Lecture VIII. Global Approximation Methods: I
|
|
- Imogene Dorothy Davidson
- 5 years ago
- Views:
Transcription
1 Lecture VIII Global Approximation Methods: I Gianluca Violante New York University Quantitative Macroeconomics G. Violante, Global Methods p. 1 /29
2 Global function approximation Global methods: function interpolation over the entire domain First choice: type of approximation 1. Interpolation I: finite element methods Splines (e.g., B-splines, cubic splines, Schumaker splines) 2. Interpolation II: spectral methods Orthogonal polynomials (e.g., Chebyshev, Lagrange, etc...) Second choice: how to minimize residuals between true function and interpolant 1. Collocation 2. Least squares 3. Galerkin method G. Violante, Global Methods p. 2 /29
3 Interpolation Today, one dimension. We want to represent a real-valued function f on [a,b] with a computationally tractable function Interpolation is a form of function approximation in which the approximating (interpolant) and the true function must agree, i.e. they must have same value, at a finite number of points. In some cases additional restrictions are imposed on the interpolant. E.g., f evaluated at a finite number of points may have to agree with that of the underlying function. Other examples include additional constraints imposed on the interpolant such as monotonicity, convexity, or smoothness requirements. In general, interpolation is any method that takes information at a finite set of points, X, and finds a function which satisfies that information at X. G. Violante, Global Methods p. 3 /29
4 Interpolation We want to approximate a known function f (x). The interpolant ˆf is chosen to be a linear combination of a collection of basis functions {φ j (x)} n j=0 where n is the order/degree of interpolation Basis functions: linearly independent functions that span the family of functions chosen for the interpolation (the function space) any function in the space can be obtained as a linear combination of basis functions, just as every vector in a vector space can be obtained as a linear combination of basis vectors. In general we are interested in the space of continuous or continuously differentiable functions. G. Violante, Global Methods p. 4 /29
5 Interpolation Thus, for a given family of basis function {φ j } n j=0 we have: f(x) ˆf(x) n w j φ j (x) j=0 We have reduced the problem of characterizing an infinite-dimensional object, f, to the problem of determining the n weights (or coefficients) {w j } There is arbitrariness in interpolation: there are arbitrarily many functions passing through a finite number of points. Which basis? Choices: spectral or finite element methods depending whether the basis functions are nonzero over the entire domain of the true function (except possibly at a finite number of points) or nonzero only on a subinterval of the domain. G. Violante, Global Methods p. 5 /29
6 Finite element methods Use local basis These are called splines An order n spline consists of a series of n th order polynomial segments spliced together so as to preserve continuity of derivatives of order n 1 or less. The points at which the pieces are spliced together are called knots or breakpoints. By convention the first and last knots are the two extremes of the function domain. G. Violante, Global Methods p. 6 /29
7 Splines An order n spline with K +1 knots has K(n+1) parameters. The simplest way to proceed (Judd calls it the kindergarten procedure of connecting the dots ) is to use linear B-splines, where B stands for basic. Suppose we have a grid with knots x 0 <...x k <... < x K and we want to approximate a function f (x). B 0 splines are defined as right-continuous step functions B 0 k(x) = { 1 if x k x < x k+1 0 otherwise So interpolant is: ˆf (x) = K 1 k=0 [ ] f (xk )+f (x k+1 ) B 0 2 k(x) G. Violante, Global Methods p. 7 /29
8 Linear B-splines B 1 splines implement piece-wise linear interpolation B 1 k(x) = x x k 1 x k x k 1 if x k 1 x < x k x k+1 x x k+1 x k if x k x < x k+1 0 elsewhere and each one looks like a tent-function with peak at x k equal to one, thus the weight of the interpolant must be f (x k ) The interpolant is therefore defined as: ˆf (x) = K f (x k )Bk(x) 1 k=0 You can also obtain the same weights from 2K conditions: ˆf at knots equal to f(x) (K + 1 conditions), and interpolant continuous at interior knots (K 1 conditions) G. Violante, Global Methods p. 8 /29
9 Linear spline Basis Functions on [0, 1] G. Violante, Global Methods p. 9 /29
10 Linear B-splines It is easy to see that for any given point x on an interval [x k,x k+1 ] there are only two nonzero basis at x, B k (x) and B k+1 (x). To locate the knots (k,k +1) that bracket x you use a bracketing algorithm that essentially proceeds by bisection. Thus ˆf (x) = f (x k ) x k+1 x x k+1 x k +f (x k+1 ) x x k x k+1 x k = f (x k ) x k+1 x x k +x k x k+1 x k +f (x k+1 ) x x k x k+1 x k = f (x k )+[f (x k+1 ) f (x k )] x x k x k+1 x k that is a familiar formula for linear interpolation of decision rules (see later...) G. Violante, Global Methods p. 10 /29
11 Linear Spline approximation (7 knots) x 0 =a x 1 x 2 x 3 x 4 x 5 x 6 =b G. Violante, Global Methods p. 11 /29
