CS 6210 Fall 2016 Bei Wang. Lecture 1 A (Hopefully) Fun Introduction to Scientific Computing
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1 CS 6210 Fall 2016 Bei Wang Lecture 1 A (Hopefully) Fun Introduction to Scientific Computing
2 About this class Technical content followed by fun investigations
3 Stay engaged in the classroom Share your SC-related research questions/projects Learn Matlab Start your HW early Things can get technical Discussion helps us learn from each other Read the textbook carefully HW typesetting preferred: PDF Think Big: How would SC algorithms and technical transform my research? What is the future of SC?
4
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6 What is Scientific Computing? And why do we care about it?
7 Scientific computing is a discipline concerned with... The development and study of numerical algorithms for solving mathematical problems in science and engineering. Roughly viewed as mathematical model + numerical analysis It is literally rocket science and beyond... From Flickr: wind.com.my
8 2,147,483,647 Why is this number important?
9 2,147,483,647 and integer overflow 1. June 4, European Space Agency (ESA). Crewless Ariane 5 rocket explosion after 39 seconds of launching, resulted in a loss of $370m. (Correction: Ariane 5 was a signed 16-bit integer overflow) 2. YouTube: We never thought a video would be watched in numbers greater than a 32-bit integer (=2,147,483,647 views), but that was before we met PSY. (This was a joke!) Integer Overflow: Maximum positive value for a 32-bit signed binary integer in computing. Now, the counters in YouTube are capable of reaching how many views? From Flickr: wind.com.my
10 9,223,373,036,854,775,808 roughly 9 quintillion (US) The number of views YouTube counter could reach From Flickr: wind.com.my From Flickr: CampusGrotto
11 observed phenomenon mathematical model discretization solution algorithm efficiency accuracy robustness Scientific Computing implementation programming environment data structure computing architecture
12 Some Brief History [Trefethen 2000]
13 Before Newton s method Gaussian elimination Gauss quadrature least squares fitting Richardson extrapolation... floating point arithmetic Fortran finite differences finite elements simplex algorithm Monte Carlo Fast Fourier Transform... quasi Newton iterations adaptivity Matlab multigrid sparse and iterative linear algebra spectral methods interior point methods...
14 After advancement in linear algebra multipole methods breakthroughs in preconditioners, spectral methods & time stepping for PDS... From Flickr: wind.com.my
15 The Best of the Computer Age Top 10 algorithms of the century in SC [Cipra 2000]
16 Monte Carlo method Metropolis algorithm: repeated random sampling to obtain numerical results of problems with many degrees of freedom. e.g. optimization, numerical integration. Manhattan Project! Simplex method for linear programming. A problem mistakenly solved as a homework solution! In terms of widespread use: one of the most successful algorithm of all time! Krylov subspace iteration methods Find a few eigenvalues of large sparse matrices or solve linear equations by avoiding matrix multiplications.
17 Decompositional approach to matrix computation. For efficient SW packages. Facilitates the analysis of rounding errors. Fortran: single most important event in the history of computer programming! It produces code of such efficiency that its output would startle the programmers who studied it. QR algorithm for computing eigenvalues. Transform the once-formidable eigenvalue problems into routine.
18 Quicksort. The posterchild of computational complexity Fast multipole algorithm gives accurate calculation of pair-pair interactions in O(n). Fast Fourier transform. Revolutionized signal processing. Integer relation detection algorithm. Given a bunch of real numbers, x1, x2,...xn, are there not all zero integers a1,a2,...an s.t. a1x1+a2x2+...anxn=0? Quantum field theory!
19 The Future of Scientific Computing 50 years from now [Trefethen 2000]
20 1 Speech and graphics everywhere We ll talk to computers more than type to them, and they ll respond with pictures more often than numbers. Computer Graphics, Visualization, HCI, 3D images
21 2 Fully intelligent, adaptive numerics Numerical computing will be adaptive, iterative, exploratory, intelligent - computational power will be beyond your wildest dreams. Everything is embedded in an iterative loop, problems solved atop an encyclopedia of numerical methods
22 3 Loss of determinism Determinism in numerical computing will be gone. It is not reasonable to ask for exactness in numerical computation...we may not ask for repeatability either.
23 4 Floating point arithmetic: best general purpose approximation The importance of floating point arithmetic will be undiminished. 128 bit plus word lengths, most numerical problems can not be solved symbolically still, still need approximations.
24 5 The quest for speed in matrix multiplication Linear systems of equations will be solved in time O(N^{2+e}) Complexity of matrix multiplication = complexity of almost all matrix problems: inverse, determinants, solve linear systems... How fast can we multiply two n by n matrices? Standard O(N^3). Strassen s algorithm O(N^2.81). Coppersmith and Winograd s algorithm O(N^2.38)...Is O(N^2) achievable?
25 6 Multipole methods Multipole methods and their descendants will be ubiquitous. Speed up the calculation of long-ranged forces in the n-body problem. Large-scale numerical computations rely more on approximate algorithms...more robust and faster than exact ones.
26 7 seamless interoperability The dream of seamless interoperability will have been achieved. No separation between numerical and symbolic calculations, work across different discretizations and grids, removing humans from the loop.
27 8 massive parallel computing The problem of massively parallel computing will have been blown open by ideas related to human brain. Understanding human brain and its implications for computing.
28 9 New programming methods Our methods of programming will have been blown open by ideas related to genomes and natural selection. Think digitally about the evolution of life in earth.
29 For your procrastination reading list 1. Wikipedia: 2. [Trfethen 2000] Lyoyd N. Trefethen: Predictions for Scientific Computing Fifty Years From Now [Cipra 2000] Barry A. Cipra: The Best of the 20th Century: Editors Name Top 10 Algorithms
30 Take home message 1. Think Big: how SC could transform your research? 2. Keep your eyes open: identify the newest advancement in SC. We will revisit the prediction by the end of the semester. 3. Master the fundamentals: practice makes perfect. 4. Have some fun while learning!
31 THANKS! Any questions? You can us at 1. Instructor: 2. TA:
32 Extra Notes So it goes.
33 Credits Special thanks to all the people who made and released these awesome resources for free: Presentation template by SlidesCarnival Photographs by Unsplash
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