DOCSLIB.ORG

Interpolation and Approximation Interpolation · Least-Squares Techniques Interpolation General & Polynomial Interpolation Interpolation Problem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Interpolation and Approximation Interpolation · Least-Squares Techniques Interpolation General & Polynomial Interpolation Interpolation Problem Geometric Modeling Summer Semester 2012 Interpolation and Approximation Interpolation · Least-Squares Techniques Interpolation General & Polynomial Interpolation Interpolation Problem First approach to modeling smooth objects: • Given a set of points along a curve or surface • Choose basis functions that span a suitable function space . Smooth basis functions . Any linear combination will be smooth, too • Find a linear combination such that the curve/surface interpolates the given points 3 / 85 Interpolation Problem Different types of interpolation: • Nearest • Linear • Polynomial 4 / 85 General Formulation Settings: ds • Domain R , mapping to R. f (x) • Looking for a function f: R. x2 • Basis set: B = {b1,...,bn}, bi: R. x1 • Represent f as linear combination of basis functions: n 1 f (x) b (x) , i.e. f is just determined by λ i i λ i1 n ds • Function values: {(x1, y1), ..., (xn, yn)}, (xi, yi) R R • We want to find such that: i {1,...,n}: fλ (xi ) yi 5 / 85 Illustration R f: R R y1 b2 y3 b1 y2 b3 x1 x2 x3 1D Example n f (x) b (x) i i i {1,...,n}: f (xi ) yi i1 6 / 85 Solving the Interpolation Problem Solution: linear system of equations • Evaluate basis functions at points xi: n i {1,...,n}: b (x ) y i i i i i1 • Matrix form: b (x ) b (x ) y 1 1 n 1 1 1 b1(xn ) bn(xn )n yn 7 / 85 Illustration y2 = R 3 b 2 y1 = y3 = b1 3 b3 2 2 3 1 2 1 1 x1 x2 x3 interpolation problem linear system 8 / 85 Illustration y2 = R 3 b 2 y1 = y3 = b1 3 b3 2 2 3 1 2 1 1 x1 x2 x3 interpolation problem linear system anything in between does not matter (determined by basis only) 9 / 85 Example Example: Polynomial Interpolation • Monomial basis B = {1, x, x2, x3, ..., xn-1} • Linear system to solve: 1 x x n1 1 1 y n1 1 1 1 x2 x2 “Vandermonde Matrix” y n1 n n 1 xn xn 10 / 85 Example with Numbers Example with numbers • Quadratic monomial basis B = {1, x, x2} • Function values: {(0,2), (1,0), (2,3)} [(x, y)] • Linear system to solve: 1 0 0 2 1 (2,3) 1 1 12 0 (0,2) 1 2 43 3 • Result: 1 = 2, 2 = -9/2, 3 = 5/2 (1,0) 11 / 85 Condition Number... The interpolation problem is ill conditioned: • For equidistant xi, the condition number of the Vandermode matrix grows exponentially with n (maximum degree+1 = number of points to interpolate) cond 2,5E+17 cond 1,0E+18 1,0E+16 2,0E+17 1,0E+14 1,0E+12 1,5E+17 1,0E+10 1,0E+08 1,0E+17 1,0E+06 (logarithmic) 5,0E+16 1,0E+04 1,0E+02 0,0E+00 1,0E+00 0 5 10 15 20 25 0 5 10 15 20 25 #points #points 12 / 85 Why ill-conditioned? • Solution with inverse Vandermonde matrix: Mx = y => x = M-1y = (V D-1 UT)y 2.5 0 0 0 0 1.1 0 0 D: 0 0 0.9 0 0 0 0 0.000000001 • Condition number defined as a ratio between largest and 2,5E+17smallest singular value: max /min cond 2,0E+17 1,5E+17 1,0E+17 5,0E+16 0,0E+00 0 5 10 15 20 25 #points Why ill-conditioned? Monomial Basis: • Functions become 2 1,8 increasingly indistinguishable 1,6 with degree (non orthogonal) 1,4 1,2 • Only differ in growing rate 1 (xi growth faster than xi-1) 0,8 0,6 • For higher degrees numerical 0,4 0,2 precision became a key factor 0 0 0,5 1 