12 Pros and cons of linear splines Pros: Preserves monotonicity and concavity of f. We can exploit information about f clustering points more closely together in areas of high curvature or areas where we know a kink exists in order to increase our accuracy. We can capture binding inequality constraints very well. Cons: The approximated function is not differentiable at the knots (they become kinks). ˆf = 0 where it exists (it does not exist at the knots). Function not smooth. G. Violante, Global Methods p. 12 /29
13 Cubic splines Cubic spline (n = 3), has 4K coefficients to be determined ˆf (xk ) = f (x k ) (K +1 conditions) Continuity of ˆf at interior points Continuity of ˆf at interior points Continuity of ˆf at interior points (K 1 conditions) (K 1 conditions) (K 1 conditions) ˆf (x 0 ) = ˆf (x K ) = 0 (2 conditions) These last two can be modified to something else depending on what is suitable in the particular case at hand, they are ad-hoc and they can be more than 2. G. Violante, Global Methods p. 13 /29
14 Cubic spline Basis Functions on [0,1] G. Violante, Global Methods p. 14 /29
15 Cubic spline Basis Functions on [0,1] Spline 1: Piecewise Linear Interpolation Spline 2: Piecewise Cubic Interpolation Data G. Violante, Global Methods p. 15 /29
16 Pros and cons of cubic splines Pros: Easy to compute: interpolant matrix very sparse, easy to invert. Smooth approximation Cons: It may not be able to handle constraints well It does not preserve monotonicity and concavity of f G. Violante, Global Methods p. 16 /29
17 Schumaker splines So far: linear interpolation preserves shape (monotonicity and concavity), but not differentiability. Cubic splines preserves differentiability, but not shape. Schumaker quadratic splines preserve both (Schumaker, 1983, SIAM Journal on Numerical Analysis). By shape we mean that in those intervals where the data is monotonically increasing or decreasing, the spline has the same property. Similarly for convexity or concavity. Idea: add knots to a subinterval by means of an algorithm G. Violante, Global Methods p. 17 /29
18 Solving the income fluctuation problem V (a,y j ) = max {c,a } u(c)+β s.t. y i Y π(y i y j )V (a,y i ) c+a Ra+y j a a The Euler equation, once we substitute in the budget constraint, reads: u c (Ra+y j a ) βr π(y i y j )u c (Ra +y i a ) 0. y i Y where the strict inequality holds when the constraint binds. G. Violante, Global Methods p. 18 /29
19 Policy function iteration with linear interpolation 1. Construct a grid on the asset space {a 0,a 2,...,a m } with a 0 = a = Guess an initial vector of decision rules for a on the grid points, call it â 0 (a i,y j ), where the subscript 0 denotes the initial iteration 3. For each point (a i,y j ) on the grid, check whether the borrowing constraint binds. I.e. check whether: u c (Ra i +y j a 0 ) βr y Y π(y y j )u c (Ra 0 +y â 0 (a 0,y )) > If this inequality holds, the borrowing constraint binds. Then, set a 0(a i,y j ) = a 0 and repeat this check for the next grid point. If the equation instead holds with the < inequality, we have an interior solution (it is optimal to save for the household) and we proceed to the next step. G. Violante, Global Methods p. 19 /29
20 Policy function iteration with linear interpolation 5. For each point (a i,y j ) on the grid, use a nonlinear equation solver to find the solution a of the nonlinear equation u c (Ra i +y j a ) βr y Y π(y y j )u c (Ra +y â 0 (a,y )) = 0 (a) Need to evaluate the function â 0 (a,y ) outside grid points: assume it is piecewise linear. (b) Every time the solver calls an a which lies between grid points, do as follows. First, find the pair of adjacent grid points {a i,a i+1 } such that a i < a < a i+1, and then compute (â0 â 0 (a,y ) = â 0 (a i,y )+(a (a i+1,y ) â 0 (a i,y ) ) a i ) a i+1 a i (c) If the solution of the nonlinear equation is a, then set a 0(a i,y j ) = a and iterate on the next grid point. G. Violante, Global Methods p. 20 /29
21 Policy function iteration with linear interpolation 6. Check convergence by comparing a 0(a i,y j ) â 0 (a i,y j ) through some pre-specified norm. For example, declare convergence at iteration n when max i,j { a n(a i,y j ) â n (a i,y j ) } < ε for some small number ε which determines the degree of tolerance in the solution algorithm. 7. If convergence is achieved, stop. Otherwise, go back to point 3 with the new guess â 1 (a i,y j ) = a 0(a i,y j ). Note that the most time-consuming step in this procedure is 5, the root-finding problem. We now discuss how to avoid it. G. Violante, Global Methods p. 21 /29
22 Endogenous grid method (EGM) EGM is much faster than the traditional method because it does not require the use of a nonlinear equation solver The essential idea of the method is to construct a grid on a, next period s asset holdings, rather than on a, as is done in the standard algorithm. The method also requires the policy function to be at least weakly monotonic in the state Recall the Euler equation: u c (c(a,y)) βr y Y π(y y)u c (c(a,y )). As usual, we start from a guess ĉ 0 (a,y) and iterate on the Euler Equation until the decision rule for consumption that we solve for is essentially identical to the one in the previous iteration. G. Violante, Global Methods p. 22 /29
