On the Optimal Field Sensing in Near-Field Characterization

We deal with the problem of characterizing a source or scatterer from electromagnetic radiated or scattered field measurements. The problem refers to the amplitude and phase measurements which has applications also to interferometric approaches at optical frequencies. From low frequencies (microwaves) to high frequencies or optics, application examples are near-field/far-field transformations, object restoration from measurements within a pupil, near-field THz imaging, optical coherence tomography and ptychography. When analyzing the transmitting-sensing system, we can define “optimal virtual" sensors by using the Singular Value Decomposition (SVD) approach which has been, since long time, recognized as the “optimal” tool to manage linear algebraic problems. The problem however emerges of discretizing the relevant singular functions, thus defining the field sampling. To this end, we have recently developed an approach based on the Singular Value Optimization (SVO) technique. To make the “virtual” sensors physically realizable, in this paper, two approaches are considered: casting the “virtual” field sensors into arrays reaching the same performance of the “virtual” ones; operating a segmentation of the receiver. Concerning the array case, two ways are followed: synthesize the array by a generalized Gaussian quadrature discretizing the linear reception functionals and use elementary sensors according to SVO. We show that SVO is “optimal” in the sense that it leads to the use of elementary, non-uniformly located field sensors having the same performance of the “virtual” sensors and that generalized Gaussian quadrature has essentially the same performance.

An Optimal Finite Element Method with Uzawa Iteration for Stokes Equations including Corner Singularities

The Uzawa method is an iterative approach to find approximated solutions to the Stokes equations. This method solves velocity variables involving augmented Lagrangian operator and then updates pressure variable by Richardson update. In this paper, we construct a new version of the Uzawa method to find optimal numerical solutions of the Stokes equations including corner singularities. The proposed method is based on the dual singular function method which was developed for elliptic boundary value problems. We estimate the solvability of the proposed formulation and special orthogonality form for two singular functions. Numerical convergence tests are presented to verify our assertion.

The art of coarse Stokes: Richardson extrapolation improves the accuracy and efficiency of the method of regularized stokeslets

The method of regularized stokeslets is widely used in microscale biological fluid dynamics due to its ease of implementation, natural treatment of complex moving geometries, and removal of singular functions to integrate. The standard implementation of the method is subject to high computational cost due to the coupling of the linear system size to the numerical resolution required to resolve the rapidly varying regularized stokeslet kernel. Here, we show how Richardson extrapolation with coarse values of the regularization parameter is ideally suited to reduce the quadrature error, hence dramatically reducing the storage and solution costs without loss of accuracy. Numerical experiments on the resistance and mobility problems in Stokes flow support the analysis, confirming several orders of magnitude improvement in accuracy and/or efficiency.