lecture_28 - Introduction to Scientific Computing Volodymyr...

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National Center for Supercomputing Applications University of Illinois at Urbana-Champaign Introduction to Scientific Computing Volodymyr Kindratenko
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± This lecture is entirely based on the first chapter of Scientific Computing: An Introductory Survey by Michael T. Heath http://www.cse.illinois.edu/heath/scicomp/
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Lecture Topics ± Scientific Computing ± Approximations in scientific computing ± Computer Arithmetic ± Survey of scientific computing methods
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Scientific Computing ± What is scientific computing ? ± Design and analysis of algorithms for numerically solving mathematical problems in science and engineering ± Traditionally called numerical analysis ± Distinguishing features of scientific computing ± Deals with continuous quantities ± 7LPH² GLVWDQFH² VSHHG² « DUH FRQWLQXRXV LQ QDWXUH ± Considers effects of approximations ± Data is not precise, digital computers perform only approximate calculations ± Why scientific computing? ± Simulation of natural phenomena ± Virtual prototyping of engineering designs
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General Strategy ± Replace difficult problem by easier one having same or closely related solution ± LQILQLWH ĺ ILQLWH ± GLIIHUHQWLDO ĺ DOJHEUDLF ± QRQOLQHDU ĺ OLQHDU ± FRPSOLFDWHG ĺ VLPSOH ± Solution obtained may only approximate that of original problem ± Knowing how accurate the solution is becomes important
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Well-Posed Problems ± Problem is well-posed if solution ± exists ± is unique ± depends continuously on problem data ± Otherwise, problem is ill-posed ± Even if problem is well posed, solution may still be sensitive to input data ± Computational algorithm should not make sensitivity worse
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Sources of Approximation ± Before computation ± Modeling ± Some features may be simplified or omitted ± Empirical measurements ± Lab instruments have finite precision ± Previous computations ± May be only approximate ± During computation ± Truncation or discretization ± Some features may be omitted or simplified ± Rounding ± Computer representation of real numbers is inexact ± Accuracy of final result reflects all these
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Example: Approximations ± Computing surface area of Earth using formula A = 4 S r 2 involves several approximations ± Earth is modeled as sphere, idealizing its true shape ± Value for radius (~6370 km) is based on empirical measurements and previous computations ± Value for S requires truncating infinite process ± Values for input data and results of arithmetic operations are rounded in computer ± The accuracy of the computation depends on all these approximations
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Data Error and Computational Error ± Some errors are due to the input data , some are due to the computational process ± Typical problem: compute value of function f : ± ĺ ± for given argument ± x = true value of input ± f(x) = desired result ± x = approximate (inexact) input ± f = approximate function actually computed ± Total error: f ( x ) í I²[³ = ( f ( x ) í f( x )) + (f( x ³ í I²[³³ ± f ( x ³ í I² x ) - computational error ± f( x ³ í I²[ ) - propagated data error
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lecture_28 - Introduction to Scientific Computing Volodymyr...

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