This Course at MIT

Below, Prof. Ian Hutchinson describes various aspects of how he taught 22.15 Essential Numerical Methods.

A Foundational Course for Graduate Study and Research

The rationale behind the course is that every advanced engineering and physical science student needs to be able to use numerical methods in their research and professional activities, and they need to know the rudiments of how those methods work. Especially in the nuclear field, many large codes are routinely used for all sorts of design and performance calculations. An MIT-trained student should have some basic understanding of the sorts of things that such codes contain, and what their constraints, strengths, weaknesses, and limitations are. Just as every serious engineer should know how an internal combustion engine works, but not the details of every car's design, so they should know, for example, how a thermal hydraulics code they might use works, but not necessarily its detailed design.

Many Nuclear Science and Engineering graduate students become not just code users, but code developers in their research activities, developing scripts and programs to analyze data, to link large calculations together, or even to implement large-scale parallel numerical calculations. Those that specialize in computational research are going to need more than is in the course Essential Numerical Methods, but the course gives a wide perspective on computational engineering. That helps a specialist to see their detailed application in context, and understand the framework of computational science and engineering.

In developing this course, I first made the decision on principle that it would be language-agnostic. That is, it does not require anyone to use a particular programming language or mathematical system. However, since I also decided (partly for lack of time) that no direct matrix algorithms (multiplication, inversion, decomposition, or eigenanalysis) would be taught, students need access to a system that has those algorithms as libraries or built in. MATLAB® and its open source alternative Octave are the easiest to use because of their focus on matrices and their use of matrix notation. However, Python, Mathematica®, and C have also been used by students, and IDL® would certainly also work fine.

Balancing Breadth and Detail in a Half-Term Course

The course is constrained to occupy only half a term, because there are several other required "modules" common to all students in their first year. That forces the course to be concise, which is a challenge to both students and instructor.

Because the course is so compressed, tough choices have to be made about material to leave out. One thing that is omitted is formal mathematical proofs. That worries some students with mathematical aspirations, but it shouldn't. I believe it is much more important in this topic to develop mathematical insight than to learn rigorous proofs. There is still a lot of mathematics in the course, because we really dig into questions like order of convergence, stability, and computational cost. So this is not just a "survey" course, floating at high altitude and never getting our boots dirty. What we do is drill down into certain topics chosen from the whole field and erect some solid knowledge posts. They are very spaced out, but they serve as an anchor framework that enables students to understand the field as a whole, and to construct more detailed knowledge structures in particular areas later as they might have need.

I have no qualms about the material or the abbreviated style of the module and how the course is taught. But I do think that less experienced students find a serious challenge in the required speed of assimilation. The intensity of the course—like a boot camp—is exciting; but sometimes a learner just needs time to let the ideas soak in and become a part of one's unconscious mental framework. A half-term module just doesn't have the leisure for that. My hope is that it gives a big push forward in understanding computational methods, and the resultant momentum carries the students beyond the end of the module so that the assimilation takes place more fully in succeeding months and years.