1、Interpolation Methods,Robert A. Dalrymple Johns Hopkins University,Why Interpolation?,For discrete models of continuous systems, we need the ability to interpolate values in between discrete points. Half of the SPH technique involves interpolation of values known at particles (or nodes).,Interpolati
2、on,To find the value of a function between known values.Consider the two pairs of values (x,y):(0.0, 1.0), (1.0, 2.0)What is y at x = 0.5? That is, whats (0.5, y)?,Linear Interpolation,Given two points, (x1,y1), (x2,y2):Fit a straight line between the points.y(x) = a x +ba=(y2-y1)/(x2-x1), b= (y1 x2
3、y2 x1)/(x2-x1), Use this equation to find y values for any x1 x x2,Polynomial Interpolants,Given N (=4) data points,Find the interpolating function that goes through the points:If there were N+1 data points, the function would bewith N+1 unknown values, ai, of the Nth order polynomial,Polynomial In
4、terpolant,Force the interpolant through the four points to get four equations:,Rewriting:The solution is found by inverting p,Example,Data are: (0,2), (1,0.3975), (2, -0.1126), (3, -0.0986).Fitting a cubic polynomial through the four points gives:,Matlab code for polynomial fitting,% the data to be
5、interpolated (in 1D)x=-0.2 .44 1.0 1.34 1.98 2.50 2.95 3.62 4.13 4.64 4.94;y=2.75 1.80 -1.52 -2.83 -1.62 1.49 2.98 0.81 -2.14 -2.93 -1.81;plot(x,y,bo)n=size(x,2) % CUBIC FITp=ones(1,n)xx.*xx.*(x.*x)a=py %same as a=inv(p)*yyp=p*ahold on;plot(x,yp,k*),Note: linear and quadratic fit: redefine p,Polynom
6、ial Fit to Example,Exact: red Polynomial fit: blue,Beware of Extrapolation,An Nth order polynomial has N roots!,Exact: red,Least Squares Interpolant,For N points, we will have a fitting polynomial of order m (N-1). The least squares fitting polynomial be similar to the exact fit form: Now p is N x m
7、 matrix. Since we have fewer unknown coefficient as data points, the interpolant cannot go through each point. Define the error as the amount of “miss”Sum of the (errors)2:,Least Squares Interpolant,Minimizing the sum with respect to the coefficients a:Solving,This can be rewritten in this form,whic
8、h introduces a pseudo-inverse.,Reminder:,for cubic fit,Question,Show that the equation above leads to the following expression for the best fit straight line:,Matlab: Least-Squares Fit,%the data to be interpolated (1d)x=-0.2 .44 1.0 1.34 1.98 2.50 2.95 3.62 4.13 4.64 4.94;y=2.75 1.80 -1.52 -2.83 -1.
9、62 1.49 2.98 0.81 -2.14 -2.93 -1.81;plot(x,y,bo)n=size(x,2) % CUBIC FITp=ones(1,n)xx.*xx.*(x.*x)pinverse=inv(p*p)*pa=pinverse*yyp=p*aplot(x,yp,k*),Cubic Least Squares Example,x: -0.2 .44 1.0 1.34 1.98 2.50 2.95 3.62 4.13 4.64 4.94 y: 2.75 1.80 -1.52 -2.83 -1.62 1.49 2.98 0.81 -2.14 -2.93 -1.81,Data
10、irregularly spaced,Least Squares Interpolant,Cubic Least Squares Fit: * is the fitting polynomialo is the given data,Exact,Piecewise Interpolation,Piecewise polynomials: fit all pointsLinear: continuity in y+, y- (fit pairs of points)Quadratic: +continuity in slopeCubic splines: +continuity in secon
11、d derivative RBFAll of the above, but smoother,Radial Basis Functions,Developed to interpolate 2-D data: think bathymetry. Given depths: , interpolate to a rectangular grid.,Radial Basis Functions,2-D data:,For each position, there is an associated value:Radial basis function (located at each point)