1,5 2 Monomial basis 14 / 85 The Cure... This problem can be fixed: • Use orthogonal polynomial basis • How to get one? e.g. Gram-Schmidt orthogonalization (see assignment sheet #1) • Legendre polynomials – orthonormal on [-1..1] • Much better condition of the linear system (converges to 1) 16 / 85 However... This does not fix all problems: • Polynomial interpolation is instable . “Runge’s phenomenon”: Oscillating behavior . Small changes in control points can lead to very different result. xi sequence important. • Weierstraß approximation theorem: . Smooth functions (C0) can be approximated arbitrarily well with polynomials . However: Need carefully chosen construction for convergence . Not useful in practice 17 / 85 Runge’s Phenomenon 18 / 85 Conclusion Conclusion: Need a better basis for interpolation For example, piecewise polynomials will work much better % Splines 19 / 85 Approximation (Reweighted) Least-squares, Scattered Data Approximation Common Situation: • We have many data points, they might be noisy • Example: Scanned data • Want to approximate the data with a smooth curve / surface What we need: • Criterion – what is a good approximation? • Methods to compute this approximation 21 / 85 Least-Squares We assume the following scenario: • We have a set of function values yi at positions xi. (1D 1D for now) • The independent variables xi are known exactly. • The dependent variables yi are known approximately, with some error. • The error is normal distributed, independent, and with the same distribution at every point (normal noise). • We know the class of functions from which the noisy samples were taken. 23 / 85 Situation y 1 yn y2 x1 x2 xn Situation: • Original sample points taken at xi from original f. • Unknown Gaussian noise added to each yi. • Want to estimated reconstructed f~. 24 / 85 Maximum Likelihood Estimation Goal: • Maximize the probability that the data originated from the reconstructed curve f~ fits the points • “Maximum likelihood estimation” 1 x 2 p (x) exp , 2 2π 2 Gaussian normal distribution 25 / 85 Maximum Likelihood Estimation ~ n ~ n 1 ( f (x ) y )2 arg max N ( f (x ) y ) arg max exp i i ~ 0, i i ~ 2 f i1 f i1 2π 2 ~ n 1 ( f (x ) y )2 arg max ln exp i i ~ 2 f i1 2π 2 ~ n 1 ( f (x ) y )2 arg max ln i i ~ 2 f i1 2π 2 ~ n ( f (x ) y )2 arg min i i ~ 2 f i1 2 n ~ arg min ( f (x ) y )2 ~ i i f i1 26 / 85 Maximum Likelihood Estimation ~ n ~ n 1 ( f (x ) y )2 arg max N ( f (x ) y ) arg max exp i i ~ 0, i i ~ 2 f i1 f i1 2π 2 ~ n 1 ( f (x ) y )2 arg max ln exp i i ~ 2 f i1 2π 2 ~ n 1 ( f (x ) y )2 arg max ln i i ~ 2 f i1 2π 2 ~ n ( f (x ) y )2 arg min i i ~ 2 f i1 2 n ~ arg min ( f (x ) y )2 ~ i i f i1 27 / 85 Least-Squares Approximation This shows: • The solution with maximum likelihood in the considered scenario (y-direction, iid Gaussian noise) minimizes the sum of squared errors. Next: Compute optimal coefficients ~ k • Linear ansatz: f x: j bj x j1 • Task: determine optimal i 29 / 85 Maximum Likelihood Estimation Compute optimal coefficients: 2 n ~ n k arg min ( f (x ) y )2 arg min b (x ) y i i j j i i λ i1 λ i1 j1 n T 2 arg minλ b(xi ) yi λ i1 n n n arg minλ T b(x )bT(x ) λ 2 y λ Tb(x ) y 2 i i i i i λ i1 i1 i1 xTAx bx c Quadratic optimization problem 30 / 85 Critical Point 1 b1(x) bi (x1 ) y1 λ : k entries , b(x): k entries , bi : n entries, y : n entries k bk (x) bi (xn ) yn n n n λ T b(x )bT (x ) λ 2 y λ Tb(x ) y 2 λ i i i i i i1 i1 i1 y Tb n 1 T 2b(xi )b (xi )λ 2 i1 T y bk We obtain a linear system of equations: y Tb n 1 T b(xi )b (xi )λ i1 T y bk 31 / 85 Critical Point This can also be written as: b1 ,b1 b1 ,bk 1 y,b1 b ,b b ,b y,b k 1 k k k k with: n bi ,b j : bi (xt )bj (xt ) t 1 n y,bi : bi (xt ) yt t 1 32 / 85 Summary Statistical model yields least-squares criterion: n n ~ ~ 2 arg max N0, ( f (xi ) yi ) arg min ( f (x ) y ) ~ ~ i i f i1 f i1 Linear function space leads to quadratic objective: 2 ~ k n k f x : b x arg min b (x ) y j j j j i i j1 λ i1 j1 Critical point: linear system n b ,b b ,b y,b b ,b : b (x )b (x ) 1 1 1 k 1 1 i j i t j t t 1 with: n y,b : b (x ) y bk ,b1 bk ,bk k y,bk i i t t t 1 33 / 85 Variants Weighted least squares: • In case the data point’s noise has different standard deviations at the different data points • This gives a weighted least squares problem • Noisier points have smaller influence 34 / 85 Same procedure as prev. slides... ~ n ~ n 1 ( f (x ) y )2 arg max N ( f (x ) y ) arg max exp i i ~ i i ~ 2 f i1 f i1 i 2π 2 i ~ n 1 ( f (x ) y )2 arg max log exp i i ~ 2 f i1 i 2π 2 i ~ n 1 ( f (x ) y )2 arg max log i i ~ 2 f i1 i 2π 2 i ~ n ( f (x ) y )2 arg min i i ~ 2 f i1 2 i n 1 ~ arg min ( f (x ) y )2 ~ 2 i i f i1 i weights 35 / 85 Result Linear system for the general case: n b ,b b ,b y,b b ,b : b (x )b (x ) 2 x 1 1 1 n 1 1 i j i i j i i with: l1 n y,b : b (x ) y 2 x bn ,b1 bn ,bn n y,bn i i i i i l1 2 1 1 xi 2 , i.e.
Load more

Recommended publications
  • On Multivariate Interpolation
    On Multivariate Interpolation Peter J. Olver† School of Mathematics University of Minnesota Minneapolis, MN 55455 U.S.A. [email protected] http://www.math.umn.edu/∼olver Abstract. A new approach to interpolation theory for functions of several variables is proposed. We develop a multivariate divided difference calculus based on the theory of non-commutative quasi-determinants. In addition, intriguing explicit formulae that connect the classical finite difference interpolation coefficients for univariate curves with multivariate interpolation coefficients for higher dimensional submanifolds are established. † Supported in part by NSF Grant DMS 11–08894. April 6, 2016 1 1. Introduction. Interpolation theory for functions of a single variable has a long and distinguished his- tory, dating back to Newton’s fundamental interpolation formula and the classical calculus of finite differences, [7, 47, 58, 64]. Standard numerical approximations to derivatives and many numerical integration methods for differential equations are based on the finite dif- ference calculus. However, historically, no comparable calculus was developed for functions of more than one variable. If one looks up multivariate interpolation in the classical books, one is essentially restricted to rectangular, or, slightly more generally, separable grids, over which the formulae are a simple adaptation of the univariate divided difference calculus. See [19] for historical details. Starting with G. Birkhoff, [2] (who was, coincidentally, my thesis advisor), recent years have seen a renewed level of interest in multivariate interpolation among both pure and applied researchers; see [18] for a fairly recent survey containing an extensive bibli- ography. De Boor and Ron, [8, 12, 13], and Sauer and Xu, [61, 10, 65], have systemati- cally studied the polynomial case.