23 EGM: Algorithm 1. Construct a grid for (a,y) 2. Guess a policy function ĉ 0 (a i,y j ). If y is persistent, a good initial guess is to set ĉ 0 (a i,y j ) = ra i +y j which is the solution under quadratic utility if income follows a random walk. 3. Fix y j. Instead of iterating over {a i }, we iterate over {a i }. For any pair {a i,y j} on the mesh construct the RHS of the Euler equation [call it B(a i,y j)] B(a i,y j ) βr y Y π(y y j )u c (ĉ 0 (a i,y )) where the RHS of this equation uses the guess ĉ 0 on the grid G. Violante, Global Methods p. 23 /29
24 EGM: Algorithm 4. Use the Euler equation to solve for the value c(a i,y j) that satisfies u c ( c(a i,y j )) = B(a i,y j ) and note that it can be done analytically, e.g. for u c (c) = c γ we have c(a i,y j) = [B(a i,y j)] 1 γ. Here algorithm becomes much more efficient because it does not require a nonlinear solver. 5. From the budget constraint: c(a i,y j )+a i = Ra i +y j solve for a (a i,y j) the value of assets today that would lead the consumer to have a i assets tomorrow if her income shock was y j today. This yields c(a i,y j) = c(a i,y j), a function not defined on the grid points. This is the endogenous grid and it changes on each iteration. G. Violante, Global Methods p. 24 /29
25 EGM: Algorithm 6. Let a 0 be the value of asset holdings that induces the borrowing constraint to bind next period, i.e., the value for a that solves that equation at the point a 0, the lower bound of the grid. 7. Now we need to update our guess defined on the original grid. To get new guess ĉ 1 (a i,y j ) on grid points a i > a 0 we can use simple linear interpolation methods using values for { c(a n,y j ),c ( )} a n+1,y j on the two most adjacent values { } a n,a n+1 that bracket ai. If some points a i are beyond a m, the upper bound of the endogenous grid, just extend linearly the function G. Violante, Global Methods p. 25 /29
26 EGM: Algorithm To update the consumption policy function on grid values a i < a 0, we use the budget constraint: ĉ 1 (a i,y j ) = Ra i +y j a 0 since we cannot use the Euler equation as the borrowing constraint is binding for sure next period: the reason is that we found it was binding at a 0, therefore a fortiori it will be binding for a i < a Check convergence as before. G. Violante, Global Methods p. 26 /29
27 Envelope condition method (ECM) It is an alternative method that, in some cases, like the EGM, avoids the use of nonlinear solvers 1. Construct a grid on a with K +1 points: call it A. 2. Guess a value function ˆV 0 (a,y j ) = K k=0 w0 kj B k(a) where B k are cubic splines. 3. Compute the derivative of the value function on the nodes of A: ˆV 0 (a i,y j ) = K wkjb 0 k(a i ) k=0 Note: if you wish to use B 1 splines instead, you need to use a different grid in the above step because ˆV 0 is not differentiable on the nodes of the original grid A G. Violante, Global Methods p. 27 /29
28 Envelope Condition Method (ECM) 5. For any pair {a i,y j } on A Y construct the envelope condition: ˆV 0 (a i,y j ) = u c (Ra i +y j â 0 (a i,y j ))R from which we can obtain: â 0 (a i,y j ) = Ra i +y j u 1 c ( ) ˆV 0 (a i,y j ) R On each point of the grid we must verify if the borrowing constraint binds and if it does, we set â 0 (a i,y j ) = Update the value function from the Bellman equation. For each point of the grid A compute: ˆV 1 (a i,y j ) = u(ra i +y j â 0 (a i,y j ))+β y j Y π(y j y j ) ˆV 0 (â 0 (a i,y j ),y j ) G. Violante, Global Methods p. 28 /29
29 Envelope Condition Method (ECM) In practice, need to solve (K +1)J equations in (K +1)J { } coefficients wkj 1 of the type: K wkjb 1 k (a i ) = u(ra i +y j â 0 (a i,y j )) k=0 +β K π(y j y j ) wkj 0 B k(â 0 (a i,y j )) y j Y k=0 for all gridpoints, i = 0,1,...K and j = 1,2,...,J It is a linear system of equations G. Violante, Global Methods p. 29 /29
Lecture X. Global Approximation Methods
Lecture X Global Approximation Methods Gianluca Violante New York University Quantitative Macroeconomics G. Violante, Global Methods p. 1 /14 Envelope condition method (ECM) It is an alternative method
More informationCS 450 Numerical Analysis. Chapter 7: Interpolation
Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80
More informationNatural Quartic Spline
Natural Quartic Spline Rafael E Banchs INTRODUCTION This report describes the natural quartic spline algorithm developed for the enhanced solution of the Time Harmonic Field Electric Logging problem As
More informationLinear Interpolating Splines
Jim Lambers MAT 772 Fall Semester 2010-11 Lecture 17 Notes Tese notes correspond to Sections 112, 11, and 114 in te text Linear Interpolating Splines We ave seen tat ig-degree polynomial interpolation
More informationComputational Physics PHYS 420
Computational Physics PHYS 420 Dr Richard H. Cyburt Assistant Professor of Physics My office: 402c in the Science Building My phone: (304) 384-6006 My email: rcyburt@concord.edu My webpage: www.concord.edu/rcyburt
More informationMar. 20 Math 2335 sec 001 Spring 2014
Mar. 20 Math 2335 sec 001 Spring 2014 Chebyshev Polynomials Definition: For an integer n 0 define the function ( ) T n (x) = cos n cos 1 (x), 1 x 1. It can be shown that T n is a polynomial of degree n.
