POSITIVE FILTERED P_N METHOD FOR LINEAR TRANSPORT EQUATIONS AND THE ASSOCIATED OPTIMIZATION ALGORITHM

We propose a positive, accurate moment closure for linear kinetic transport equations based on a filtered spherical harmonic (FP_N) expansion in the angular variable. The FP_N moment equations are accurate approximations to linear kinetic equations, but they are known to suffer from the occurrence of unphysical, negative particle concentrations. The new positive filtered P_N (FP_N+) closure is developed to address this issue. The FP_N+ closure approximates the kinetic distribution by a spherical harmonic expansion that is non-negative on a finite, predetermined set of quadrature points. With an appropriate numerical PDE solver, the FP_N+ closure generates particle concentrations that are guaranteed to be non-negative. Under an additional, mild regularity assumption, we prove that as the moment order tends to infinity, the FP_N+ approximation converges, in the L2 sense, at the same rate as the FP_N approximation; numerical tests suggest that this assumption may not be necessary. By numerical experiments on the challenging line source benchmark problem, we confirm that the FP_N+ method indeed produces accurate and non-negative solutions. To apply the FP_N+ closure on problems at large temporal-spatial scales, we develop a positive asymptotic preserving (AP) numerical PDE solver. We prove that the propose AP scheme maintains stability and accuracy with standard mesh sizes at large temporal-spatial scales, while, for generic numerical schemes, excessive refinements on temporal-spatial meshes are required. We also show that the proposed scheme preserves positivity of the particle concentration, under some time step restriction. Numerical results confirm that the proposed AP scheme is capable for solving linear transport equations at large temporal-spatial scales, for which a generic scheme could fail. Constrained optimization problems are involved in the formulation of the FP_N+ closure to enforce non-negativity of the FP_N+ approximation on the set of quadrature points. These optimization problems can be written as strictly convex quadratic programs (CQPs) with a large number of inequality constraints. To efficiently solve the CQPs, we propose a constraint-reduced variant of a Mehrotra-predictor-corrector algorithm, with a novel constraint selection rule. We prove that, under appropriate assumptions, the proposed optimization algorithm converges globally to the solution at a locally q-quadratic rate. We test the algorithm on randomly generated problems, and the numerical results indicate that the combination of the proposed algorithm and the constraint selection rule outperforms other compared constraint-reduced algorithms, especially for problems with many more inequality constraints than variables.

Integration of virus-like particle macromolecular bioreceptors in electrochemical biosensors

Rapid, sensitive and selective detection of chemical hazards and biological pathogens has shown growing importance in the fields of homeland security, public safety and personal health. In the past two decades, efforts have been focusing on performing point-of-care chemical and biological detections using miniaturized biosensors. These sensors convert target molecule binding events into measurable electrical signals for quantifying target molecule concentration. However, the low receptor density and the use of complex surface chemistry in receptors immobilization on transducers are common bottlenecks in the current biosensor development, adding to the cost, complexity and time. This dissertation presents the development of selective macromolecular Tobacco mosaic virus-like particle (TMV VLP) biosensing receptor, and the microsystem integration of VLPs in microfabricated electrochemical biosensors for rapid and performance-enhanced chemical and biological sensing. Two constructs of VLPs carrying different receptor peptides targeting at 2,4,6-trinitrotoluene (TNT) explosive or anti-FLAG antibody are successfully bioengineered. The VLP-based TNT electrochemical sensor utilizes unique diffusion modulation method enabled by biological binding between target TNT and receptor VLP. The method avoids the influence from any interfering species and environmental background signals, making it extremely suitable for directly quantifying the TNT level in a sample. It is also a rapid method that does not need any sensor surface functionalization process. For antibody sensing, the VLPs carrying both antibody binding peptides and cysteine residues are assembled onto the gold electrodes of an impedance microsensor. With two-phase immunoassays, the VLP-based impedance sensor is able to quantify antibody concentrations down to 9.1 ng/mL. A capillary microfluidics and impedance sensor integrated microsystem is developed to further accelerate the process of VLP assembly on sensors and improve the sensitivity. Open channel capillary micropumps and stop-valves facilitate localized and evaporation-assisted VLP assembly on sensor electrodes within 6 minutes. The VLP-functionalized impedance sensor is capable of label-free sensing of antibodies with the detection limit of 8.8 ng/mL within 5 minutes after sensor functionalization, demonstrating great potential of VLP-based sensors for rapid and on-demand chemical and biological sensing.