffusion model and certain interior data, is it possible to do the inversion? This is highly relevant to the multi-photon microscopy as well.

6. Black-body radiation model is a nonlinear diffusion coupled with the linear Boltzmann, the fast forward solver has not yet been studied.

7. Besides the diffusion, the super-diffusion/sub-diffusion types of equations are also interesting, and the non-local PDEs are attracting more and more attention in these years. Are the models applicable to practical applications like biomedical imaging? The Levy process is related to the super-diffusion due to the famous Levy flights, which is useful to depict the migration of species or animals, financial flow, etc. However, it is not commonly seen in medical imaging with the non-local models.

8. How to model the nonlinear (super-/sub-) diffusion equation on geometric random graphs correctly? [Short answer: For abnormal diffusion, the system is always dynamic, the temperature is not well-defined, and probably the question is not well-posed. ]

9. Follow-up of Problem 5, the multi-photon process, is there a way to show the uniqueness of optical tomography with multi-photon without a linearization approach? The linearization approach has been studied already. 

10. For the linear inverse heat problem (no matter in time or in space), the results are quite complete, while for nonlinear thermal diffusivity, there are very limited results. [If the solution is in Gevrey space, it is possible to derive some zero stability.]

11. In the two-photon transport model for optical tomography [SZ21],  there is a quite interesting Gronwall-type inequality (see the paper for a proof) involving supremum over angular space, which is used to derive uniqueness for a distributional source. [It should be able to deal with small scattering cases as well, but cannot solve the general case.]

12. Can we extend the Grownwall inequality in [SZ21]  to the large e scattering case? [The Gronwall-type inequality is ready to solve the photoacoustic problem (absorption only) in the transport regime.]