More informationlecture 10: B-Splines
9 lecture : -Splines -Splines: a basis for splines Throughout our discussion of standard polynomial interpolation, we viewed P n as a linear space of dimension n +, and then expressed the unique interpolating
More informationFebruary 2017 (1/20) 2 Piecewise Polynomial Interpolation 2.2 (Natural) Cubic Splines. MA378/531 Numerical Analysis II ( NA2 )
f f f f f (/2).9.8.7.6.5.4.3.2. S Knots.7.6.5.4.3.2. 5 5.2.8.6.4.2 S Knots.2 5 5.9.8.7.6.5.4.3.2..9.8.7.6.5.4.3.2. S Knots 5 5 S Knots 5 5 5 5.35.3.25.2.5..5 5 5.6.5.4.3.2. 5 5 4 x 3 3.5 3 2.5 2.5.5 5
More informationLecture 6: Interpolation
Lecture 6: Interpolation Fatih Guvenen January 10, 2016 Fatih Guvenen ( ) Lecture 6: Interpolation January 10, 2016 1 / 25 General Idea Suppose you are given a grid (x 1, x 2,..., x n ) and the function
More informationInterpolation by Spline Functions
Interpolation by Spline Functions Com S 477/577 Sep 0 007 High-degree polynomials tend to have large oscillations which are not the characteristics of the original data. To yield smooth interpolating curves
More informationAPPM/MATH Problem Set 4 Solutions
APPM/MATH 465 Problem Set 4 Solutions This assignment is due by 4pm on Wednesday, October 16th. You may either turn it in to me in class on Monday or in the box outside my office door (ECOT 35). Minimal
More informationPolynomials tend to oscillate (wiggle) a lot, even when our true function does not.
AMSC/CMSC 460 Computational Methods, Fall 2007 UNIT 2: Spline Approximations Dianne P O Leary c 2001, 2002, 2007 Piecewise polynomial interpolation Piecewise polynomial interpolation Read: Chapter 3 Skip:
More informationValue Function Iteration versus Euler equation methods
Value Function Iteration versus Euler equation methods Wouter J. Den Haan London School of Economics c by Wouter J. Den Haan Overview 1 How to do value function iteration (VFI) 2 equation methods 1 convergence
More informationConsider functions such that then satisfies these properties: So is represented by the cubic polynomials on on and on.
1 of 9 3/1/2006 2:28 PM ne previo Next: Trigonometric Interpolation Up: Spline Interpolation Previous: Piecewise Linear Case Cubic Splines A piece-wise technique which is very popular. Recall the philosophy
More informationA Linear Programming Approach to Concave Approximation and Value Function Iteration
A Linear Programming Approach to Concave Approximation and Value Function Iteration Ronaldo Carpio Takashi Kamihigashi May 18, 2015 Abstract A basic task in numerical computation is to approximate a continuous
More informationMaximizing an interpolating quadratic
Week 11: Monday, Apr 9 Maximizing an interpolating quadratic Suppose that a function f is evaluated on a reasonably fine, uniform mesh {x i } n i=0 with spacing h = x i+1 x i. How can we find any local
More informationInteractive Graphics. Lecture 9: Introduction to Spline Curves. Interactive Graphics Lecture 9: Slide 1
Interactive Graphics Lecture 9: Introduction to Spline Curves Interactive Graphics Lecture 9: Slide 1 Interactive Graphics Lecture 13: Slide 2 Splines The word spline comes from the ship building trade
More informationRational Bezier Surface
Rational Bezier Surface The perspective projection of a 4-dimensional polynomial Bezier surface, S w n ( u, v) B i n i 0 m j 0, u ( ) B j m, v ( ) P w ij ME525x NURBS Curve and Surface Modeling Page 97
More informationInterpolation - 2D mapping Tutorial 1: triangulation
Tutorial 1: triangulation Measurements (Zk) at irregular points (xk, yk) Ex: CTD stations, mooring, etc... The known Data How to compute some values on the regular spaced grid points (+)? The unknown data
More informationLecture 9. Curve fitting. Interpolation. Lecture in Numerical Methods from 28. April 2015 UVT. Lecture 9. Numerical. Interpolation his o
Curve fitting. Lecture in Methods from 28. April 2015 to ity Interpolation FIGURE A S Splines Piecewise relat UVT Agenda of today s lecture 1 Interpolation Idea 2 3 4 5 6 Splines Piecewise Interpolation
More informationLecture 25: Bezier Subdivision. And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10
Lecture 25: Bezier Subdivision And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10 1. Divide and Conquer If we are going to build useful
More informationLecture 9: Introduction to Spline Curves
Lecture 9: Introduction to Spline Curves Splines are used in graphics to represent smooth curves and surfaces. They use a small set of control points (knots) and a function that generates a curve through
More informationWhat is Multigrid? They have been extended to solve a wide variety of other problems, linear and nonlinear.
AMSC 600/CMSC 760 Fall 2007 Solution of Sparse Linear Systems Multigrid, Part 1 Dianne P. O Leary c 2006, 2007 What is Multigrid? Originally, multigrid algorithms were proposed as an iterative method to
More information99 International Journal of Engineering, Science and Mathematics
Journal Homepage: Applications of cubic splines in the numerical solution of polynomials Najmuddin Ahmad 1 and Khan Farah Deeba 2 Department of Mathematics Integral University Lucknow Abstract: In this
More informationJustify all your answers and write down all important steps. Unsupported answers will be disregarded.
