An adaptive kernel-split quadrature method for parameter-dependent layer potentials

Advances in Computational Mathematics - It is well known in the literature that standard hierarchical matrix ( ${\mathscr{H}}$ -matrix)-based methods, although very efficient for asymptotically...     

Kinetic fluorescence studies of 5-aminolaevulinic acid-induced protoporphyrin IX accumulation in basal cell carcinomas.

Laser-induced fluorescence (LIF) investigations have been performed in connection with photodynamic therapy (PDT) of basal cell carcinomas and adjacent normal skin following topical application of 5-aminolaevulinic acid (ALA) in order to study the kinetics of the protoporphyrin IX (PpIX) build-up. Five superficial and 10 nodular lesions in 15 patients are included in the study. Fluorescence measurements are performed prior to the application of ALA, 2, 4 and 6 h post ALA application, immediately post PDT (60 J cm-2 at 635 nm), and 2 h after the treatment. Hence, the build-up, photobleaching and re-accumulation of PpIX can be followed. Superficial lesions show a maximum PpIX fluorescence 6 h post ALA application, whereas the intensity is already the highest 2-4 h after the application in nodular lesions. Immediately post PDT, the fluorescence contribution at 670 nm from the photoproducts is about 2% of the pre-PDT PpIX fluorescence at 635 nm. Two hours after the treatment, a uniform distribution of PpIX is found in the lesion and surrounding normal tissue. During the whole procedure, the autofluorescence of the lesions and the normal skin does not vary significantly from the values recorded before the application of ALA.     

Fast Ewald summation for Stokesian particle suspensions

We present a numerical method for suspensions of spheroids of arbitrary aspect ratio, which sediment under gravity. The method is based on a periodized boundary integral formulation using the Stokes double layer potential. The resulting discrete system is solved iteratively using generalized minimal residual accelerated by the spectral Ewald method, which reduces the computational complexity to , where N is the number of points used to discretize the particle surfaces. We develop predictive error estimates, which can be used to optimize the choice of parameters in the Ewald summation. Numerical tests show that the method is well conditioned and provides good accuracy when validated against reference solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

Error estimation for quadrature by expansion in layer potential evaluation

In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.