We will be concerned with the mathematical modeling, numerical simulation, and shape optimization of microfluidic biochips that are used for various biomedical applications. A particular feature is that the fluid flow in the fluidic network on top of the biochips is induced by surface acoustic waves generated by interdigital transducers. We are thus faced with a multiphysics problem that will be modeled by coupling the equations of piezoelectricity with the compressible Navier-Stokes equations. Moreover, the fluid flow exhibits a multiscale character that will be taken care of by a homogenization approach. We will discuss and analyze the mathematical models and deal with their numerical solution by space-time discretizations featuring appropriate finite element approximations with respect to hierarchies of simplicial triangulations of the underlying computational domains. Simulation results will be given for the propagation of the surface acoustic waves on top of the piezoelectric substrate and for the induced fluid flow in the microchannels of the fluidic network. The performance of the operational behavior of the biochips can be significantly improved by shape optimization. In particular, for such purposes we present a multilevel interior point method relying on a predictor-corrector strategy with an adaptive choice of the continuation step length along the barrier path. As a specific example, we will consider the shape optimization of pressure driven capillary barriers between microchannels and reservoirs.

Fixed-point continuation (FPC) is an approach, based on operator-splitting and continuation, for solving minimization problems with $\ell_1$-regularization:

We investigate the application of this algorithm to compressed sensing signal recovery, in which $f(x) = \frac{1}{2}\|Ax-b\|_M^2$, $A \in \mathbb{R}^{m \times n}$ and $m \leq n$.  In particular, we extend the original algorithm to obtain better practical results, derive appropriate choices for $M$ and $\bar{\mu}$ under a given measurement model, and present numerical results for a variety of compressed sensing problems. The numerical results show that the performance of our algorithm compares favorably with that of several recently proposed algorithms.

Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiff ordinary differential equations. Making use of special properties of Chebyshev-like polynomials, these methods have favorable stability properties compared to standard explicit methods while remaining explicit. A new class of such methods, called ROCK, introduced in [Numer. Math., 90, 1-18, 2001] has recently been extended to stiff stochastic differential equations under the name S-ROCK [C. R. Acad. Sci. Paris, 345(10), 2007 and Commun. Math. Sci, 6(4), 2008]. In this paper we discuss the extension of the S-ROCK methods to systems with discrete noise and propose a new class of methods for such problems, the $\tau$-ROCK methods. One motivation for such methods is the simulation of multi-scale or stiff chemical kinetic systems and such systems are the focus of this paper, but our new methods could potentially be interesting for other stiff systems with discrete noise. Two versions of the $\tau$-ROCK methods are discussed and their stability behavior is analyzed on a test problem. Compared to the $\tau$-leaping method, a significant speed-up can be achieved for some stiff kinetic systems. The behavior of the proposed methods are tested on several numerical experiments.

Nonlinear rank-one modification of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical fiber. In this paper, we first study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method (SLAM). The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method.

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions, and numerical results are given to examine their feasibility and effectiveness. In addition, the advantages of the Newton-HSS methods over the Newton-USOR, the Newton-GMRES and the Newton-GCG methods are shown through solving systems of nonlinear equations arising from the finite difference discretization of a two-dimensional convection-diffusion equation perturbed by a nonlinear term. The numerical implementations also show that as preconditioners for the Newton-GMRES and the Newton-GCG methods the HSS iteration outperforms the USOR iteration in both computing time and iteration step.

As an important model in quantum semiconductor devices, the Schrödinger-Poisson equations have generated widespread interests in both analysis and numerical simulations in recent years. In this paper, we present Gaussian beam methods for the numerical simulation of the one-dimensional Schrödinger-Poisson equations. The Gaussian beam methods for high frequency waves outperform the geometrical optics method in that the former are accurate even around caustics. The purposes of the paper are first to develop the Gaussian beam methods, based on our previous methods for the linear Schrödinger equation, for the Schrödinger-Poisson equations, and then check their validity for this weakly-nonlinear system.

In this paper, we are concerned with uniform superconvergence of Galerkin methods for singularly perturbed reaction-diffusion problems by using two Shishkin-type meshes. Based on an estimate of the error between spline interpolation of the exact solution and its numerical approximation, an interpolation post-processing technique is applied to the original numerical solution. This results in approximation exhibit superconvergence which is uniform in the weighted energy norm. Numerical examples are presented to demonstrate the effectiveness of the interpolation post-processing technique and to verify the theoretical results obtained in this paper.