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:
Lagrange interpolating polynomial scheme is just a reformulation of Newton scheme that avoids computation of divided differences
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
RBF#
- INTEGRATION BY RBF OVER THE SPHERE: https://www.math.unipd.it/~marcov/pdf/AMR05_17.pdf
- RBF for Scientific computing: https://math.boisestate.edu/~wright/montestigliano/RBFsForScientificComputingPartOne.pdf
- Interpolation and Best Approximation for Spherical Radial Basis Function Networks: https://www.hindawi.com/journals/aaa/2013/206265
- Spherical Radial Basis Functions, Theory and Applications (SpringerBriefs in Mathematics)
- Transport schemes on a sphere using radial basis functions: https://www.math.utah.edu/~wright/misc/msFinal_Grady.pdf
- On choosing a radial basis function and a shape parameterwhen solving a convective PDE on a sphere: https://amath.colorado.edu/faculty/fornberg/Docs/Fornberg_Piret_2.pdf
- A FAST ALGORITHM FOR SPHERICAL BASISAPPROXIMATION: https://www.math.uni-luebeck.de/mitarbeiter/prestin/ps/sharma.pdf
Spherical Splines#
- Spline Representations of Functions on a Sphere forGeopotential Modeling: https://kb.osu.edu/bitstream/handle/1811/78653/1/SES_GeodeticScience_Report_475.pdf
- Fitting scattered data on sphere-like surfaces using spherical splines: https://math.vanderbilt.edu/schumake/ans4.pdf
- Bernstein-Bézier polynomials on spheres and sphere-like surfaces: https://math.vanderbilt.edu/neamtum/papers/ans2.pdf
- Survey on Spherical Spline Approximation: https://pdfs.semanticscholar.org/63eb/efb9cbdc248371e2fe4f09fa7e70b89c5008.pdf
- scattered data fitting on the sphere: scattered data fitting on the sphere