Diana C. Resasco

Diffusion on a curved surface coupled to diffusion in the volume

An algorithm is presented for solving a diffusion equation on a curved surface coupled to diffusion in the volume, a problem often arising in cell biology. It applies to pixilated surfaces obtained from experimental images and performs at low computational cost. In the method, the Laplace-Beltrami operator is approximated locally by the Laplacian on the tangential plane and then a finite volume discretization scheme based on a Voronoi decomposition is applied. Convergence studies show that mass conservation built in the discretization scheme and cancellation of sampling error ensure convergence of the solution in space with an order between 1 and 2. The method is applied to a cell-biological problem where a signaling molecule, G-protein Rac, cycles between the cytoplasm and cell membrane thus coupling its diffusion in the membrane to that in the cell interior. Simulations on realistic cell geometry are performed to validate, and determine the accuracy of, a recently proposed simplified quantitative analysis of fluorescence loss in photobleaching. The method is implemented within the Virtual Cell computational framework freely accessible at www.vcell.org.

Generalized deflated block-elimination

A stable algorithm is presented to solve a nonsingular bordered system of the form \[ \left( {\begin{array}{*{20}c} A & B \\ {C^T } & D \\ \end{array} } \right)\left( {\begin{array}{*{20}c} x \\ y \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} f \\ g \\ \end{array} } \right), \] where B and C are n by m matrices and the n by n matrix A could be nearly singular with at most

Analysis of nonlinear dynamics on arbitrary geometries with the Virtual Cell

\mu

Virtual Cell: computational tools for modeling in cell biology.

small singular values. The algorithm needs only a solver for A and the solution to an

A domain-decomposed fast poisson solver on a rectangle

m + \mu

On the condition of nearly singular matrices under rank-1 perturbations

by

A domain-decomposed fast Poisson solver on a rectangle

m + \mu

A framework for the analysis and construction of domain decomposition preconditioners

dense linear system. It is, thus, well suited for problems for which A has easily exploitable structures and

A Survey of Preconditioners for Domain Decomposition.

m + \mu \ll n

Analysis of domain decomposition preconditioners on irregular regions

, such as in continuation methods, bifurcation problems and constrained optimization.