log k = ∑j yj j−s ψj: the sliders set the first coefficients and the decay exponent s, the rest of the 200 coefficients are drawn at random. Small s means a rough coefficient. All demos
$$-\nabla\!\cdot\big(k(y,x)\,\nabla u(y,x)\big) = 1 \ \text{ in } \Omega=(0,1)^2,\qquad u = 0 \ \text{ on } \partial\Omega,$$
$$k = e^{a},\qquad a(y,x) = \sum_{j=1}^{200} y_j\, j^{-s}\, \psi_j(x),\qquad \psi_j(x) = \sin(k_j\pi x_1)\,\sin(l_j\pi x_2)\ \text{ ordered by } k_j^2+l_j^2 .$$
Parameters: \(y = (y_1,\dots,y_{200}) \in [-1,1]^{200}\) (sliders \(y_1,\dots,y_{10}\); the rest 0 or random) and the decay exponent \(s \in [0.25, 1]\).