Notes on scipy.optimize.minimize

Problem: evenly distributed points on ellipsoid surfaces.
Solution: numerically minimizing energy of charges constrained on the ellipsoid surface.
Assuming the ellipsoid satisfies equation:
$$\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2+\left(\frac{z}{c}\right)^2=1$$
the ratios of 3 axes of the ellipsoid surface are $(a,b,c)^T$. The minimization process is:
  1. let
    $$\mathbf{x}^\prime:=\frac{1}{\sqrt{\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2+\left(\frac{z}{c}\right)^2}}(x, y, z)^T$$
    then $\mathbf{x}^\prime$ satisfy the ellipsoid equation;
  2. minimize the energy function
    $$u=\sum_i \sum_{j>i} \frac{1}{\sqrt{(x^\prime_i-x^\prime_j)^2+(y^\prime_i-y^\prime_j)^2+(z^\prime_i-z^\prime_j)^2}}$$
  3. the gradient vector is, e.g. the $x$ component of ith particle:
    $$\partial u/\partial x_i=\sum_{j\ne i} - ((x^{\prime 2}_i/a^2-1)(x^\prime_j-x^\prime_i) + (x^{\prime}_i y^{\prime}_i(y^{\prime}_j-y^{\prime}_i)+x^{\prime}_i z^{\prime}_i (z^{\prime}_j-z^{\prime}_i))/a^2)/d^3$$ with $d=\sqrt{(x^{\prime}_i -x^{\prime}_j)^2+(y^{\prime}_i -y^{\prime}_j)^2+(z^{\prime}_i -z^{\prime}_j)^2}$
    ***NOTE THE GRADIENT IS CALCULATED WITH RESPECT TO $\mathbf{x}$, $\nabla_\mathbf{x}U$, NOT $\mathbf{x}^\prime$, $\nabla_{\mathbf{x}^\prime}U$***
The scipy.optimize.minimize function takes jac=True parameter if the function provides gradient vector, this would boost up the program dramatically.

Comments

Popular posts from this blog

Veni, vidi, vici

Toeplitz, Circulant matrix and discrete convolution theorem

Di-block copolymer analysis: GPC