Research Projects

The Applied mathematics section covers research on applied mathematics, computation, and computer science. Under these categories fall the areas of methodology in software design, analysis of algorithms, operations research, probability, biostatistics, numerical analysis, and statistics. Research in the section can then be divided into basic research in those areas that come under it and projects within other disciplines where mathematical modeling and mathematical techniques often give good results. In the last few years those projects have related to marine science and fisheries, ocean currents, avalanches, and meteorology. Many of the projects are in association with other institutions, such as the Icelandic Meteorological Office, the Marine Research Institute and engineering firms, e.g. Vatnaskil.

The projects in computers science come from a variety of areas. Among them is work in programming languages and large scale programming. That involves programming in units and methods to divide large projects safely into smaller units ensuring that all the pieces of the projects work correctly together. The research into software design concerns the design of programming languages, development of graphical user interfaces, and improved software design methods. Also being investigated is the influence of the new methods and how best to utilize them.

Among the new and promising activities in computer science at the Institute are cryptography and computer security, the future effects of the information revolution, the use of computer algorithms in molecular biology, and groupware systems. Some of the projects are in association with software companies, such as Softis h.f. and Hugvit h.f.

The applied mathematics section employs two research associates and supplies research facilities to 11 professors from the mathematics and computer science departments at the University of Iceland. In addition a few assistants are employed for shorter periods.

Below are short descriptions of some of the projects that members of the section were working on in 1996.

Ebba Þóra Hvannberg

• AMUSE (Advanced Multimedia Services for Residential users)
  The goal of the AMUSE research project is to do a trial with interactive TV over a high speed ATM network to residential users. Among the services that are offered are Movies on demand, News on demand and TV series on demand and WWW browsing. During the second project year the trial in Reykjavik was organized and planned, that is software and hardware installed, content was aquired and converted to digital form. Furthermore, a plan for the experiments to be done during the 3 month trial was prepared. The measurements that were planned were for example measuring usability and usage. Dissemination of the trial and its technology was also a part of the research project.

Hermann Þórisson
Probability

• Mainly general coupling theory, Palm theory, and regenerative processes and their generalizations. Also the application of these fields to Markov processes and in queueing and simulation.
• The year 1996 was mostly devoted to writing a book entitled Coupling, Stationarity and Regeneration to be published by Springer Verlag in 1998.
• Research centered on the following topics related to this book project. Coupling of random elements under a locally compact transformation group. Extension of the concept of point-stationarity to d > 1 dimensions with applications in Palm theory. Convergence of a taboo-regenerative process and the structure of the taboo-stationary limit process (quasi-stationarity) with simulation applications. Extension methods for probability spaces. Applications to Markov processes of general coupling theory for stochastic processes.

Hjálmtýr Hafsteinsson

• Computer science and molecular biology
  The use of computer algorithms to solve problems from molecular biology.
• Random number generator testing
  Simulation and statistical testing of a lottery random number generator.

Kjartan G. Magnússon
Biomathematics - Mathematical models of fish stock interactions

• Stochastic feeding models
  Stochastic models of the feeding processe for predatory fish and methods to estimate parameters (including average meal frequency and average meal size) and to test feeding models using the frequency distribution of stomach contents
• Structured predator-prey model with cannibalism
  The predator stock is divided into mature and immature components and the mature predator feeds on the immatures in addition to the prey. The effect of cannibalism on long term stock levels was investigated and the existence of stable oscillatory solutions resulting from the incorporation of cannibalism was demonstrated

Kristján Jónasson

• Snow avalanche risk-zoning on the basis of physical models. Methods that make use of simple avalanche models in order to transfer historical avalanche records between avalanche sites have been developed. The aim is to make risk-zoning at particular sites more reliable. This work is carried out in collaboration with Sven Þ. Sigurðsson and Guðmundur Guðmundsson both at the division of applied mathematics and computer science.

Magnús M. Halldórsson

• Approximation algorithms
  Design and analysis of approximation algorithms for computionally intractable problems. Hard optimization problems arise in all areas of computing. Exact algorithms can only solve small instances of such problems, thus the main hope is to develop heuristics that yield solutions close to optimal. Of particular interest are algorithms that come with a guarantee on the quality of the solutions they obtain. The focus of this development is on problems of fundamental importance. Results for such problems often find surprising applications in highly disparate application domains.

  One example is the Tree Editing distance problem, where we are given two trees and are to find a minimum number of changes, node insertions, or deletions that transform one tree into the other. This is important in computational molecular biology, that data sets invariably contain noisy elements necessitating approximate matching. In particular, RNA proteins are typically represented as trees. In cooperation with Keisuke Tanaka of NTT, we have developed the first algorithm that yields non-trivial bounds on the quality of the solutions obtained.

• Orthogonal matrix partition
  For the solution of numerical problems on parallel machines, the partition of a matrix onto the processors is essential for efficient computation. Orthogonal partition yields rectangular blocks with regular communication patterns. Such a primitive has been implemented in HPF Fortran. With Bengt Aspvall and Fredrik Manne of University of Bergen, have we analyzed efficient natural heuristics and given the first approximation results.

Oddur Benediktsson

• Software development methodology
• Quality control of software processes and products
• Object oriented methodology
• Personal software processes

Ottó J. Björnsson
Probability and statistics

• The concept of probability in physics; quantum probability.
• The history of (elementary) mathematics in Iceland: The life and work of Björn Gunnlaugsson.

Sven Þ. Sigurðsson

• Numerical solutions of partial differential equations for groundwater- and shallow water flow and material transport in such flows. The project involves the design and analysis of new approximation methods based on Galerkin finite element methods. It is carried out in collaboration with Vatnaskil Consulting Engineers and focuses on problems that have arisen in actual modelling situations.

• Snow avalanche risk-zoning on the basis of physical models. Methods that make use of simple avalanche models in order to transfer historical avalanche records between avalanche sites have been developed. The aim is to make risk-zoning at particular sites more reliable. This work is carried out in collaboration with Kristján Jónasson, formerly at the section of applied mathematics but presently at The Icelandic Meteorological Office.