Scientific Computing

Jul 20, 2014

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Scientific Computing. Dr. Guy Tel- Zur Lecture 8. Agenda. MHJ Chapter 8: The Metropolis Algorithm HPC in the Cloud VisIt mini course http:// visitusers.org/index.php?title=SC10_Tutorial Parallel methods in Matlab R mini course ( + Elastic-R). Administration. Home assignment #1

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Scientific Computing

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Scientific Computing Dr. Guy Tel-Zur Lecture 8
Agenda • MHJ Chapter 8: The Metropolis Algorithm • HPC in the Cloud • VisItmini course http://visitusers.org/index.php?title=SC10_Tutorial • Parallel methods in Matlab • R mini course ( + Elastic-R)
Administration • Home assignment #1 • The Final Projects • Next week – holiday - no class

Scientific Computing

Scientific Computing. Lecture 6 Dr. Guy Tel- Zur. Butterfly in Africa by Anna Cervova , http://www.publicdomainpictures.net. Agenda. MPI demo Mandelbrot set (again)

343 views • 19 slides

Scientific Computing

Scientific Computing. Partial Differential Equations Poisson Equation Calculus of Variations. Finite Differences.

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“Scientific Computing”

Simulation. Application area. Visualization. “Scientific Computing”. Mathematics. // - computing. Numerics. Software- engineering. Computer Science. Alternative 1 Real World Simplified model Analytical solution Exact solution to “simple model”. Alternative 2 Real World

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Scientific Computing

Scientific Computing. Introduction to Matlab Programming. Today. We will introduce the programming environment for our course: Matlab We will cover: Basic Operations, Functions, Plotting Vectors, Looping Creating a new Function in a M-file. Why Matlab?.

498 views • 26 slides

Scientific Computing

Scientific Computing. Dr. Guy Tel- Zur. Version 28-05-2011.12:40. Agenda. Administration Visualization Visualization in Sage VisIt : Tutorial The Silo file format A demo program Animation Demo : AWS – HTC Instance for Parallel Computing and Cellular Automata / NKS demo on the cloud

471 views • 29 slides

Scientific Computing

Scientific Computing. Linear Least Squares. Interpolation vs Approximation. Recall : Given a set of (x,y) data points, Interpolation is the process of finding a function (usually a polynomial) that passes through these data points. . Interpolation vs Approximation.

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Scientific Computing

Scientific Computing. Numerical Solution Of Ordinary Differential Equations - Further Analysis . Solving First Order ODE’s. A general form for a first order ODE is Or alternatively We desire a solution x(t) which satisfies this ODE and one specified boundary condition.

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Scientific Computing

Scientific Computing. Numerical Solution Of Ordinary Differential Equations - Runge-Kutta Methods. Runge-Kutta Methods. If we take more terms in the Taylor’s Series expansion of f(t,x), we would get a more accurate solution algorithm.

626 views • 30 slides

Scientific Computing

Scientific Computing. QR Factorization Part 2 – Algorithm to Find Eigenvalues. Similar Matrices. Definition : A and B are similar matrices if and only if there exists a nonsingular matrix P such that B = P -1 AP . (or PBP -1 = A )

281 views • 19 slides

Scientific Computing

Scientific Computing. Singular Value Decomposition SVD. SVD - Overview. SVD is a way to decompose singular (or nearly singular) matrices, i.e. matrices that do not have inverses. This includes square matrices whose determinant is zero (or nearly zero) and all rectangular matrices.

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Scientific Computing

Scientific Computing. Linear Systems – Gaussian Elimination. Linear Systems. Linear Systems. Solve Ax=b, where A is an n  n matrix and b is an n  1 column vector We can also talk about non-square systems where A is m  n , b is m  1, and x is n  1

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Scientific Computing

Scientific Computing. Interpolation – Divided Differences Efficiency and Error Analysis. How is Interpolation used?. Here is a table of values for the temperature at a given depth for a lake. We want a good estimate of the temperature at a depth of 7.5 meters. Newton’s Nested Formula.

369 views • 24 slides

Scientific Computing

Scientific Computing. By: Fatima Hallak To: Dr. Guy Tel-Zur. SPIN Spin home page. Spin targets efficient software verification. Supports a high level language called PROMELA (a PROcess MEta LAnguage). ً Written in ANSI standard C

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Scientific Computing

Scientific Computing. Matrix Norms, Convergence, and Matrix Condition Numbers. Vector Norms. A vector norm is a quantity that measures how large a vector is (the magnitude of the vector). For a number x, we have |x| as a measurement of the magnitude of x.

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Scientific Computing

Scientific Computing. Ron Elber. Keshav Pingali. Charlie Van Loan. Steve Vavasis. Time Scale Problems in the Popular Molecular Dynamics Approach. seconds. Action optimization generates approximate trajectories. Channel Gating. Fast folding. Protein Activation. Slow folding.

208 views • 12 slides

Scientific Computing

Scientific Computing. QR Factorization Part 1 – Orthogonal Matrices, Reflections and Rotations. Orthogonal Matrices. Matrix Factoring : So far, we have seen that we can factor a non-singular matrix A as A = LU, where L is lower triangular and U is upper triangular.

411 views • 19 slides

Scientific Computing

Scientific Computing. Multi-Step and Predictor-Corrector Methods. Overview. One-Step Methods – use only info from previous step Euler Runge-Kutta Multistep Methods- use info from several prior steps Adam Bashforth Adam Moulton Method Predictor-Corrector Method. Multi-Step Principle.

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Scientific Computing

Scientific Computing. Cubic Splines. Cubic Spline. Definition : A function S is a cubic spline on [a,b] if The domain of S is [a,b] S is continuous on [a,b] S’ is continuous on [a,b] S’’ is continuous on [a,b]

270 views • 12 slides

Scientific Computing

Scientific Computing. Numerical Solution Of Ordinary Differential Equations - Euler’s Method. Bungee Jumper – Coupled ODEs. Bungee Jumper – Coupled ODEs. We want to model the vertical dynamics of a jumper connected to a stationary platform with a bungee cord. F=ma Forces :

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Scientific Computing

Scientific Computing. Algorithm Convergence and Root Finding Methods. Algorithm Convergence.

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Scientific Computing

Scientific Computing. Matrix Condition Numbers. Matrix Condition Number. Multiplication of a vector x by a matrix A results in a new vector A x that can have a very different norm from x . The range of the possible change can be expressed by two numbers,

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Scientific Computing

Scientific Computing. Linear and Quadratic Splines. Splines Overview. We have looked at two methods of finding interpolating polynomials for a set of data points – Lagrange polynomials and Newton’s method .

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