0th-order theory

order of magnitude physics:

  • dimensional analysis
  • back-of-envelope calculation

numerical approximation to one/two number of significance

1st-order theory

  • (single factor) linear theory
  • linear elasticity theory
  • Fick’s law
  • Darcy’s law
  • Newton–Raphson method
  • asymptotic homogenization
  • linearization

2nd-order theory

Forchheimer equation

Higher-order theory: Asymptotics

所有渐进展开方法都是一个思路: "分解" (可分空间上对一组基投影, 不可分空间上的核变换), (排序), 截断近似。

  • Taylor expansion
  • Spectral methods
    • PCA (Principal Component Analysis) of matrices
    • K-L expansion, PCE (polynomial chaos expansion)
    • Fourier analysis
      • Fourier series
      • Fourier transform
      • generalized Fourier transforms

General non-linear theory

Navier-Stokes equation

Not-categorized

  • coupling of mechanisms (multi-physics)
  • relaxation of assumptions
  • generalization:

Generalization

Countability

  • single - multiple - countably infinite - continuous (bounded) - continuous (unbounded)
  • scalar- vector - sequence - function

Compactness

  • scalar - vector - tensor (matrix)
  • real number - complex number
  • linear equation - system of linear equations
  • linear function - linear transformation - linear functional - linear operator

Complexity of function

  • algebraic equation - transcendental equation - ...

Constant to variable

Deterministic to stochastic

  • random variable - random vector - random sequence - random waveform
  • ODE - SDE; PDE - SPDE

Point to space

  • vector - vector space
  • matrix - matrix analysis
  • function - functional analysis (Hilbert space, Sobolev space)
  • random variable - space of random variables (Gaussian Hilbert space)

Comments

  • Nonlinear models seems to be residues, as common phenomena are well approximated by linear models.
  • 定量学科是个细活，娇贵得很。粗糙的问题用不上它。

🏷 Category=Modeling