12、where is the distance from xj,The radial basis function interpolant is:,RBF,To find the unknown coefficients i, force the interpolantto go through the data points: whereThis gives N equations for the N unknown coefficients.,RBF,Morse et al., 2001,Multiquadric RBF,MQ:RMQ:,Hardy, 1971; Kansa, 1990,R
13、BF Example,11 (x,y) pairs: (0.2, 3.00), (0.38, 2.10), (1.07, -1.86), (1.29, -2.71), (1.84, -2.29), (2.31, 0.39), (3.12, 2.91), (3.46, 1.73), (4.12, -2.11), (4.32, -2.79), (4.84, -2.25) SAME AS BEFORE,RBF Errors,Log10 sqrt (mean squared errors) versus c: Multiquadric,RBF Errors,Log10 sqrt (mean squar
14、ed errors) versus c: Reciprocal Multiquadric,Consistency,Consistency is the ability of an interpolating polynomial to reproduce a polynomial of a given order. The simplest consistency is constant consistency: reproduce unity.where, again, If gj(0) = 1, then a constraint results:,Note: Not all RBFs h
15、ave gj(0) = 1,RBFs and PDEs,Solve a boundary value problem:,The RBF interpolant is:N is the number of arbitrarily spaced points; the j are unknown coefficients to be found.,RBFs and PDEs,Introduce the interpolant into the governing equation and boundary conditions:,These are N equations for the N un
16、known constants, j,RBFs and PDEs (3),Problem with many RBF is that the N x N matrix that has to be inverted is fully populated.RBFs with small footprints (Wendland, 2005),1D:3D:,His notation:Advantages: matrix is sparse, but still N x N,Wendland 1-D RBF with Compact Support,h=1 Max=1,Moving Least Sq
17、uares Interpolant,are monomials in x for 1D (1, x, x2, x3) x,y in 2D, e.g. (1, x, y, x2, xy, y2 .)Note aj are functions of x,Moving Least Squares Interpolant,Define a weighted mean-squared error:where W(x-xi) is a weighting function that decays with increasing x-xi. Same as previous least squares ap
18、proach, except for W(x-xi),Weighting Function,q=x/h,Moving Least Squares Interpolant,Minimizing the weighted squared errors for the coefficients:,Moving Least Squares Interpolant,SolvingThe final locally valid interpolant is:,Moving Least Squares (1),% generate the data to be interpolated (1d) x=-0.
19、2 .44 1.0 1.34 1.98 2.50 2.95 3.62 4.13 4.64 4.94; y=2.75 1.80 -1.52 -2.83 -1.62 1.49 2.98 0.81 -2.14 -2.93 -1.81; plot(x,y,bo) n=size(x,2) % QUADRATIC FIT p=ones(1,n) x x.*x xfit=0.30; sum=0.0 % compute msq error for it=1:18, % fiting at 18 points xfit=xfit+0.25; d=abs(xfit-x) for ic=1:n q=d(1,ic)/
20、51; % note 0.3 works for linear fit; 0.51 for quadratic if q = 1. Wd(1,ic)=0.66*(1-1.5*q*q+0.75*q3); elseif q = 2. Wd(1,ic)=0.66*0.25*(2-q)3; else Wd(1,ic)=0.0; end end,MLS (2),Warray=diag(Wd); A=p*(Warray*p) B=p*Warray acoef=(inv(A)*B)*y % QUADRATIC FIT yfit=acoef*1 xfit xfit*xfit hold on; plot(xf
21、it, yfit,k*) sum=sum+(3.*cos(2.*pi*xfit/3.0)-yfit)2; end,MLS Fit to (Same) Irregular Data,Given data: circles; MLS: *; exact: line,h=0.51,.3,.5,1.0,1.5,Varying h Values,Conclusions,There are a variety of interpolation techniques for irregularly spaced data:Polynomial FitsBest Fit PolynomialsPiecewise PolynomialsRadial Basis FunctionsMoving Least Squares,
