What are some good graduate level books on applied mathematics which explain in-depth the general modern problem-solving methods of the real-world typical hard problems? 


There is a lot of books on numerical methods, engineering math, but I do not know any good modern book, which emphasizes algorithmic complexity of the discussed problems.



Applied mathematics Books (graduate level)
Since the question was tagged with "algorithms", I will give an algorithms recommendation. (You don't say specifically what type of problems you want to solve, but you do mention "algorithmic complexity.") For a book that was written to motivate the theory of algorithms from real-world problems, I would recommend Algorithm Design by Kleinberg and Tardos. It discusses many problem-solving methods. From the website for the book:



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> Algorithm Design introduces algorithms by looking at the real-world problems that motivate them. In a clear, straight-forward style, Kleinberg and Tardos teaches students to analyze and define problems for themselves and from this, to recognize which design principles are appropriate for a given situation. The text encourages a greater understanding of the algorithm design process and an appreciation of the role of algorithms in the broader field of computer science. 
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Amazon link: [http://www.amazon.com/Algorithm-Design-Jon-Kleinberg/dp/0321295358](http://rads.stackoverflow.com/amzn/click/0321295358)


