The research projects have been organized within the SCAT (Scientific Computing and Applications in Technology) program. The SCAT program consists of projects in five main research areas. In the year 1995, these areas were
Within each research area, a reasonable level of synergy between different projects is generally provided either by the use of similar methods or by the study of similar problems with different types of methods.
The different research areas have a common mathematical background which makes the communication possible between projects in different areas. Moreover, much effort has been invested in a common computational environment. On one hand, it reduces the routine work in the projects, and, on the other hand, makes it easier for the projects to communicate with each other.
The project deals with the finite element analysis of some important, mostly nonlinear problems. As there is no general numerical method capable of solving an arbitrary nonlinear problem, the attention has to be restricted to concrete problems arising from important applications. Currently, the fields of active interest include problems in electromagnetics, free boundary problems and situations where finite elements are used in connection with integral equations (for example, heat radiation).
This project consists of two parts. The first part deals with nonsmooth analysis and optimization. Subdifferential calculus has been applied to generalize the optimality conditions of nondifferentiable functions. This theoretical basis has made it possible to develop effective bundle-type methods for nonsmooth and nonconvex optimization problems. The methods have been implemented as a subroutine library NSOLIB (see section 4.6).
The second part deals with multiobjective optimization. A methodological state-of-the art survey with firm theoretical foundations has been prepared. The actual aim has been to generate effective and user-friendly methods for complicated real-world problems. As a concrete result, an interactive method, NIMBUS, for nondifferentiable multiobjective optimization problems has been developed. The algorithm is based on the classification of the objective functions. According to the classification, a new (multiobjective) optimization problem is formed and solved employing NSOLIB. Some optimal control problems arising from optimal shape design and continuous casting processes have been solved as applications.
Three implementations of NIMBUS are available. They are versions for mainframe, MS-Windows and WWW (World-Wide Web) environments. The idea of centralized computing and distributed interface in the WWW-NIMBUS is unique and, thus, the WWW version is developed in the most intensive way. The URL is http://nimbus.mit.jyu.fi/NIMBUS.
The goal of the project is to construct simulation models for some key parts of a paper machine, as well as to study the properties of paper. A so-called former section of a paper machine (where fluidlike suspension of water and fibres gradually solidifies to a porous fibre net) has been modelled. Mathematical properties of the model are studied and numerical methods are being developed.
Also the modelling of copying behaviour of paper is considered. The research focusses on modelling the heat and moisture transfer in paper during hot roll nip actions which are common in copying and laser printing machines. Experimental tests have also been carried out in order to obtain verification data for the developed model.
Free boundary problems form one of the main research fields of the laboratory. In the past, the work on free boundary problems and related control problems has initiated the research in nonsmooth optimization and in state constrained control problems. Currently, the laboratory takes part in the ESF programme ``Mathematical Treatment of Free Boundary Problems''.
The theoretical work on free boundary problems concentrates now on modelling the evaporation process. The goal is to find adequate free boundary conditions for a coupled system of Navier-Stokes and Stefan equations, as well as to develop suitable techniques for mathematical analysis and numerical solution of such problems.
Another main line in the research is the use of techniques developed for shape optimization in solving free boundary problems and analysing the solution methods. Often, the free boundary conditions can be formulated as optimality conditions of a shape optimization problem. The methods are developed for applications in naval hydrodynamics in co-operation with the Maritime Institute of Finland.
There is also work going on in numerical simulation of an industrial process for producing silicon crystals. A numerical model has been constructed which takes into account all the major mechanisms of heat transfer including solidification and heat radiation.
The goal in this project is to analyse the thermodynamical models for heat radiation from the mathematical point of view. The models lead to nonlinear integro-differential equations. Currently, the work concentrates on the problems related to diffuse and gray radiation on opaque surfaces and in semi-transparent materials. Co-operation with the University of Stuttgart and with CSC (heat radiation in crystal growth).
The goal has been to introduce efficient and reliable methods for the synthesis of magnetic and electric fields. The results have been documented in a monograph ``Inverse Problems and Optimal Design in Electricity and Magnetism''.
Special attention has been paid to an algorithmic approach. The book deals with the solution of real-life synthesis problems in electricity and magnetism, using the numerical techniques (for solving partial differential equations, optimization, inverse, and shape design problems) introduced. Inverse problems have been classified from an engineering viewpoint. Classical and simple examples have been found from the following categories: synthesis of sources, synthesis of boundary conditions, synthesis of material properties, and synthesis of shapes. A survey of solved problems which have appeared in the literature has been done.
Some practical topics have been handled. In particular, implementations of the finite-element method, optimal shape design and sensitivity analysis, including automatic differentiation of computer programs, are available. Finally, a survey of subroutine libraries for the solution of partial differential equations, electromagnetic problems and optimization problems including nonsmooth and multiobjective optimization, and shape optimal design has been made.
