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Awhile ago, I and other VNQF members posted some of the most famous books in Quantitative Finance and related fields (including Partial Differential Equations, Numerical Methods, Optimization, Parallel Computing). Recently, I’ve posted a few classic books in Applied Math, based on the request of a member in the VEF’s VietnamBookDrive project. Even more recently, anh Ngo Quang Hung (and “đồng bọn”) have created a list of great textbooks in Computer Science.

Here, I merely recompose the list of some bible books in Applied Math that I already did, but maybe more in depth and not limited to the scope of VEF. By Applied Math, I mean a wide spectrum from extremely mathematical topics (they’re essentially subsets of “Pure Math”) such as Partial Differential Equations to Computations (Numerical Analysis, Multiscale Modeling,…) to Probabilistic/Statistical Sciences to Computer Science and to a wide range of Applications (Physics, Chemistry, Biology, Engineering, Finance, Economics, Business, Medicines, etc.). Due to the breath and depth of this giant discipline, the list is by no mean complete and will be updated here continuously (possibly until I die).

For some of the books here, I have gone through or half-through or quarter-through. For the rest I haven’t but this doesn’t prevent them from being invaluable sources of reference. Another important note: the words in my description shouldn’t be quantified or taken superficially. For example: if I say something basic, it doesn’t necessarily mean easy stuff (well, may be for researchers in the field) and can be far beyond my understanding.

1. Analysis and Partial Differential Equations

• “Partial Differential Equations” by L. Evans: this is considered the standard text in many many PDE classes. It covers most basic aspects in the study of PDEs. A notable plus of this book compared to the one below are chapters in Part III: Calculus of Variations, Hamiltonian Systems and Optimal Control. These topics are quite important for many people but can be irrelevant for other PDEsers (well, there are tons of directions in PDEs).
• “Partial Differential Equations” (3 volumes) by M. Taylor: quite technical. Neither of the 3 volumes has a chapter that dedicates to the topics (in italic) that I mentioned above although each volume is as thick as Evans’ book. All of the volumes focus exclusively and in real depth on 3 major classes of PDEs: Elliptic, Parabolic, and Hyperbolic equations. If someone wants to focus on some particular problems such as Wave Scattering or Navier-Stokes equation, the second volume is a good treatment.
• “Methods of Modern Mathematical Physics (volume 1): Functional Analysis” by M. Reed and B. Simon: I view it as the best text in Functional Analysis although some other students may prefer “Functional Analysis” by Rudin or by Yoshida. Since the book has a flavor for Mathematical Physics, the chapters on Hilbert Space and Spectral Theory are very well-written.
• “Real Analysis” and “Functional Analysis” by W. Rudin
• For Harmonic Analysis (my understanding on the field is very limited), I’d read the lecture notes by Terry Tao. His adviser, Elias Stein, is also a celebrated Harmonic Analysts and has written some Introductory Harmonic Analysis books: they should be good.

2. Numerical Analysis/Numerical Methods

• “Matrix Computations” by Golub and Van Loan. Although there area numerous state-of-the-art numerical linear algebra libraries (like LAPACK, uBLAS) and we probably don’t want to reinvent the wheel, Numerical Linear Algebra is still the first course to take in any Numerical Analysis program. This book is by far the most common text for such courses.
• “Iterative Methods for Sparse Linear Systems” by Y. Saad: a standard textbook used in Iterative Linear Algebra courses. Nevertheless, not all methods are best described in this book. For Conjugate Gradient Method (and Steepest Descent), I would point to the paper “Painless Conjugate Gradient” by Jonathan Shewchuk. The Multigrid method is better described in “A Multigrid Tutorial” by Briggs et al.
• “Numerical Recipes in C (or C++): the Art of Scientific Computing” by Press, Flannery, Teukolsky, and Vetterling. This book is actually the closest to being a bible for engineers and scientists. A figure is worth a thousand words so let me give you some number: up to this point, “Numerical Recipes” in C and C++ (let alone Fortran and other languages) combined already has over 30000 citations.
• “Numerical methods for nonlinear conservation laws” by Randall LeVeque
• “Level Set Methods and Dynamic Implicit Surfaces” by S. Osher or “Level Set Methods and Fast Marching Methods” by J. Sethian. You may wonder why I don’t mention any book on the 2 popular (classes of) methods: Finite Difference and Finite Elements, while there are tons of books exclusively on Finite Element Methods (FEM). The reason is that any textbook in Numerical Analysis has at least 1 chapter in Finite Difference Methods (FDM) and at least 2 chapters (due to more mathematical foundations) on FEM. Similarly, the Handbook of Numerical Analysis has 2 parts on FDM and 3 on FEM (each part counts 1/3 to 1/2 of a volume). Level Set Methods (LSM) is an important (not so new) technique that is not part of a regular Numerical Analysis textbook. FYI: unfortunately, the parents of the method (Osher and Sethian) have since not produced any more paper together.

3. Stochastic Calculus, Stochastic Optimal Control, and Quantitative Finance

• “Stochastic Differential Equations: an Introduction with Applications” by B. Oksendal: arguably the best textbook on Stochastic Differential Equations
• “Stochastic Calculus for Finance” by S. Shreve: a great one for ones who want to combine rigorous Math and Finance
• “Financial Modelling with Jump Processes” by R. Cont and P. Tankov: modeling stock prices by Lévy processes (loosely speaking, it generalizes the notion of Brownian motion by allowing jumps) seems trendy nowadays. I started looking at this book (with the reading speed of epsilon minutes per day) a few days ago and the reason is that Peter Pankov is giving a series of lectures on the subject at UT Austin. But hey, the book has a good rating on Amazon.
• Again, for this subject, I would better point you to http://www.vnqf.org/forums/books_and_papers/696-gioi_thieu_sach.html

4. Optimization

• “Numerical Optimization” by Noceldal & Wright. It is famous and highly recommended (not by me but by many Professors) for anyone studying Optimization (i.e. Operations Research). State-of-the-art techniques are described in depth here, such as (Quasi-)Newton with Conjugate Gradient method (that’s what I looked at).
• “Convex Optimization” by S. Boyd and L. Vandenberghe. Similarly, it’s highly recommended by many experts but I haven’t had chance to worry about Convex Optimization
• “Practical Methods of Optimization” by Fletcher: recommended by Tuyen Huynh

In the following fields, my understanding is upper-bounded by epsilon but still, I put these on my bookshelf (for future read, hopefully). I’m sure they are useful stuffs to learn and to master (even in the context of Applied Math).

• “Algebra” by S. Lang (the standard text by algebraists)
• “Thermodynamics and Statistical Mechanics” by Greiner
• “Differential Geometry of Curves and Spaces” by do Carmo

To be updated…

I apologize for my handwaving manner on some books because I haven’t really read. Anyhow, I can’t omit them because they’re great texts/references on important subjects. Now, comments or questions?