    [Show full text]
  • Polynomial Interpolation
    Numerical Methods 401-0654 6 Polynomial Interpolation Throughout this chapter n N is an element from the set N if not otherwise specified. ∈ 0 0 6.1 Polynomials D-ITET, ✬ ✩D-MATL Definition 6.1.1. We denote by n n 1 n := R t αnt + αn 1t − + ... + α1t + α0 R: α0, α1,...,αn R . P ∋ 7→ − ∈ ∈ the R-vectorn space of polynomials with degree n. o ≤ ✫ ✪ ✬ ✩ Theorem 6.1.2 (Dimension of space of polynomials). It holds that dim n = n +1 and n P P ⊂ 6.1 C∞(R). p. 192 ✫ ✪ n n 1 ➙ Remark 6.1.1 (Polynomials in Matlab). MATLAB: αnt + αn 1t − + ... + α0 Vector Numerical − Methods (αn, αn 1,...,α0) (ordered!). 401-0654 − △ Remark 6.1.2 (Horner scheme). Evaluation of a polynomial in monomial representation: Horner scheme p(t)= t t t t αnt + αn 1 + αn 2 + + α1 + α0 . ··· − − ··· Code 6.1: Horner scheme, polynomial in MATLAB format 1 function y = polyval (p,x) 2 y = p(1); for i =2: length (p), y = x y+p( i ); end ∗ D-ITET, D-MATL Asymptotic complexity: O(n) Use: MATLAB “built-in”-function polyval(p,x); △ 6.2 p. 193 6.2 Polynomial Interpolation: Theory Numerical Methods 401-0654 Goal: (re-)constructionofapolynomial(function)frompairs of values (fit). ✬ Lagrange polynomial interpolation problem ✩ Given the nodes < t < t < < tn < and the values y ,...,yn R compute −∞ 0 1 ··· ∞ 0 ∈ p n such that ∈P j 0, 1,...,n : p(t )= y . ∀ ∈ { } j j ✫ ✪ D-ITET, D-MATL 6.2 p. 194 6.2.1 Lagrange polynomials Numerical Methods 401-0654 ✬ ✩ Definition 6.2.1.
    [Show full text]
  • Sparse Polynomial Interpolation and Testing
    Sparse Polynomial Interpolation and Testing by Andrew Arnold A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philsophy in Computer Science Waterloo, Ontario, Canada, 2016 © Andrew Arnold 2016 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Interpolation is the process of learning an unknown polynomial f from some set of its evalua- tions. We consider the interpolation of a sparse polynomial, i.e., where f is comprised of a small, bounded number of terms. Sparse interpolation dates back to work in the late 18th century by the French mathematician Gaspard de Prony, and was revitalized in the 1980s due to advancements by Ben-Or and Tiwari, Blahut, and Zippel, amongst others. Sparse interpolation has applications to learning theory, signal processing, error-correcting codes, and symbolic computation. Closely related to sparse interpolation are two decision problems. Sparse polynomial identity testing is the problem of testing whether a sparse polynomial f is zero from its evaluations. Sparsity testing is the problem of testing whether f is in fact sparse. We present effective probabilistic algebraic algorithms for the interpolation and testing of sparse polynomials. These algorithms assume black-box evaluation access, whereby the algorithm may specify the evaluation points. We measure algorithmic costs with respect to the number and types of queries to a black-box oracle.
    [Show full text]
  • Sparse Polynomial Interpolation Codes and Their Decoding Beyond Half the Minimum Distance *
    Sparse Polynomial Interpolation Codes and their Decoding Beyond Half the Minimum Distance * Erich L. Kaltofen Clément Pernet Dept. of Mathematics, NCSU U. J. Fourier, LIP-AriC, CNRS, Inria, UCBL, Raleigh, NC 27695, USA ÉNS de Lyon [email protected] 46 Allée d’Italie, 69364 Lyon Cedex 7, France www4.ncsu.edu/~kaltofen [email protected] http://membres-liglab.imag.fr/pernet/ ABSTRACT General Terms: Algorithms, Reliability We present algorithms performing sparse univariate pol- Keywords: sparse polynomial interpolation, Blahut's ynomial interpolation with errors in the evaluations of algorithm, Prony's algorithm, exact polynomial fitting the polynomial. Based on the initial work by Comer, with errors. Kaltofen and Pernet [Proc. ISSAC 2012], we define the sparse polynomial interpolation codes and state that their minimal distance is precisely the code-word length 1. INTRODUCTION divided by twice the sparsity. At ISSAC 2012, we have Evaluation-interpolation schemes are a key ingredient given a decoding algorithm for as much as half the min- in many of today's computations. Model fitting for em- imal distance and a list decoding algorithm up to the pirical data sets is a well-known one, where additional minimal distance. information on the model helps improving the fit. In Our new polynomial-time list decoding algorithm uses particular, models of natural phenomena often happen sub-sequences of the received evaluations indexed by to be sparse, which has motivated a wide range of re- an arithmetic progression, allowing the decoding for a search including compressive sensing [4], and sparse in- larger radius, that is, more errors in the evaluations terpolation of polynomials [23,1, 15, 13,9, 11].
