Global Interpolation. Think globally. Act locally -- L. N. Trefethen, Spectral Methods in Matlab (SIAM, 2000)

Transcription

1 Global Interpolation Think globally. Act locally -- L. N. Trefethen, Spectral Methods in Matlab (SIAM, 2000)

2 Interpolation Interpolation is the process of fitting a smooth function to pass through smooth data points This allows us to do various things: Evaluate the function between the points or anywhere interpolate Numerically differentiate the function Numerically integrate the function Evaluate the function beyond the range of the data extrapolate (can be dangerous!) interpolate extrapolate Lecture 10 2

3 Types of Interpolation Interpolation can be local ( piecewise ) or global Local use just data surrounding the x value that you want to evaluate the function Global use all the data Interpolation can be done by fitting data to a variety of smooth functional forms: Polynomial interpolation Fourier interpolation local global c i coefficients, e i (x) basis functions Lecture 10 3

4 Polynomial interpolation Given N+1 data points (x j,y j ), there is a unique polynomial of degree N that goes through all the points Even though the polynomial is unique, it can be expressed many different ways, e.g. Monomial form Newton s form Lagrange s form Chebyshev form Others Most important form for today s lecture is: Chebyshev polynomial expansion Recursion formula: Chebyshev polynomials We obtain the expansion coefficients {c} by collocation at the N+1 data points: Lecture 10 4

5 Local vs. Global Polynomial Interpolation Fitting a low-order polynomial to a few points, N 5, of a data set that span a point x is a good way to locally interpolate a function y(x) near x Now suppose we want to evaluate y(x) throughout the whole domain of x, [-1,1] i.e. develop a global interpolant Can we just interpolate with a higher-order (degree N >>1) polynomial through all of the N+1 points?? Interpolate at x = Lecture 10 5

6 Global Interpolation Example Let s try global interpolation by fitting an N=16 polynomial to a smooth function sampled at 17 equispaced points: This is a disaster! The error, while small in the middle, is huge near the boundaries. This is the socalled: Runge phenomenon Runge phenomenon Lecture 10 6

7 Chebyshev to the rescue We see that global polynomial interpolation of a large data set of equispaced points can produce disastrous Runge oscillations near the boundaries Remarkably, this problem can be fixed by simply choosing to collocate using unequally spaced data points the Chebyshev points These points are the extrema of the polynomial T N (x) over [-1,1] Plot of T 16 (x) Lecture 10 7

8 Our example revisited Let s repeat our example and compare the use of uniformly spaced points versus Chebyshev points for N = 16 Notice that the Chebyshev points are clustered near the boundaries at x = -1, 1 See ChebyInterpL10.m Problem solved! We now have a global approximant that is uniformly accurate! Lecture 10 8

9 Errors and Other Intervals A remarkable feature of global interpolation with Chebyshev polynomials is the rapid rate of convergence. If the function being described is suitably smooth (analytic), the error in the approximant can be shown to decay as fast as This exponentially fast convergence with the number of data points is referred to as spectral accuracy and is highly desirable To achieve these results, it is very important that the independent (x) variable of your data be scaled to the interval [-1,1] or Chebyshev interpolation will not work! A simple change of variable will do the trick: Lecture 10 9

10 Another Remarkable Fact about Chebyshev Interpolation Recall that we obtain the Chebyshev polynomial expansion coefficients {c} by collocation and solving a linear system (polyfit did this for us in our examples) For large N, this looks expensive, requiring O(N 3 ) flops However, if we use the Chebyshev points: Here we have used the property [DCT] ji Thus, {y} and {c} are related by a Discrete Cosine Transform (DCT): {y} = [DCT] {c} The DCT can be performed in O(N log N) operations by a Fast Fourier Transform (FFT) algorithm: much faster! (See Lecture 16) Lecture 10 10

11 Spline Interpolation An alternative way to develop a global interpolant for a data set is to use splines Spline interpolation involves using a different low-order polynomial (<4) in each interval between points ( knots ) The polynomial coefficients are determined by matching the function values and low-order derivatives (<3) at the knots Spline interpolants are thus piece-wise continuous functions that are differentiable (smooth) only to low order Linear spline: y continuous Quadratic spline: y,y continuous Cubic spline: y, y,y continuous Linear Spline Fit: N=16 Notice discontinuous derivative at knots Lecture 10 11

12 Constructing a Cubic Spine With N+1 data points, have N intervals between points. In the ith interval, the approximant is: We thus have 4N coefficients to determine These are fixed by the following 4N linear conditions: The y i (x) must match the data points at each end of each interval: 2N conditions The first derivatives must match in adjacent intervals for each interior point (knot): N-1 conditions The second derivatives much match in adjacent intervals for each interior point (knot): N-1 conditions The second derivatives of the two end points are set to zero (free end conditions): 2 conditions The result is a banded linear system of equations to solve for the 4N coefficients efficient O(N) algorithms are available (c.f. tridiagonal case) The banded nature of the spline matrix shows that the spline method is only weakly global information about the approximant is only tenuously propagated from interval to interval. This is unlike Chebyshev interpolants, where the T i (x j ) matrix is dense: Lecture 10 12

13 An Example Let s use MatLab s interp1 function to construct linear and cubic spline approximants to the function y(x) = exp(x)*sin(5x) N=16 as before. See SplineL10.m yi = Interp1(x,y,xi, spline ) interpolates a cubic spline from data (x,y) at the points xi The cubic spline results look good, but Chebyshev interpolation with N=16 gives a much smaller error of 6.64x10-8! Lecture 10 13

14 Global Interpolation Summary Both Spline and Chebyshev interpolation are powerful tools for developing a global approximant to a smooth function sampled at discrete points: Chebyshev enjoys spectral accuracy (if the function is analytic) and can be efficiently implemented using FFT methods. The data points have to be sampled at the Chebyshev nodes or roots Cubic Spline interpolation converges less rapidly, with errors decaying algebraically as ~N -4, but is easily and efficiently implemented with arbitrary spaced data Lecture 10 14