Lect02-2010 - ProjectionanditsImportance...

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  Slide 1 / 39 Projection and its Importance in Scientific Computing ________________________________________________ CS 594 Lecture Notes 01/20/2010 Stan Tomov EECS Department The University of Tennessee, Knoxville January 20, 2010
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  Slide 2 / 39 Contact information     office :  Claxton 330     phone : (865) 974-6317     email  :  tomov@cs.utk.edu Additional reference materials: [1] R.Barrett, M.Berry, T.F.Chan, J.Demmel, J.Donato, J. Dongarra, V. Eijkhout,       R.Pozo, C.Romine, and H.Van der Vorst,  Templates for the Solution of Linear Systems:       Building Blocks for Iterative Methods (2 nd  edition)       http://netlib2.cs.utk.edu/linalg/html_templates/Templates.html [2] Yousef Saad , Iterative methods for sparse linear systems (1st edition)        http://www-users.cs.umn.edu/~saad/books.html       
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  Slide 3 / 39 Topics (as related to  scientific computing )        Projection in Scientific Computing               PDEs, Numerical  solution, Tools, etc.             Sparse matrices,  parallel implementations                     Iterative Methods  
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  Slide 4 / 39 Outline Part I – Fundamentals Part II – Projection in Linear Algebra Part III – Projection in Functional Analysis (e.g. PDEs) HPC with Multicore and GPUs 
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  Slide 5 / 39 Part I Fundamentals
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  Slide 6 / 39 What is Projection? Here are two examples        (from linear algebra)                                       (from functional analysis)   u Pu e P : orthogonal projection of        vector u on e 0 1 1 e = 1 Pu u = f(x) P : best approximation (projection) of f(x)      in span{ e }   C[0,1] The error (u – Pu) to be orthogonal to vector e
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  Slide 7 / 39 Definition Projection  is a linear transformation P from                       a linear space V to itself such that                                P 2  = P equivalently               Let V is direct sum of subspaces V 1  and V 2                                V = V 1    V 2               i.e. for   u   V there are unique u 1  V 1  and u 2 V 2  s.t.                                   u = u 1 +u 2                        Then P: V V 1  is defined for    u   V as  Pu   u 1  
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  Slide 8 / 39 Importance in Scientific  Computing To compute approximations Pu   u where   dim V 1  <<  dim V                           V = V 1    V 2 When computation directly in V is not feasible or even possible.
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This note was uploaded on 04/01/2010 for the course COMPUTER S cs202 taught by Professor Jiuhui during the Spring '08 term at 東京国際大学.

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Lect02-2010 - ProjectionanditsImportance...

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