Optimization of parameters in coherent spin dynamics of radical pairs in quantum biology.

Identification of the external electromagnetic fields and internal hyperfine parameters which optimize the quantum singlet-triplet yield of simplified radical pairs modeled by Schrödinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms are analyzed. A method that combines sensitivity analysis with Tikhonov regularization is implemented. Numerical results demonstrate that the quantum singlet-triplet yield of the radical pair system can be significantly reduced if optimization is pursued simultaneously for both external magnetic fields and internal hyperfine parameters. The results may contribute towards understanding the structure-function relationship of a putative magnetoreceptor to manipulate and enhance quantum coherences at room temperature and leveraging biofidelic function to inspire novel quantum devices.

Cancer detection through Electrical Impedance Tomography and optimal control theory: theoretical and computational analysis.

The Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the $ m $ electrodes for a given electrode current is analyzed. A PDE constrained optimal control framework in Besov space is developed, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm difference of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The novelty of the control-theoretic model is its adaptation to the clinical situation when additional "voltage-to-current" measurements can increase the size of the input data from $ m $ up to $ m! $ while keeping the size of the unknown parameters fixed. The existence of the optimal control and Fréchet differentiability in the Besov space along with optimality condition is proved. Numerical analysis of the simulated model example in the 2D case demonstrates that by increasing the number of input boundary electrode currents from $ m $ to $ m^2 $ through additional "voltage-to-current" measurements the resolution of the electrical conductivity of the body identified via gradient method in Besov space framework is significantly improved.

Identification of parameters in systems biology.

We consider the inverse problem for the identification of the finite dimensional set of parameters for systems of nonlinear ordinary differential equations (ODEs) arising in systems biology. A numerical method which combines Bellman's quasilinearization with sensitivity analysis and Tikhonov's regularization is implemented. We apply the method to various biological models such as the classical Lotka-Volterra system, bistable switch model in genetic regulatory networks, gene regulation and repressilator models from synthetic biology. The numerical results and application to real data demonstrate the quadratic convergence.