Let $z=x+\sqrt{-1}y$ represent a complex variable with $x$ and $y$ being the real and imaginary part of $z$ respectively. $f$ is a complex function defined on $D$ which is a domain in $\mathbb{C}$. $\mu(f)$ is the Beltrami coefficient which satisfies the following Beltrami equation \begin{equation} \begin{split} \frac{\partial{f}}{\partial{\bar{z}}}=\mu(f)\frac{\partial{f}}{\partial{z}} \end{split} \end{equation}
Suppose $f(z) = u(x,y)+\sqrt{-1}v(x,y)$,
where $u(x,y)=\sum_{i=0}^{m}\sum_{j=0}^{n}x_{ij}B_{ij}(x,y),v(x,y)=\sum_{i=0}^{m}\sum_{j=0}^{n}y_{ij}B_{ij}(x,y),x_{ij},y_{ij} \text{are real numbers}$.
From the Beltrami equation, the Beltrami coefficient can be computed as \begin{equation} \begin{split} \mu(f) = \frac{(u_x-v_y)+\sqrt{-1}(v_x+u_y)}{(u_x+v_y)+\sqrt{-1}(v_x-u_y)} \end{split} \end{equation} Now I want to solve the following optimization problem: \begin{equation}\label{original_model} \begin{split} &\arg\min_{f} \int_{D}|\mu(f)|^2 \mathrm{d}z + \omega\int_{D}|\nabla \mu(f)|^2\mathrm{d}z\\ s.t. \quad &x_{i0},y_{i0},x_{in},y_{in},x_{0j},y_{0j},x_{mj},y_{mj}(i=0,1,\cdots,m,j=0,1,\cdots,n)\quad \text{are given}\\ \end{split} \end{equation} This model seems hard to solve, instead I transform it into the following model: \begin{equation} \begin{split} &\arg\min_{\nu,f} \int_{D}|\nu|^2 \mathrm{d}z + \omega\int_{D}|\nabla \nu|^2 \mathrm{d}z\\ s.t. \quad &x_{i0},y_{i0},x_{in},y_{in},x_{0j},y_{0j},x_{mj},y_{mj}(i=0,1,\cdots,m,j=0,1,\cdots,n)\quad \text{are given},\\ &\nu = \mu(f). \end{split} \end{equation} The above problem can be converted into \begin{equation} \begin{split} &\arg\min_{\nu,f} \int_{D}|\nu|^2 \mathrm{d}z + \omega_1\int_{D}|\nabla \nu|^2 \mathrm{d}z+\omega_2\int_{D}|\nu-\mu(f)|^2\mathrm{d}z\\ s.t. \quad &x_{i0},y_{i0},x_{in},y_{in},x_{0j},y_{0j},x_{mj},y_{mj}(i=0,1,\cdots,m,j=0,1,\cdots,n)\quad \text{are given}. \end{split} \end{equation}
I have a rough idea to solve this model, which is to divide it into two sub problems: We set $\nu_0=0$ initially. Suppose $\nu_n$ is obtained at the $n$th iteration. Fixing $\nu_n$, we first minimize over $f$ to obtain $f_n$. Once $f_n$ is obtained, by fixing $f_n$, we minimize over $\nu$ to obtain $\nu_{n+1}$.
But I don't know how to solve these two sub problems?