Матти Вуоринен

Harri Hakula, Antti Rasila, Matti Vuorinen

We study the problem of computing the exterior modulus of a bounded quadrilateral. We reduce this problem to the numerical solution of the Dirichlet-Neumann problem for the Laplace equation. Several experimental results, with error estimates, are reported. Our main method makes use of an $hp$-FEM algorithm, which enables computations in the case of complicated geometry. For simple geometries, good agreement with computational results based on the SC Toolbox, is observed. We also use the reciprocal error estimation method introduced in our earlier paper to validate our numerical results. In particular, exponential convergence, in accordance with the theory of Babu\vska and Guo, is demonstrated.

Sergei Bezrodnykh, Andrei Bogatyrev, Sergei Goreinov et al.

Making use of two different analytical-numerical methods for capacity computation, we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers. These two methods are based respectively on the use of the Lauricella function and Riemann theta functions. We apply these results to benchmark the performance of numerical algorithms, which are based on adaptive $hp$--finite element method and boundary integral method.

Harri Hakula, Antti Rasila, Matti Vuorinen

We study the problem of computing the conformal modulus of rings and quadrilaterals with strong singularities and cusps on their boundary. We reduce this problem to the numerical solution of the associated Dirichlet and Dirichlet-Neumann type boundary values problems for the Laplace equation. Several experimental results, with error estimates, are reported. In particular, we consider domains with dendrite like boundaries, in such cases where an analytic formula for the conformal modulus can be derived. Our numerical method makes use of an $hp$-FEM algorithm, written for this very complicated geometry with strong singularities.

Antti Rasila, Matti Vuorinen

The numerical performance of the AFEM method of K. Samuelsson is studied in the computation of moduli of quadrilaterals.

Oona Rainio, Mohamed M. S. Nasser, Matti Vuorinen et al.

For augmentation of the square-shaped image data of a convolutional neural network (CNN), we introduce a new method, in which the original images are mapped onto a disk with a conformal mapping, rotated around the center of this disk and mapped under such a Möbius transformation that preserves the disk, and then mapped back onto their original square shape. This process does not result the loss of information caused by removing areas from near the edges of the original images unlike the typical transformations used in the data augmentation for a CNN. We offer here the formulas of all the mappings needed together with detailed instructions how to write a code for transforming the images. The new method is also tested with simulated data and, according the results, using this method to augment the training data of 10 images into 40 images decreases the amount of the error in the predictions by a CNN for a test set of 160 images in a statistically significant way (p-value=0.0360).

Shuliang Gao, Anni Hakanen, Antti Rasila et al.

We study planar domains $G$ equipped with a hyperbolic type metric and approximate geodesics that join two points $x,y \in G$ and their lengths. We present an algorithm that enables one to approximate the shortest distance in polygonal domains taken with respect to the quasihyperbolic metric. The method is based on Dijkstra's algorithm, and we give several examples demonstrating how the algorithm works and analyze its accuracy. We experimentally demonstrate several previously theoretically observed features of geodesics, such as the relationship between hyperbolic and quasihyperbolic distance in the unit disk. We also investigate bifurcation of geodesics and the connection of this phenomenon to the medial axis of the domain.

Lauri Ruotsalainen, Matti Vuorinen

Numpy and SciPy are program libraries for the Python scripting language, which apply to a large spectrum of numerical and scientific computing tasks. The Sage project provides a multiplatform software environment which enables one to use, in a unified way, a large number of software components, including Numpy and Scipy, and which has Python as its command language. We review several examples, typical for scientific computation courses, and their solution using these tools in the Sage environment.

Antti Rasila, Matti Vuorinen

Basic facts and definitions of conformal moduli of rings and quadrilaterals are recalled. Some computational methods are reviewed. For the case of quadrilaterals with polygonal sides, some recent results are given. Some numerical experiments are presented.