Numerical Analysis FMN011 2017/05/30 The exam lasts 5 hours and has 15 questions. A minimum of 35 points out of the total 70 are required to get a passing grade. These points will be added to those you
More information8 Piecewise Polynomial Interpolation
Applied Math Notes by R. J. LeVeque 8 Piecewise Polynomial Interpolation 8. Pitfalls of high order interpolation Suppose we know the value of a function at several points on an interval and we wish to
More informationConvergence of C 2 Deficient Quartic Spline Interpolation
Advances in Computational Sciences and Technology ISSN 0973-6107 Volume 10, Number 4 (2017) pp. 519-527 Research India Publications http://www.ripublication.com Convergence of C 2 Deficient Quartic Spline
More informationScientific Computing: Interpolation
Scientific Computing: Interpolation Aleksandar Donev Courant Institute, NYU donev@courant.nyu.edu Course MATH-GA.243 or CSCI-GA.22, Fall 25 October 22nd, 25 A. Donev (Courant Institute) Lecture VIII /22/25
More informationEvaluating the polynomial at a point
Evaluating the polynomial at a point Recall that we have a data structure for each piecewise polynomial (linear, quadratic, cubic and cubic Hermite). We have a routine that sets evenly spaced interpolation
More informationPRIMAL-DUAL INTERIOR POINT METHOD FOR LINEAR PROGRAMMING. 1. Introduction
PRIMAL-DUAL INTERIOR POINT METHOD FOR LINEAR PROGRAMMING KELLER VANDEBOGERT AND CHARLES LANNING 1. Introduction Interior point methods are, put simply, a technique of optimization where, given a problem
More informationMath 226A Homework 4 Due Monday, December 11th
Math 226A Homework 4 Due Monday, December 11th 1. (a) Show that the polynomial 2 n (T n+1 (x) T n 1 (x)), is the unique monic polynomial of degree n + 1 with roots at the Chebyshev points x k = cos ( )
More informationCurve and Surface Fitting with Splines. PAUL DIERCKX Professor, Computer Science Department, Katholieke Universiteit Leuven, Belgium
Curve and Surface Fitting with Splines PAUL DIERCKX Professor, Computer Science Department, Katholieke Universiteit Leuven, Belgium CLARENDON PRESS OXFORD 1995 - Preface List of Figures List of Tables
More informationToday s class. Roots of equation Finish up incremental search Open methods. Numerical Methods, Fall 2011 Lecture 5. Prof. Jinbo Bi CSE, UConn
Today s class Roots of equation Finish up incremental search Open methods 1 False Position Method Although the interval [a,b] where the root becomes iteratively closer with the false position method, unlike
More informationFour equations are necessary to evaluate these coefficients. Eqn
1.2 Splines 11 A spline function is a piecewise defined function with certain smoothness conditions [Cheney]. A wide variety of functions is potentially possible; polynomial functions are almost exclusively
More informationAdvanced Operations Research Techniques IE316. Quiz 1 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 1 Review Dr. Ted Ralphs IE316 Quiz 1 Review 1 Reading for The Quiz Material covered in detail in lecture. 1.1, 1.4, 2.1-2.6, 3.1-3.3, 3.5 Background material
More informationA new 8-node quadrilateral spline finite element
Journal of Computational and Applied Mathematics 195 (2006) 54 65 www.elsevier.com/locate/cam A new 8-node quadrilateral spline finite element Chong-Jun Li, Ren-Hong Wang Institute of Mathematical Sciences,
More informationDeficient Quartic Spline Interpolation
International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 3, Number 2 (2011), pp. 227-236 International Research Publication House http://www.irphouse.com Deficient Quartic
More informationIterative Algorithms I: Elementary Iterative Methods and the Conjugate Gradient Algorithms
Iterative Algorithms I: Elementary Iterative Methods and the Conjugate Gradient Algorithms By:- Nitin Kamra Indian Institute of Technology, Delhi Advisor:- Prof. Ulrich Reude 1. Introduction to Linear
More informationIntroduction to Optimization Problems and Methods
Introduction to Optimization Problems and Methods wjch@umich.edu December 10, 2009 Outline 1 Linear Optimization Problem Simplex Method 2 3 Cutting Plane Method 4 Discrete Dynamic Programming Problem Simplex
More informationCurve fitting using linear models
Curve fitting using linear models Rasmus Waagepetersen Department of Mathematics Aalborg University Denmark September 28, 2012 1 / 12 Outline for today linear models and basis functions polynomial regression
More informationGeneralised Mean Averaging Interpolation by Discrete Cubic Splines
Publ. RIMS, Kyoto Univ. 30 (1994), 89-95 Generalised Mean Averaging Interpolation by Discrete Cubic Splines By Manjulata SHRIVASTAVA* Abstract The aim of this work is to introduce for a discrete function,
More informationMathematical Programming and Research Methods (Part II)
Mathematical Programming and Research Methods (Part II) 4. Convexity and Optimization Massimiliano Pontil (based on previous lecture by Andreas Argyriou) 1 Today s Plan Convex sets and functions Types
More informationSolving Systems of Spline Equations: A Linear Programming-based Approach
Engineering, Test & Technology Boeing Research & Technology Solving Systems of Spline Equations: A Linear Programming-based Approach Thomas Grandine Senior Technical Fellow Support and Analytics Technology
