Modeling of First-Order Photobleaching Kinetics Using Krylov Subspace Spectral Methods

We solve the first order 2-D reaction–diffusion equations which describe binding-diffusion kinetics using the photobleaching scanning profile of a confocal laser scanning microscope, approximated by a Gaussian laser profile. We show how to solve the first-order photobleaching kinetics partial differential equations (PDEs) using a time-stepping method known as a Krylov subspace spectral (KSS) method. KSS methods are explicit methods for solving time-dependent variable-coefficient partial differential equations. They approximate Fourier coefficients of the solution using Gaussian quadrature rules in the spectral domain. In this paper, we show how a KSS method can be used to obtain not only an approximate numerical solution, but also an approximate analytical solution when using initial conditions that come from pre-bleach steady states and also general initial conditions, to facilitate asymptotic analysis. Analytical and numerical results are presented. It is observed that although KSS methods are explicit, it is possible to use a time step that is far greater than what the CFL condition would indicate

On the Treatment of Bilinear Forms Involving Matrix Functions as Perturbations of Quadratic Forms

Optimizing the approximation of bilinear forms using a one-sided perturbation, where the computation of the approximate solution is generated faster while using less storage or reusing information is possible. Standard methods for approximation like Lanczos and others exist; however, separating the work a little bit and having the bilinear form relate to the quadratic form can yield a more efficient algorithm. When we have an application where the needs are different, it is helpful to understand how the processes like symmetric Lanczos and unsymmetric Lanczos relate to one another so that we can find a way to break things down to accommodate specific cases. The objective of this research is to build an efficient algorithm to approximate bilinear forms with matrix functions utilizing the quadrature rule for approximating quadratic forms without applying the standard methods of approximation

Solution of PDES For First-Order Photobleaching Kinetics Using Krylov Subspace Spectral Methods

We solve the first order reaction-diffusion equations which describe binding-diffusion kinetics using a photobleaching scanning profile of a confocal laser scanning microscope approximated by a Gaussian laser profile. We show how to solve these equations with prebleach steady-state initial conditions using a time-domain method known as a Krylov Subspace Spectral (KSS) method. KSS methods are explicit methods for solving time- dependent variable-coefficient partial differential equations (PDEs). KSS methods are advantageous compared to other methods because of their stability and their superior scalability. These advantages are obtained by applying Gaussian quadrature rules in the spectral domain developed by Golub and Meurant. We present a simple approximate analytical solution to the reaction-diffusion equations, as well as a computational solution that is first-order accurate in time. We then use this solution to examine short- and long-time behaviors