    [Show full text]
  • Chinese Remainder Theorem
    THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens 1. Introduction The Chinese remainder theorem says we can uniquely solve every pair of congruences having relatively prime moduli. Theorem 1.1. Let m and n be relatively prime positive integers. For all integers a and b, the pair of congruences x ≡ a mod m; x ≡ b mod n has a solution, and this solution is uniquely determined modulo mn. What is important here is that m and n are relatively prime. There are no constraints at all on a and b. Example 1.2. The congruences x ≡ 6 mod 9 and x ≡ 4 mod 11 hold when x = 15, and more generally when x ≡ 15 mod 99, and they do not hold for other x. The modulus 99 is 9 · 11. We will prove the Chinese remainder theorem, including a version for more than two moduli, and see some ways it is applied to study congruences. 2. A proof of the Chinese remainder theorem Proof. First we show there is always a solution. Then we will show it is unique modulo mn. Existence of Solution. To show that the simultaneous congruences x ≡ a mod m; x ≡ b mod n have a common solution in Z, we give two proofs. First proof: Write the first congruence as an equation in Z, say x = a + my for some y 2 Z. Then the second congruence is the same as a + my ≡ b mod n: Subtracting a from both sides, we need to solve for y in (2.1) my ≡ b − a mod n: Since (m; n) = 1, we know m mod n is invertible.
    [Show full text]
  • Anl-7826 Chinese Remainder Ind Interpolation
    ANL-7826 CHINESE REMAINDER IND INTERPOLATION ALGORITHMS John D. Lipson UolC-AUA-USAEC- ARGONNE NATIONAL LABORATORY, ARGONNE, ILLINOIS Prepared for the U.S. ATOMIC ENERGY COMMISSION under contract W-31-109-Eng-38 The facilities of Argonne National Laboratory are owned by the United States Govern­ ment. Under the terms of a contract (W-31 -109-Eng-38) between the U. S. Atomic Energy 1 Commission. Argonne Universities Association and The University of Chicago, the University employs the staff and operates the Laboratory in accordance with policies and programs formu­ lated, approved and reviewed by the Association. MEMBERS OF ARGONNE UNIVERSITIES ASSOCIATION The University of Arizona Kansas State University The Ohio State University Carnegie-Mellon University The University of Kansas Ohio University Case Western Reserve University Loyola University The Pennsylvania State University The University of Chicago Marquette University Purdue University University of Cincinnati Michigan State University Saint Louis University Illinois Institute of Technology The University of Michigan Southern Illinois University University of Illinois University of Minnesota The University of Texas at Austin Indiana University University of Missouri Washington University Iowa State University Northwestern University Wayne State University The University of Iowa University of Notre Dame The University of Wisconsin NOTICE This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontrac­ tors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately-owned rights.
    [Show full text]
  • Polynomial Interpolation: Summary ∏
    Polynomial Interpolation 1 Polynomial Interpolation: Summary • Statement of problem. We are given a table of n + 1 data points (xi,yi): x x0 x1 x2 ... xn y y0 y1 y2 ... yn and seek a polynomial p of lowest degree such that p(xi) = yi for 0 ≤ i ≤ n Such a polynomial is said to interpolate the data. • Finding the interpolating polynomial using the Vandermonde matrix. 2 n Here, pn(x) = ao +a1x +a2x +...+anx where the coefficients are found by imposing p(xi) = yi for each i = 0 to n. The resulting system is 2 n 1 x0 x0 ... x0 a0 y0 1 x x2 ... xn a y 1 1 1 1 1 1 x x2 ... xn a y 2 2 2 2 = 2 . . .. . . . . . 2 n 1 xn xn ... xn an yn The matrix above is called the Vandermonde matrix. If this was singular it would imply that for some nonzero set of coefficients the associated polynomial of degree ≤ n would have n + 1 zeros. This can’t be so this matrix equation can be solved for the unknown coefficients of the polynomial. • The Lagrange interpolation polynomial. n x − x p (x) = y ` (x) + y ` (x) + ... + y ` (x) where ` (x) = ∏ j n 0 0 1 1 n n i x − x j = 0 i j j 6= i • The Newton interpolation polynomial (Divided Differences) pn(x) = c0 + c1(x − x0) + c2(x − x0)(x − x1) + ... + cn(x − x0)(x − x1)···(x − xn−1) where these coefficients will be found using divided differences. It is fairly clear however that c0 = y0 and c = y1−y0 .
