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Cliff Notes#

Condensed Cheatsheet/mnemonics of stuff I forget

Overview: Map of Mathematics#

  • Algebra: abstraction of numbers
  • Group Theory: abstraction of symmetry
  • Ring Theory: abstraction of arithmetic
  • Graph Theory: abstraction of relationships
  • Category Theory: abstraction of composition

Linear Algebra#

Differential Geometry#

  • Inner Product: Angle
  • Norm: Length
  • Metric: Distance
  • Measure: Volume/Size
  • \(L_p\) norm: max() Vector Component

  • \(L_0\) norm: Counting Norm

  • Gradient: Derivative of scalar field

  • Divergence: Sink vs Source aka volume density of outward flux
  • Curl: Rotation Rate around point
  • Laplacian: Difference average of neighborhood at a point - value at point
  • Jacobian: Gradient of vector field
  • describes skew/rotation/distortion of differential patch around \(f(\vec p)\)

  • analogue of 1st order Taylor polynomial i.e. best linear approximation rate of change
    \(f(\vec{p} + \varepsilon \vec{h})\approx f(\vec{p} )+\mathbf{J}_{f} (\vec{p})\cdot \varepsilon \vec{h}\)

  • Laplace Equation: Maximal smoothness/mean curvature is zero

  • intuitive as equilibrium steady-state state e.g. diffuse heat flow
  • intuitive as surface has no bumps or local minimas
  • intuition from CMU Discrete Differential Geometry Course: Lecture 18
  • Poisson Equation: Generalization of Laplace Equation

  • intuitive as soap film (pde solution) covering a wire (boundary condition)

  • Manifold: fancy name of a curved space

  • Reimannian manifold: manifold with geodesic metric (Reimannian metric)

  • Functionals: Functions that take functions as inputs (derivative/integral operators)

  • Spaces: Banach Space ⊇ Hilbert Space ⊇ Sobolev Space

  • Banach Space: norm+completeness
  • Hilbert Space: inner-product norm

  • Sobolev Space: nice derivatives up to order S

  • Group: closed under multiplication, commutative, identity function, inverse

  • Lie Group: curved space with a group structure i.e. a group that is a manifold where multiplication is smooth/infinitely differentiable
  • Lie Algebra: tangent space of Lie group
  • Tangent space: linear approximation of a curved space
  • Non-abelian group: non-commutative group i.e. \(a*b \neq b*a\) (e.g. SO(3) rotation group)
  • Dual number: convenient for computation of Lie algebra

Equations#

\[ \begin{align*} del := & & \nabla & = \left(\frac{\partial}{\partial x_1}, \dotsc, \frac{\partial}{\partial x_n}\right)\\ & & & \\ Gradient := & & \nabla f(\boldsymbol{\vec{x}}) & = \left(\frac{\partial f(\boldsymbol{\vec{x}})}{\partial x_1} ,\dotsc ,\frac{\partial f (\boldsymbol{\vec{x}})}{\partial x_n}\right)\\ & & & = \begin{bmatrix} \frac{\partial f}{\partial x_1} (\boldsymbol{\vec{x}})\\ \vdots \\ \frac{\partial f}{\partial x_n} (\boldsymbol{\vec{x}}) \end{bmatrix}\\ & & & \\ Divergence := & & \operatorname{div}\mathbf{F} & = \nabla \cdot \mathbf{F}\\ & & & = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right) \cdot (F_x ,F_y ,F_z)\\ & & & = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}\\ & & & \\ Curl := & & \nabla \times \mathbf{F} & = \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \boldsymbol{\hat{k}}\\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z}\\ F_x & F_y & F_z \end{vmatrix}\\ & & & = \left(\frac{\partial F_z}{\partial y} -\frac{\partial F_y}{\partial z}\right)\boldsymbol{\hat{\imath}} + \left(\frac{\partial F_x}{\partial z} -\frac{\partial F_z}{\partial x}\right)\boldsymbol{\hat{\jmath}} + \left(\frac{\partial F_y}{\partial x} -\frac{\partial F_x}{\partial y}\right)\boldsymbol{\hat{k}}\\ & & & = \begin{bmatrix} \frac{\partial F_z}{\partial y} -\frac{\partial F_y}{\partial z}\\ \frac{\partial F_x}{\partial z} -\frac{\partial F_z}{\partial x}\\ \frac{\partial F_y}{\partial x} -\frac{\partial F_x}{\partial y} \end{bmatrix}\\ & & & \\ Jacobian := & & \mathbf{f} & :\mathbb{R}^{n}\to \mathbb{R}^{m}\\ & & \mathbf{J}_f & = \begin{bmatrix} \dfrac{\partial f(\boldsymbol{\vec{x}})}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{f} (\boldsymbol{\vec{x}})}{\partial x_n} \end{bmatrix}\\ & & & = \begin{bmatrix} \nabla f_1 (\boldsymbol{\vec{x}})\\ \vdots \\ \nabla f_m (\boldsymbol{\vec{x}}) \end{bmatrix}\\ & & & = \begin{bmatrix} \dfrac{\partial f_1 (\boldsymbol{\vec{x}})}{\partial x_1} & \cdots & \dfrac{\partial f_1 (\boldsymbol{\vec{x}})}{\partial x_n}\\ \vdots & \ddots & \vdots \\ \dfrac{\partial f_m (\boldsymbol{\vec{x}})}{\partial x_1} & \cdots & \dfrac{\partial f_m (\boldsymbol{\vec{x}})}{\partial x_n} \end{bmatrix}\\ & & & \\ Laplacian := & & \Delta f & = \nabla^2 f = \nabla \cdot \nabla f = \sum_{i=1}^n \frac{\partial^2 f}{\partial x_i^2}\\ & & & \\ Laplace Equation := & & \Delta f & = 0\\ Poisson Equation := & & \Delta f & = h \end{align*} \]

Spectral Theory#

Legendre polynomial#

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 => \(f : \mathbb{R}^{n} \to \mathbb{R}^{n}, \ f(\lambda \mathbf{v})=\lambda^{k} f(\mathbf{v})\) where \(k,\lambda \in \mathbb{R}\)

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

Approximation Schemes#

  • 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, lagrangian/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 an integral that represents the bending energy of a surface
  • the idea of thin-plate splines is to choose a function f(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 Splines#

  • Gross reduction summary: b-splines with slerp instead of lerp between control points

RBF#

Spherical Splines#