More informationAsymptotic Error Analysis
Asymptotic Error Analysis Brian Wetton Mathematics Department, UBC www.math.ubc.ca/ wetton PIMS YRC, June 3, 2014 Outline Overview Some History Romberg Integration Cubic Splines - Periodic Case More History:
More informationCS321 Introduction To Numerical Methods
CS3 Introduction To Numerical Methods Fuhua (Frank) Cheng Department of Computer Science University of Kentucky Lexington KY 456-46 - - Table of Contents Errors and Number Representations 3 Error Types
More informationRemark. Jacobs University Visualization and Computer Graphics Lab : ESM4A - Numerical Methods 331
Remark Reconsidering the motivating example, we observe that the derivatives are typically not given by the problem specification. However, they can be estimated in a pre-processing step. A good estimate
More informationSPLINE APPROXIMATION VIA THE CONTROL POLYGON
SPLINE APPROXIMATION VIA THE CONTROL POLYGON by Håkon Mørk THESIS for the degree of MASTER S DEGREE IN APPLIED MATHEMATICS AND MECHANICS (Master i Anvendt matematikk og mekanikk) Faculty of Mathematics
More informationSolve Non-Linear Parabolic Partial Differential Equation by Spline Collocation Method
Solve Non-Linear Parabolic Partial Differential Equation by Spline Collocation Method P.B. Choksi 1 and A.K. Pathak 2 1 Research Scholar, Rai University,Ahemdabad. Email:pinalchoksey@gmail.com 2 H.O.D.
More informationAn introduction to interpolation and splines
An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve
More informationMoving Beyond Linearity
Moving Beyond Linearity The truth is never linear! 1/23 Moving Beyond Linearity The truth is never linear! r almost never! 1/23 Moving Beyond Linearity The truth is never linear! r almost never! But often
More informationHandout 4 - Interpolation Examples
Handout 4 - Interpolation Examples Middle East Technical University Example 1: Obtaining the n th Degree Newton s Interpolating Polynomial Passing through (n+1) Data Points Obtain the 4 th degree Newton
More informationFall CSCI 420: Computer Graphics. 4.2 Splines. Hao Li.
Fall 2014 CSCI 420: Computer Graphics 4.2 Splines Hao Li http://cs420.hao-li.com 1 Roller coaster Next programming assignment involves creating a 3D roller coaster animation We must model the 3D curve
More information60 2 Convex sets. {x a T x b} {x ã T x b}
60 2 Convex sets Exercises Definition of convexity 21 Let C R n be a convex set, with x 1,, x k C, and let θ 1,, θ k R satisfy θ i 0, θ 1 + + θ k = 1 Show that θ 1x 1 + + θ k x k C (The definition of convexity
More informationReals 1. Floating-point numbers and their properties. Pitfalls of numeric computation. Horner's method. Bisection. Newton's method.
Reals 1 13 Reals Floating-point numbers and their properties. Pitfalls of numeric computation. Horner's method. Bisection. Newton's method. 13.1 Floating-point numbers Real numbers, those declared to be
More informationPS Geometric Modeling Homework Assignment Sheet I (Due 20-Oct-2017)
Homework Assignment Sheet I (Due 20-Oct-2017) Assignment 1 Let n N and A be a finite set of cardinality n = A. By definition, a permutation of A is a bijective function from A to A. Prove that there exist
More informationCOMPUTATIONAL INTELLIGENCE (INTRODUCTION TO MACHINE LEARNING) SS18. Lecture 2: Linear Regression Gradient Descent Non-linear basis functions
COMPUTATIONAL INTELLIGENCE (INTRODUCTION TO MACHINE LEARNING) SS18 Lecture 2: Linear Regression Gradient Descent Non-linear basis functions LINEAR REGRESSION MOTIVATION Why Linear Regression? Simplest
More informationIntroduction to Programming for Engineers Spring Final Examination. May 10, Questions, 170 minutes
Final Examination May 10, 2011 75 Questions, 170 minutes Notes: 1. Before you begin, please check that your exam has 28 pages (including this one). 2. Write your name and student ID number clearly on your
More informationContents. I The Basic Framework for Stationary Problems 1
page v Preface xiii I The Basic Framework for Stationary Problems 1 1 Some model PDEs 3 1.1 Laplace s equation; elliptic BVPs... 3 1.1.1 Physical experiments modeled by Laplace s equation... 5 1.2 Other
More information1.2 Numerical Solutions of Flow Problems
1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian
More informationCS-184: Computer Graphics
CS-184: Computer Graphics Lecture #12: Curves and Surfaces Prof. James O Brien University of California, Berkeley V2007-F-12-1.0 Today General curve and surface representations Splines and other polynomial
More informationNOTATION AND TERMINOLOGY
15.053x, Optimization Methods in Business Analytics Fall, 2016 October 4, 2016 A glossary of notation and terms used in 15.053x Weeks 1, 2, 3, 4 and 5. (The most recent week's terms are in blue). NOTATION
More informationModule 2: Single Step Methods Lecture 4: The Euler Method. The Lecture Contains: The Euler Method. Euler's Method (Analytical Interpretations)
The Lecture Contains: The Euler Method Euler's Method (Analytical Interpretations) An Analytical Example file:///g /Numerical_solutions/lecture4/4_1.htm[8/26/2011 11:14:40 AM] We shall now describe methods
