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I am going to start my first year of a research-oriented master program on inverse problem.

From what I know, unlike pure math students, applied math students usually don't spend the first year reading certain textbook (such that Evans' PDE book). Instead, applied math students start by reading certain papers, and if they don't understand some concepts, they refer to certain books/papers, read specific sections which can solve the problem and proceed the original paper. And they spend a lot of time implementing numerical schemes. Hence I got a feeling that applied math students may not have enough time to systematically build up a solid theoretical background (such as functional analysis) by finishing an advanced textbook, taking their own notes and solving exercises.

May I know whether my understanding is correct? As graduate students/ researchers on numerical analysis, have you spent some time reading a entire (or a large part of) pure math book (especially at the beginning of your graduate program)? May you share your experience about how to balance reading textbooks and doing research with me?

In particular, currently I am interested in reading Brezis book Functional analysis, sobolev space and partial differential equation. But I am not sure whether it's useful and how to allocate the time.

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  • $\begingroup$ Read: never... Skim as needed... Always! $\endgroup$ – Paul Feb 23 '15 at 22:46
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I think your premise is flawed: If you wish to do research on a topic, you first need to learn the fundamentals -- no matter whether it's "pure" or "applied" math (a distinction which is not really useful in general) -- and the best place is from a good textbook (or, even better, attending a good lecture). Once you get to the cutting edge, you need to switch to current publications (again, both in pure and applied math). If your research bridges several fields (say, PDEs, numerical analysis, optimization in your case, or number theory and abstract algebra in algebraic number theory), you of course need to be more selective to fit the fundamentals of different fields into the same time frame.

As for which books to read and how much, this depends on your specific topic. Surely your advisor is a better person to ask than random strangers on the internet -- after all, this is exactly his job?

(But I'd be surprised if this did not involve a solid foundation in functional analysis, and Brezis' is certainly a good book.)

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To add my own experience to ChristianClason's post: it's not uncommon to for graduate students in computational science to read pure mathematics textbooks and take pure mathematics classes. I graduated with an engineering degree and have read large portions of several pure math books. Part of the reason I did so is because my adviser required me to take a certain amount of pure mathematics, since it was a prerequisite for optimization classes at MIT.

There are tradeoffs. It's certainly possible to read from textbooks about things like optimization, PDE methods, and numerical analysis that don't require much in the way of pure math knowledge. These textbooks are useful for rapid introduction to a subject, and will help you hit the ground running. However, you will be severely limited when it comes to developing novel numerical methods -- you'll be limited to more basic methods presented in those books, and won't have as good an understanding of when certain methods fail, and why. Without any pure math knowledge, you would likely be unable to demonstrate basic facts that would make such methods more compelling, such as convergence proofs or error bounds. Learning pure mathematics will give you the skills to understand and eventually write proofs that demonstrate rigorously mathematical properties of numerical methods that you are investigating. It also enables you to understand a wider selection of papers: after learning pure math, you'll better understand more papers in computational journals like SISC, SINUM, SIMAX, BIT Numerical Mathematics, Numerische Mathematik, etc., and it will be easier to write papers geared towards those journals.

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  • $\begingroup$ Thanks for your answer. So how do you allocate the time? Do you spend time reading those textbooks at the beginning or at the same time when you are doing your research? $\endgroup$ – John Feb 23 '15 at 23:10
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    $\begingroup$ @JohnZHANG: I tend to allocate the time as I need to. A semester course on real analysis was a prerequisite for doing any research in optimization, so I studied that before my research. After that, I tended to allocate time when I felt like learning additional mathematics was sufficiently worth it. For the equivalent of a semester-long class, I know I couldn't get away with doing less than 10 hours of studying a week (and often, more than that). If I'm self-studying, the absolute bare minimum I'd study would be every other day, probably an hour a day, but I try for 10 hours over 6 days/week. $\endgroup$ – Geoff Oxberry Feb 24 '15 at 1:36

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