Tuomo Rossi

Dirk Pauly, Tuomo Rossi

We present both, theory and an algorithm for solving time-harmonic wave problems in a general setting. The time-harmonic solutions will be achieved by computing time-periodic solutions of the original wave equations. Thus, an exact controllability technique is proposed to solve the time-dependent wave equations. We discuss a first order Maxwell type system, which will be formulated in the framework of alternating differential forms. This enables us to investigate different kinds of classical wave problems in one fell swoop, such as acoustic, electro-magnetic or elastic wave problems. After a sufficient theory is established, we formulate our exact controllability problem and suggest a least-squares optimization procedure for its solution, which itself is solved in a natural way by a conjugate gradient algorithm operating in the canonical Hilbert space. Therefore, it might be one of the biggest advances of this approach that the proposed conjugate gradient algorithm does not need any preconditioning.

Dirk Pauly, Sergey Repin, Tuomo Rossi

In this paper, we consider an initial boundary value problem for Maxwell's equations. For this hyperbolic type problem, we derive guaranteed and computable upper bounds for the difference between the exact solution and any pair of vector fields in the space-time cylinder that belongs to the corresponding admissible energy class. For this purpose, we use a method suggested earlier for the wave equation.

Joonas Hämäläinen, Tommi Kärkkäinen, Tuomo Rossi

In this work, two new initialization methods for K-means clustering are proposed. Both proposals are based on applying a divide-and-conquer approach for the K-means|| type of an initialization strategy. The second proposal also utilizes multiple lower-dimensional subspaces produced by the random projection method for the initialization. The proposed methods are scalable and can be run in parallel, which make them suitable for initializing large-scale problems. In the experiments, comparison of the proposed methods to the K-means++ and K-means|| methods is conducted using an extensive set of reference and synthetic large-scale datasets. Concerning the latter, a novel high-dimensional clustering data generation algorithm is given. The experiments show that the proposed methods compare favorably to the state-of-the-art. We also observe that the currently most popular K-means++ initialization behaves like the random one in the very high-dimensional cases.