More informationand the crooked shall be made straight, and the rough ways shall be made smooth; Luke 3:5
ecture 8: Knot Insertion Algorithms for B-Spline Curves and Surfaces and the crooked shall be made straight, and the rough ways shall be made smooth; uke 3:5. Motivation B-spline methods have several advantages
More informationMonotonic Cubic Spline Interpolation
Monotonic Cubic Spline Interpolation George Wolberg Itzik Alfy Department of Computer Science City College of New York / CUNY New York, NY wolberg@cs-mailengrccnycunyedu Abstract This paper describes the
More informationLecture 8. Divided Differences,Least-Squares Approximations. Ceng375 Numerical Computations at December 9, 2010
Lecture 8, Ceng375 Numerical Computations at December 9, 2010 Computer Engineering Department Çankaya University 8.1 Contents 1 2 3 8.2 : These provide a more efficient way to construct an interpolating
More informationSplines. Parameterization of a Curve. Curve Representations. Roller coaster. What Do We Need From Curves in Computer Graphics? Modeling Complex Shapes
CSCI 420 Computer Graphics Lecture 8 Splines Jernej Barbic University of Southern California Hermite Splines Bezier Splines Catmull-Rom Splines Other Cubic Splines [Angel Ch 12.4-12.12] Roller coaster
More informationChapter 13. Boundary Value Problems for Partial Differential Equations* Linz 2002/ page
Chapter 13 Boundary Value Problems for Partial Differential Equations* E lliptic equations constitute the third category of partial differential equations. As a prototype, we take the Poisson equation
More informationSung-Eui Yoon ( 윤성의 )
CS480: Computer Graphics Curves and Surfaces Sung-Eui Yoon ( 윤성의 ) Course URL: http://jupiter.kaist.ac.kr/~sungeui/cg Today s Topics Surface representations Smooth curves Subdivision 2 Smooth Curves and
More informationCOMPUTATIONAL INTELLIGENCE (CS) (INTRODUCTION TO MACHINE LEARNING) SS16. Lecture 2: Linear Regression Gradient Descent Non-linear basis functions
COMPUTATIONAL INTELLIGENCE (CS) (INTRODUCTION TO MACHINE LEARNING) SS16 Lecture 2: Linear Regression Gradient Descent Non-linear basis functions LINEAR REGRESSION MOTIVATION Why Linear Regression? Regression
More informationDiffusion Wavelets for Natural Image Analysis
Diffusion Wavelets for Natural Image Analysis Tyrus Berry December 16, 2011 Contents 1 Project Description 2 2 Introduction to Diffusion Wavelets 2 2.1 Diffusion Multiresolution............................
More informationAn introduction to mesh generation Part IV : elliptic meshing
Elliptic An introduction to mesh generation Part IV : elliptic meshing Department of Civil Engineering, Université catholique de Louvain, Belgium Elliptic Curvilinear Meshes Basic concept A curvilinear
More informationINTERPOLATION BY QUARTIC SPLINE
INTERPOLATION BY QUARTIC SPLINE K.K.Nigam 1, Y.P. Dubey 2, Brajendra Tiwari 3, Anil Shukla 4 1 Mathematics Deptt., LNCT Jabalpur, MP, (India) 2 Sarswati Institution of Engg. & Technology, Jabalpur MP,
More informationConcept of Curve Fitting Difference with Interpolation
Curve Fitting Content Concept of Curve Fitting Difference with Interpolation Estimation of Linear Parameters by Least Squares Curve Fitting by Polynomial Least Squares Estimation of Non-linear Parameters
More informationEC422 Mathematical Economics 2
EC422 Mathematical Economics 2 Chaiyuth Punyasavatsut Chaiyuth Punyasavatust 1 Course materials and evaluation Texts: Dixit, A.K ; Sydsaeter et al. Grading: 40,30,30. OK or not. Resources: ftp://econ.tu.ac.th/class/archan/c
More informationA general matrix representation for non-uniform B-spline subdivision with boundary control
A general matrix representation for non-uniform B-spline subdivision with boundary control G. Casciola a, L. Romani a a Department of Mathematics, University of Bologna, P.zza di Porta San Donato 5, 40127
More informationx n x n stepnumber k order r error constant C r+1 1/2 5/12 3/8 251/720 abs. stab. interval (α,0) /11-3/10
MATH 573 LECTURE NOTES 77 13.8. Predictor-corrector methods. We consider the Adams methods, obtained from the formula xn+1 xn+1 y(x n+1 y(x n ) = y (x)dx = f(x,y(x))dx x n x n by replacing f by an interpolating
More informationLecture 2 September 3
EE 381V: Large Scale Optimization Fall 2012 Lecture 2 September 3 Lecturer: Caramanis & Sanghavi Scribe: Hongbo Si, Qiaoyang Ye 2.1 Overview of the last Lecture The focus of the last lecture was to give
More informationChapter 18. Geometric Operations
Chapter 18 Geometric Operations To this point, the image processing operations have computed the gray value (digital count) of the output image pixel based on the gray values of one or more input pixels;
More informationMATH 2400, Analytic Geometry and Calculus 3
MATH 2400, Analytic Geometry and Calculus 3 List of important Definitions and Theorems 1 Foundations Definition 1. By a function f one understands a mathematical object consisting of (i) a set X, called
More informationLecture 2.2 Cubic Splines
Lecture. Cubic Splines Cubic Spline The equation for a single parametric cubic spline segment is given by 4 i t Bit t t t i (..) where t and t are the parameter values at the beginning and end of the segment.
