Bjørn Frederik Nielsen

Elements of Scientific Computing

Science used to be experiments and theory, now it is experiments, theory and computations. The computational approach to understanding nature and technology is currently flowering in many fields such as physics, geophysics, astrophysics, chemistry, biology, and most engineering disciplines. This book is a gentle introduction to such computational methods where the techniques are explained through examples. It is our goal to teach principles and ideas that carry over from field to field. You will learn basic methods and how to implement them. In order to gain the most from this text, you will need prior knowledge of calculus, basic linear algebra and elementary programming.

On the use of st-segment shifts and mathematical models for identifying ischemic heart disease

It is well known that myocardial ischemia can be observed as a shift in the ST segment of an ECG recording. In this paper we will discuss how ST-segment shift, along with advanced mathematical models, can be used to compute the size and location of the ischemic region-an inverse problem. Traditionally, the inverse approach to localizing ischemia has been based on the calculation of epicardial potentials. We present an alternative approach where the location and size of the ischemic region are computed directly by solving a parameter identification problem based on a level set framework and partial differential equations (PDEs). More precisely, the ischemic region is identified by minimizing an objective function measuring the difference between the recorded and simulated ECG signals. Our results show, at least for synthetic ECG data, that values of ST shift recorded at the body surface are capable of identifying the position and (roughly) the size of the ischemia

A Glimpse of Parallel Computing

It should not be too difficult to imagine that applications of scientific computing in the real world can require huge amounts of computation.

About Scientific Software

When a science problem is solved with the aid of numerical computations, the solution procedure involves several steps: 1. Understanding the problem and formulating a mathematical model 2. Using numerical methods to solve the mathematical problems 3. Implementing the numerical methods in a computer program 4. Verifying that the results from the program are mathematically correct 5. Applying the program to the scientific problem and interpreting the results

The Diffusion Equation

This chapter treats the numerical simulation of diffusion processes. Diffusion takes place in space and time simultaneously and is an important phenomenon in nature and technology. Scientific computations of physical quantities such as temperature, pollution, and velocity are frequently based on models for diffusion. Diffusion is often coupled with other processes in physical problems, but in this chapter we will only consider diffusion as an isolated process.

Analysis of the Diffusion Equation

In Chap. 7 we studied several aspects of the theory of diffusion processes. We saw how these equations arise in models of several physical phenomena and how they can be approximately solved by suitable numerical methods. The analysis of diffusion equations is a classic subject of applied mathematics and of scientific computing. Its impact on the field of partial differential equations (PDEs) has been very important, both from a theoretical and practical point of view. The purpose of this chapter is to dive somewhat deeper into this field and thereby increase our understanding of this important topic.

Systems of Ordinary Differential Equations

In Chap. 2, we saw that models of the form\(y{\prime}(t) = F(y),\,\,\,y(0) = y_{0,}\)can be used to model natural processes.

Differential Equations: The First Steps

If you know the characteristics of something today, and you know the laws of change, then you can figure out what the characteristicswill be tomorrow. This is the basic idea of modeling lots of natural processes. Since we know the weather today and we know the equations modeling the changes of the weather, we can predict it some days ahead. We know the heat of an object now and we know the equations describing how heat changes; thus we can predict how warm an object will be later on. This is really at the heart of science and has been so for quite a while. But how do we express change? How do we make these vague statements precise and suitable for computer simulations? We do so by expressing the laws of change in terms of differential equations.