Numerical conformal mapping onto the exterior unit disk with a straight slit and logarithmic spiral slits

This paper presents a fast boundary integral equation method for numerical conformal mapping of unbounded multiply connected regions onto a disk with an infinite straight slit and finite logarithmic spiral slits. Some numerical examples are given to show the effectiveness of the proposed method

Numerical conformal mapping onto the exterior unit disk with a straight slit and logarithmic spiral slits

This paper presents a fast boundary integral equation method for numerical conformal mapping of unbounded multiply connected regions onto a disk with an infinite straight slit and finite logarithmic spiral slits. Some numerical examples are given to show the effectiveness of the proposed method.This study was supported partially through the Research Management Centre (RMC), Universiti Teknologi Malaysia (Research Grant Q.J130000.2526.16H70). This support is gratefully acknowledged.Scopu

Rapid methods for the conformal mapping of multiply connected regions

AbstractWe present fast methods for the conformal mapping of simply, doubly and multiply connected regions onto certain canonical regions in the plane. Our mapping procedure consists of two parts. First we solve an integral equation on the boundary of the region we wish to map. The solution of this integral equation is needed to determine the boundary correspondence. We have chosen to use the integral equation formulation of Mikhlin. Although it is not widely used, this formulation has the advantage that it leads to integral equations of the second kind with unique solutions and bounded kernels. The solutions are also periodic, allowing for effective use of the trapezoid rule. Once we have solved the integral equation we use a rapid method we have previously developed to determine the mapping function in the interior of the region. This method makes use of fast Poisson solvers, and thereby circumvents the difficulties associated with computing integrals at points near the boundary of the region, and avoids the expense of computing many integrals. We also provide results of numerical experiments

Circular slit maps of multiply connected regions with application to brain image processing

In this paper, we present a fast boundary integral equation method for the numerical conformal mapping and its inverse of bounded multiply connected regions onto a disk and annulus with circular slits regions. The method is based on two uniquely solvable boundary integral equations with Neumann-type and generalized Neumann kernels. The integral equations related to the mappings are solved numerically using combination of Nyström method, GMRES method, and fast multipole method. The complexity of this new algorithm is O((M+ 1) n) , where M+ 1 stands for the multiplicity of the multiply connected region and n refers to the number of nodes on each boundary component. Previous algorithms require O((M+ 1) 3n3) operations. The numerical results of some test calculations demonstrate that our method is capable of handling regions with complex geometry and very high connectivity. An application of the method on medical human brain image processing is also presented

Numerical computation of the conformal map onto lemniscatic domains

We present a numerical method for the computation of the conformal map from unbounded multiply-connected domains onto lemniscatic domains. For $\ell$-times connected domains the method requires solving $\ell$ boundary integral equations with the Neumann kernel. This can be done in $O(\ell^2 n \log n)$ operations, where $n$ is the number of nodes in the discretization of each boundary component of the multiply connected domain. As demonstrated by numerical examples, the method works for domains with close-to-touching boundaries, non-convex boundaries, piecewise smooth boundaries, and for domains of high connectivity.Comment: Minor revision; simplified Example 6.1, and changed Example 6.2 to a set without symmetr