The purpose is to study optimal control problems, especially, systems arising from free boundary problems. The main interest has been in optimal control problems governed by nonlinear parabolic systems, including, among others, parabolic variational inequalities and systems with phase transitions. The aim is twofold: firstly, to give a theoretical approach to the subject and, secondly, to present detailed algorithms (with convergence proofs) that are necessary in computerizing the optimal control processes. Several practical examples are being worked out in detail in order to demonstrate the usefulness of the proposed methods.
Collaboration with researchers from France, Germany, USA, Japan and Romania.
In this project, hemivariational inequalities, generalized variational inequalities involving nonmonotone, multivalued inclusion, are studied. The existence results have been shown for a constrained stationary hemivariational inequality and for a parabolic hemivariational inequality. In addition, a stable and convergent FEM approximation has been developed for these equations. This approximation has been used in the numerical solutions of nonmonotone contact problems of linear elasticity by nonsmooth, nonconvex optimization methods.
Collaboration with Aristotle University, Greece (prof. P.D. Panagiotopoulos).
The aim of the project is to develop methods for the identification of functional coefficients. It is assumed that one has a distributed observation of the solution of an elliptic or parabolic partial differential equation. These observations are used to determine an unknown coefficient in the equation. In physical systems, unknown parameters can represent the heat conductivity or the diffusion coefficient, for example.
Several different discretization schemes based on variants of finite element, spectral and multigrid methods are considered. In addition to new efficient numerical algorithms, one of the purposes is to improve the techniques for convergence analysis of different approaches. At the moment, there exist only a few error estimates for parameter identification problems, and mainly for elliptic partial differential equations. One goal is also the treatment of quasilinear and nonlinear problems with some linearization techniques.
Fast direct methods for solving linear systems with separable matrices have been considered. They have many applications, for example, in the fictitious domain preconditioning. A new method, the so-called Divide & Conquer algorithm, has been developed. It can be applied for solving higher-order partial differential equations, as well. Anther research topic has been the development of (approximate) partial solution techniques.
The aim is to develop and analyze efficient numerical methods based on the fictitious domain approach for solving elliptic partial differential equations with mixed boundary conditions. Similar approach is applied also in acoustic scattering problems. Special emphasis is given to new research areas such as the use of Lagrange multipliers and nonmatching discretization meshes. The methods are further applied, for example, in shape optimization problems and in convection-diffusion problems.
The numerical solution of unilateral variational inequalities (obstacle problems) has been considered. Based on finite difference or finite element discretization, various iterative methods have been proposed for the resulting algebraic system. Special emphasis has been given to the development of the methods, which can be (theoretically) treated within the unifying framework of the block relaxation methods. Also, using the theoretical results, implementation algorithms based on domain decomposition and fictitious domain approaches for large-scale algebraic obstacle problems have been studied.
Efficient and robust methods for multidisciplinary shape optimization problems governed by state equations modelling problems in computational fluid dynamics and electromagnetics have been studied and developed. Typically, two-dimensional state equations are described by the (full) potential flow and the time-harmonic Helmholtz equations. The shape optimization problem is formulated as a nonlinear minimization problem. Each objective function evaluation requires one solution of a state equation, which is obtained by using a fictitious domain method or a multigrid method. In the minimization of the objective function, parallel implementations of genetic algorithms and gradient-based methods such as SQP have been employed.
The aim of the project is to adapt some basic deterministic and stochastic methods of dynamic programming onto neural networks (feedback-recurrent and feedforward). The actual goal is to get a powerful tool for handling large-scale optimization problems arising from telecommunication planning. The main topics have been sequential decision processes (like Markov decision problems and other approaches which can be finally set as Markovian decision problems) with a special emphasis on telecommunication network flow control and routing. The theory of diffusion processes and Brownian control are used to be able to describe more accurately the stochastic nature of the arrival and flow processes of the customers in small and medium sized communication networks. The intention is to construct a soft computing tool for planning.
Collaboration with the Telecommunication Laboratory of the Technical Research Centre of Finland.
The aim of the research activity is to develop some new statistical algorithms for stochastic control and data processing problems arising in the ATM flow control processes. Next, these algorithms are intended to be mapped onto neural computation techniques (based on feedback-recurrent and feedforward neural networks) in order to get a more powerful tool for real time solution of the large-scale stochastic control and data processing issues assumed by the switching/multiplexing processes taking place in the ATM networks. The selection of the neural computing strategies as the environment of embedding both the stochastic control and the data processing solutions found by theoretical development is motivated by the computational flexibility in implementing the predictive (estimate) rather than the reactive (computationally intensive) traffic control mechanisms in the ATM networks.
