The art and science of making approximations

Clearly, the title of this course is meant to be provocative. There isn't one answer to this question. A large part of it, however, is the incredibly general-purpose and practical machinery of making approximations. Why make approximations? Because exact solutions of all but a handful of equations aren't accessible. Even for the ones that we believe we have exact solutions, for example, dy/dx=y > y=exp(x), is merely shorthand for saying go to somewhere where you either can numerically integrate the equation or where you can look up a table of numbers of the exponential function. Naming something, as Feynman said, isn't the same as understanding. Even this function is only really understood when you make approximations of what it looks like close to x=0, or what its behavior is as x takes on very large positive and negative values. But the importance of being able to make approximations is far more general-purpose than those instances where you are presented with a complex equation.

One of the most famous examples of using approximations methods and its mindset is what G.I. Taylor did during WWII. For a more detailed description watch this video. Another description of what Taylor did can be found here. Below were the published still images in Times Magazine of the Trinity explosion in New Mexico. Take note that the publication included the time stamps (top left) and a scale bar in the published images. Using just these images Taylor was able to make an incredibly accurate estimate for the energy released by the device -- Taylor's estimate was about 10 kilotons of TNT, whereas the official Manhatten Project value was around 18-20 kilotons. How did Taylor do this? And, how can you learn to do this when faced with complex problems?

So what is this general-purpose machinery/mindset/tool that applied mathematicians like to harp on about? Its the idea of dominant balance. Said simply, that any complex phenomena can be broken down into a set of simpler phenomena where only two of the effects on the menu of effects compete with each other. A competition of just two effects generically gives rise to situations where a quantitative understanding can be had. Often, if you're lucky, the phenomena of interest fall into a regime where one dominant balance "wins". Often, if you're unlucky, the phenomena falls into a regime involving the conflict between two dominant balances and the system can be complex and hard to understand. All hope, however, is not lost because the method quantitatively demarcates the regimes that are easy and hard, and even why the hard regimes are hard to understand and which require additional approaches/resources to understand. A very complex process can thus be seen as an outcome of various dominant balances between two elemental processes. Thus, the approach is now clear. Take a complex process, enumerate all possible head-to-head dominant balances, quantitatively understand the scenarios where there is a hands-down winner, thereby breaking it up into simpler manageable pieces.

Said this way, the method of dominant balance is far more general than the way to solve equations. For many of us it is how we think about a complex world, and how we go about breaking it up into manageable and understandable pieces. Since this is not a course in philosophy we will teach this approach through the numerical and analytical study of equations. In particular, starting with simple algebraic equations we will move onto ordinary differential equations, integrals, and finally partial differential equations. Numerical integration and exploration of these equations will be conducted in MATLAB.

What will you leave knowing:

1) That analytical and numerical work is guilty until proven innocent, and that the two can, and should, be used in tandem to investigate until the two agree.

2) That there exists a very general-purpose and practical toolkit to solve mathematical equations, indeed any complex problem -- the method of dominant balance. Even those that aren't classical equations with exact solutions. In fact, even those can only be understood through dominant balance.

3) A much deeper understanding of equations, integrals, ODEs, and PDEs.

4) Numerical integration of these equations.

Prereqs: This is a class for undergraduates interested in applied mathematics, and graduate students interested in modeling. Experience with coding is required -- experience and comfort with manipulating arrays, mathematical manipulation of vectors, plotting, and numerical intergration of ODEs  would be incredibly helpful. You can find many resources to help you get good at coding. We generated one for another course I teach that you can find here. Please email me (madhav.mani@northwestern.edu) if you have questions regarding prereqs. For graduate students, this is an excellent primer on how to model. Typically graduate students have more exposure to coding and Taylor Expansions but still learn a lot of how to problem solve complex problems.

Syllabus

Books/resources: This course is built upon a course developed by M.P. Brenner at Harvard. All credit should go to Michael and his team. Useful books are by Bender and Orzag, and by Hinch. However, this course in no way will follow either of these books. They are simply good resources to give alternate points of views on certain topics.

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Course Instructions and Material

Instructions

1. Pre-recorded videos and lecture notes: Lecturing of core course material will be pre-recorded in videos. All of the course material is based on the course notes that can be downloaded below. The pre-recorded material and course notes are not substitutes for each other. I believe the best way to learn and take the most out of the course is to use both of them. Watch the lectures, then read the course notes, and iterate.

2. Problem-solving, assignments, and assessment: Problem-solving, not memorization, is the way to make the most of this class. This is different from classes where memorization plays a big role. There is only one thing to memorize in this class -- Taylor Expansions, which is the idea that you can make polynomial approximations to an arbitrarily complex function, locally. The rest of the class involves learning the art of problem-solving using a very small toolkit of mathematical ideas. Assignments can be found below. A pdf of solutions will be handed out following each completed assignment so that students can learn from their mistakes. Class time will be spent on addressing student questions regarding concepts from lectures and problems in assignments etc. It will be up to the students to make sure they come to class prepared, having read the notes and watched pre-recorded lectures.

Material

Introduction

Course Notes

Assignments1, 2, 3, 4, 5 + Hodgkin

(Ignore course title and due date on these assignments)