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Random Reminders#

The nth Legendre polynomial, \(\boldsymbol{L}_{n}\), is orthogonal to every polynomial with degrees less than n i.e.

  • \(\boldsymbol{L}_{n} \perp \boldsymbol{P}_{i}, \ \forall i\in [0..n-1]\)
  • ex: \(\boldsymbol{L}_{n} \perp x^{3}\)

\(\boldsymbol{L}_{n}\) has n real roots and they are all \(\in [-1,1]\)

Harmonic functions => \(\Delta u(x) = 0\)

Homogenous function => \(f : \reals^{n} \rightarrow \reals^{n}, \ f(\lambda \mathbf{v})=\lambda^{k} f(\mathbf{v})\) where \(k,\lambda \in \reals\)

General form of Newton’s divided-difference polynomial interpolation:

\[ \begin{aligned} f_{n}(x)=& f\left(x_{0}\right)+\left(x-x_{0}\right) f\left[x_{1}, x_{0}\right]+\left(x-x_{0}\right)\left(x-x_{1}\right) f\left[x_{2}, x_{1}, x_{0}\right] \\ &+\cdots+\left(x-x_{0}\right)\left(x-x_{1}\right) \cdots\left(x-x_{n-1}\right) f\left[x_{n}, x_{n-1}, \ldots, x_{0}\right] \end{aligned} \]

Lagrange interpolating polynomial scheme is just a reformulation of Newton scheme that avoids computation of divided differences

\[ f_{n}(x)=\sum_{i=0}^{n} L_{i}(x) f\left(x_{i}\right)\newline L_{i}(x)=\prod_{j=0 \atop j \neq i}^{n} \frac{x-x_{j}}{x_{i}-x_{j}} \]

Gaussian quadrature

  • allows for accurately approximating functions where \(f(x) \in P_{2n-1}\) with only n coefficients

Regression Schemes (Linear or nonlinear)

  • Curves do not necessarily go through sample points so error at said points might be large
  • Round-off error becomes pronounced for higher order versions and ill-conditioned matrices are a problem
  • Orthogonal polynomials do not necessarily suffer from this

Interpolation Schemes (splines, lagriangina/newtonian, etc)

  • Curves must go through sample points so error at said points is small
  • Not ill conditioned

Thin plate splines

  • construction is based on choosing a function that minimizes anintegral that represents the bending energy of a surface
  • the idea of thin-plate splines is to choose a functionf(x) that exactly interpolates the datapoints (xi,yi), say,yi=f(xi), and that minimizes the bending energy \(E[f]=\int_{\mathbf{R}^{n}}\left|D^{2} f\right|^{2} d X\)
  • Can also choose function that doesn’t exactly interpolate all control points by using smoothing parameter for regularization \(E[f]=\sum_{i=1}^{m}\left|f\left(\mathbf{x}_{i}\right)-y_{i}\right|^{2}+\lambda \int_{\mathbb{R}^{n}}\left|D^{2} f\right|^{2} d X\)

Spherical Basis Spliens:

  • Gross reduction summary: bsplines with slerp instead of lerp between control points

Laplacian => Avg of neighbors at a point - point value

  • Maximal smoothness/mean curvature is zero
  • Poisson equation = \(\Delta u\) = 0
  • Think of boundary condition being a wire and a soap film covering the wire. That’s a \(\Delta u(x,y) = 0\)
  • Another interpretation is equilibriam state. Think of temperature
  • Another interpretation is that there are no bumps or local minimas in that surface


Spherical Splines#

Last update: November 14, 2019