A reasonable approach is used in order to get the statistical description of the users as a proper estimate on their own transmission behaviour. The major theme addressed through the design effort is the traffic control and network resource management. For a system with statistic load, the primary performance control is to control the system utilization, therefore the network must have a control handle on user data transmission. The diversity in service requirements suggests that a quantitative service specification be submitted to the network together with data transmission tasks. With this knowledge, the network can then manage resources accordingly. This specification should also serve as an estimate of the user's transmission behaviour to facilitate the traffic control.
Mathematical hypermedia enables the creation of a mathematical virtual reality on a computer where mathematics can be studied with the aid of hypertext, graphics, animation, digital videos, etc. The aim of the project is to prepare hypermedia teaching applications in mathematics. They are to support the traditional lecture teaching and partly replace it at universities. These kinds of applications would afford the students an opportunity to revise their knowledge in mathematics and to study mathematics on their own. A future intention is to add intelligence and adaptiveness to bring individual tutoring and individual feedback.
Collaboration with Tampere University of Technology, Helsinki University of Technology and Lappeenranta University of Technology.
The aim of the project is to develop an MS-Windows-based hypermedia application to help the operational development and the problem solving of a steel company. The system contains three modules. The first module, called Operational Development, contains all kinds of principles and tools for the development of operations in steel industry. The second module, called Problem Solving, contains a lot of tools and techniques for problem solving. Both of these two modules include also interactive processes. These processes give advice for users in their work and utilize all the tools and techniques included in the system. The third module, called Key Figures, contains the key figures of the company involved: financing, feedback, delivery, etc. These key figures come from the process and show the quality of the production.
With this system, the company can improve the quality of its production and products by improving the knowledge of the employees and they working readiness. The system offers tools and techniques to help the employees in their everyday work, and teach how to use different equipments.
Collaboration with Rautaruukki Raahe Steel Works and Rautaruukki Thin Sheet Division in Hämeenlinna.
We consider the short and long term planning and optimization problems of energy companies in the new deregulated common market. Based on demand forecasts, energy companies have to balance their hourly heat and electricity production, purchase, sales and transmission optimally during the planning horizon. The hourly planning models are coupled by dynamic constraints resulting from on/off-constraints of power plants, energy constraints in purchase contracts, and energy storages, such as hydro power and heat networks.
The EHTO energy modelling and optimization environment is an interactive, graphical tool for the energy manager of an energy company in the new deregulated common market. The energy system can be defined and configured graphically. An optimization model for the planning problem is automatically formed. The model can be solved using efficient embedded optimization algorithms based on Lagrangian relaxation and decomposition techniques. The optimization results are presented graphically. Parameters and results can also be stored and read from a relational database, which serves as an open interface to other information systems.
Pilot system have been developed for three Finnish energy companies. Collaboration with Process Vision Ltd. and Helsinki University of Technology.
Program development has a major role in many of the projects. In this respect, many projects have shared interests and needs. This has led to the development of program libraries common to the whole laboratory.
FEMPAK 1.0 is a portable software package for the numerical solution of two-dimensional partial differential equations. It is based on the finite element method. The package consists of a set of low-level subroutines for the assembly and solution of the discrete finite element equations, a set of example driver programs and software to visualize the results. The graphical devices supported are X-windows and PostScript.
NSOLIB is a Fortran subroutine library for nonsmooth and nonconvex optimization problems with single or multiple objective functions. The methods are able to handle either simple bounds for variables, linear, nonlinear or nonsmooth constraints, or all of them at the same time. NSOLIB subroutines are implementations of the proximal bundle method. They have been tested with various standard test examples and in several research projects of the laboratory in different computing environments (microcomputers, workstations, mainframe and supercomputers). There is a need for NSOLIB since commercial subroutine packages do not contain efficient codes for nonsmooth optimization.
SIM2++ is an object-oriented simulation library in C++ for discrete event systems. It supports the process-oriented simulation paradigm in a similar manner as the Simula language. SIM2++ is implemented in C++, and an earlier version in the object-oriented Turbo Pascal exists. Virtually, concurrent processes are implemented using a low-level coroutine mechanism. The processes are run in simulated real time and they can exchange information using messages (methods) or shared memory. The system is particularly useful for modelling discrete event systems with active and passive components, such as transaction generators, server processes and queues.
SIM2++ is in educational use at the University of Jyväskylä and Helsinki University of Technology.
The Laboratory of Scientific Computing and the Department of Mathematics had 30 HP-9000 series workstations and X-terminals, 30 Pentium-level and 30 486-level PCs and one Silicon Graphics Indigo2 for researchers and students in 1995. In addition, the facilities at the Finnish National Supercomputer Centre (Cray C90, Convex C3840 and IBM SP2) and the Computer Centre of the University of Jyväskylä (VAX 4000-300 and SUN 4/670 computers connected to terminals and PCs with a local area network) are available.