    [Show full text]
  • Polynomial Interpolation: Standard Lagrange Interpolation, Barycentric Lagrange Interpolation and Chebyshev Interpolation
    Polynomial 9 Interpolation Lab Objective: Learn and compare three methods of polynomial interpolation: standard Lagrange interpolation, Barycentric Lagrange interpolation and Chebyshev interpolation. Explore Runge's phe- nomenon and how the choice of interpolating points aect the results. Use polynomial interpolation to study air polution by approximating graphs of particulates in air. Polynomial Interpolation Polynomial interpolation is the method of nding a polynomial that matches a function at specic points in its range. More precisely, if f(x) is a function on the interval [a; b] and p(x) is a poly- nomial then p(x) interpolates the function f(x) at the points x0; x1; : : : ; xn if p(xj) = f(xj) for all j = 0; 1; : : : ; n. In this lab most of the discussion is focused on using interpolation as a means of approximating functions or data, however, polynomial interpolation is useful in a much wider array of applications. Given a function and a set of unique points n , it can be shown that there exists f(x) fxigi=0 a unique interpolating polynomial p(x). That is, there is one and only one polynomial of degree n that interpolates f(x) through those points. This uniqueness property is why, for the remainder of this lab, an interpolating polynomial is referred to as the interpolating polynomial. One approach to nding the unique interpolating polynomial of degree n is Lagrange interpolation. Lagrange interpolation Given a set n of points to interpolate, a family of basis functions with the following property fxigi=1 n n is constructed: ( 0 if i 6= j Lj(xi) = : 1 if i = j The Lagrange form of this family of basis functions is n Y (x − xk) k=1;k6=j (9.1) Lj(x) = n Y (xj − xk) k=1;k6=j 1 2 Lab 9.
    [Show full text]
  • Change of Basis in Polynomial Interpolation
    NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2000; 00:1–6 Prepared using nlaauth.cls [Version: 2000/03/22 v1.0] Change of basis in polynomial interpolation W. Gander Institute of Computational Science ETH Zurich CH-8092 Zurich Switzerland SUMMARY Several representations for the interpolating polynomial exist: Lagrange, Newton, orthogonal polynomials etc. Each representation is characterized by some basis functions. In this paper we investigate the transformations between the basis functions which map a specific representation to another. We show that for this purpose the LU- and the QR decomposition of the Vandermonde matrix play a crucial role. Copyright c 2000 John Wiley & Sons, Ltd. key words: Polynomial interpolation, basis change 1. REPRESENTATIONS OF THE INTERPOLATING POLYNOMIAL Given function values x x , x , ··· x 0 1 n , f(x) f0, f1, ··· fn with xi 6= xj for i 6= j. There exists a unique polynomial Pn of degree less or equal n which interpolates these values, i.e. Pn(xi) = fi, i = 0, 1, . , n. (1) Several representations of Pn are known, we will present in this paper the basis transformations among four of them. 1.1. Monomial Basis We consider first the monomials m(x) = (1, x, x2, . , xn)T and the representation n−1 n Pn(x) = a0 + a1x + ... + an−1x + anx . †Please ensure that you use the most up to date class file, available from the NLA Home Page at http://www.interscience.wiley.com/jpages/1070-5325/ Contract/grant sponsor: Publishing Arts Research Council; contract/grant number: 98–1846389 Received ?? Copyright c 2000 John Wiley & Sons, Ltd.