More informationLecture 23: Starting to put it all together #2... More 2-Point Boundary value problems
Lecture 23: Starting to put it all together #2... More 2-Point Boundary value problems Outline 1) Our basic example again: -u'' + u = f(x); u(0)=α, u(l)=β 2) Solution of 2-point Boundary value problems
More informationLecture 7: Support Vector Machine
Lecture 7: Support Vector Machine Hien Van Nguyen University of Houston 9/28/2017 Separating hyperplane Red and green dots can be separated by a separating hyperplane Two classes are separable, i.e., each
More information(Sparse) Linear Solvers
(Sparse) Linear Solvers Ax = B Why? Many geometry processing applications boil down to: solve one or more linear systems Parameterization Editing Reconstruction Fairing Morphing 2 Don t you just invert
More informationLacunary Interpolation Using Quartic B-Spline
General Letters in Mathematic, Vol. 2, No. 3, June 2017, pp. 129-137 e-issn 2519-9277, p-issn 2519-9269 Available online at http:\\ www.refaad.com Lacunary Interpolation Using Quartic B-Spline 1 Karwan
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 24
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 24 So in today s class, we will look at quadrilateral elements; and we will
More informationThe Fundamentals of Economic Dynamics and Policy Analyses: Learning through Numerical Examples. Part II. Dynamic General Equilibrium
The Fundamentals of Economic Dynamics and Policy Analyses: Learning through Numerical Examples. Part II. Dynamic General Equilibrium Hiroshi Futamura The objective of this paper is to provide an introductory
More informationREGULAR GRAPHS OF GIVEN GIRTH. Contents
REGULAR GRAPHS OF GIVEN GIRTH BROOKE ULLERY Contents 1. Introduction This paper gives an introduction to the area of graph theory dealing with properties of regular graphs of given girth. A large portion
More informationTopology optimization in B-spline space
Topology optimization in B-spline space Xiaoping Qian Mechanical, Materials, and Aerospace Engineering Department Illinois Institute of Technology Chicago, IL 60062, USA Email: qian@iit.edu Highlights
More informationKnow it. Control points. B Spline surfaces. Implicit surfaces
Know it 15 B Spline Cur 14 13 12 11 Parametric curves Catmull clark subdivision Parametric surfaces Interpolating curves 10 9 8 7 6 5 4 3 2 Control points B Spline surfaces Implicit surfaces Bezier surfaces
More informationFinite Element Methods
Chapter 5 Finite Element Methods 5.1 Finite Element Spaces Remark 5.1 Mesh cells, faces, edges, vertices. A mesh cell is a compact polyhedron in R d, d {2,3}, whose interior is not empty. The boundary
More informationFitting to a set of data. Lecture on fitting
Fitting to a set of data Lecture on fitting Linear regression Linear regression Residual is the amount difference between a real data point and a modeled data point Fitting a polynomial to data Could use
More informationTO DUY ANH SHIP CALCULATION
TO DUY ANH SHIP CALCULATION Ship Calculattion (1)-Space Cuvers 3D-curves play an important role in the engineering, design and manufature in Shipbuilding. Prior of the development of mathematical and computer
More informationEECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation
More informationImportant Properties of B-spline Basis Functions
Important Properties of B-spline Basis Functions P2.1 N i,p (u) = 0 if u is outside the interval [u i, u i+p+1 ) (local support property). For example, note that N 1,3 is a combination of N 1,0, N 2,0,
More informationComputational Economics and Finance
Computational Economics and Finance Part I: Elementary Concepts of Numerical Analysis Spring 2016 Outline Computer arithmetic Error analysis: Sources of error Error propagation Controlling the error Rates
More informationParameterization of triangular meshes
Parameterization of triangular meshes Michael S. Floater November 10, 2009 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to
More informationPolynomial Approximation and Interpolation Chapter 4
4.4 LAGRANGE POLYNOMIALS The direct fit polynomial presented in Section 4.3, while quite straightforward in principle, has several disadvantages. It requires a considerable amount of effort to solve the
More informationIntroduction to Mathematical Programming IE406. Lecture 20. Dr. Ted Ralphs
Introduction to Mathematical Programming IE406 Lecture 20 Dr. Ted Ralphs IE406 Lecture 20 1 Reading for This Lecture Bertsimas Sections 10.1, 11.4 IE406 Lecture 20 2 Integer Linear Programming An integer
More informationCS-184: Computer Graphics. Today
CS-84: Computer Graphics Lecture #5: Curves and Surfaces Prof. James O Brien University of California, Berkeley V25F-5-. Today General curve and surface representations Splines and other polynomial bases
More informationThe use of the Spectral Properties of the Basis Splines in Problems of Signal Processing
The use of the Spectral Properties of the Basis Splines in Problems of Signal Processing Zaynidinov Hakim Nasiritdinovich, MirzayevAvazEgamberdievich, KhalilovSirojiddinPanjievich Doctor of Science, professor,
More information