    [Show full text]
  • Projection and Its Importance in Scientific Computing ______
    Projection and its Importance in Scientific Computing ________________________________________________ Stan Tomov EECS Department The University of Tennessee, Knoxville March 22, 2017 COCS 594 Lecture Notes Slide 1 / 41 03/22/2017 Contact information office : Claxton 317 phone : (865) 974-6317 email : [email protected] 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 (2nd 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 Slide 2 / 41 Topics as related to high-performance scientific computing‏ Projection in Scientific Computing Sparse matrices, PDEs, Numerical parallel implementations solution, Tools, etc. Iterative Methods Slide 3 / 41 Topics on new architectures – multicore, GPUs (CUDA & OpenCL), MIC Projection in Scientific Computing Sparse matrices, PDEs, Numerical parallel implementations solution, Tools, etc. Iterative Methods Slide 4 / 41 Outline Part I – Fundamentals Part II – Projection in Linear Algebra Part III – Projection in Functional Analysis (e.g. PDEs) HPC with Multicore and GPUs Slide 5 / 41 Part I Fundamentals Slide 6 / 41 Projection in Scientific Computing [ an example – in solvers for PDE discretizations ] A model leading to self-consistent iteration with need for high-performance diagonalization
    [Show full text]
  • Polynomial, Lagrange, and Newton Interpolation
    Polynomial, Lagrange, and Newton Interpolation Mridul Aanjaneya November 14, 2017 Interpolation We are often interested in a certain function f(x), but despite the fact that f may be defined over an entire interval of values [a; b] (which may be the entire real line) we only know its precise value at select point x1; x2; : : : ; xN . select value samples the actual function f(x) There may be several good reasons why we could only have a limited number of values for f(x), instead of its entire graph: • Perhaps we do not have an analytic formula for f(x) because it is the result of a complex process that is only observed experimentally. For example, f(x) could correspond to a physical quantity (temperature, den- sity, concentration, velocity, etc.) which varies over time in a laboratory experiment. Instead of an explicit formula, we use a measurement device to capture sample values of f at predetermined points in time. 1 • Or, perhaps we do have a formula for f(x), but this formula is not trivially easy to evaluate. Consider for example: x Z 2 f(x) = sin(x) or f(x) = ln(x) or f(x) = e−t dt 0 Perhaps evaluating f(x) with such a formula is a very expensive oper- ation and we want to consider a less expensive way to obtain a \crude approximation". In fact, in years when computers were not as ubiquitous as today, trigonometric tables were very popular. For example: Angle sin(θ) cos(θ) ::: ::: ::: 44◦ 0:695 0:719 45◦ 0:707 0:707 46◦ 0:719 0:695 47◦ 0:731 0:682 ::: ::: ::: If we were asked to approximate the value of sin(44:6◦), it would be natural to consider deriving an estimate from these tabulated values, rather than attempting to write an analytic expression for this quantity.
    [Show full text]
  • Interpolation
    Chapter 11 Interpolation So far we have derived methods for analyzing functions f , e.g. finding their minima and roots. Evaluating f (~x) at a particular ~x 2 Rn might be expensive, but a fundamental assumption of the methods we developed in previous chapters is that we can obtain f (~x) when we want it, regardless of ~x. There are many contexts when this assumption is not realistic. For instance, if we take a pho- tograph with a digital camera, we receive an n × m grid of pixel color values sampling the contin- uum of light coming into a camera lens. We might think of a photograph as a continuous function from image position (x, y) to color (r, g, b), but in reality we only know the image value at nm separated locations on the image plane. Similarly, in machine learning and statistics, often we only are given samples of a function at points where we collected data, and we must interpolate it to have values elsewhere; in a medical setting we may monitor a patient’s response to different dosages of a drug but only can predict what will happen at a dosage we have not tried explicitly. In these cases, before we can minimize a function, find its roots, or even compute values f (~x) at arbitrary locations ~x, we need a model for interpolating f (~x) to all of Rn (or some subset thereof) given a collection of samples f (~xi). Of course, techniques solving this interpolation problem are inherently approximate, since we do not know the true values of f , so instead we seek for the interpolated function to be smooth and serve as a “reasonable” prediction of function values.
    [Show full text]