Ensemble Kalman Filter mininizing Rosenbrock function
We apply EnKF to minimize the Rosenbrock function where \(d=2\) and \(\theta=(x,y)\). \[ L(x,y) = (1-x)^2 + 5 \left(y-x^2\right)^2. \] It has a unique global minimizer \((x,y)=(1,1)\). We choose \(J=10\) such that \[ \theta_0^{(1)},\cdots,\theta_0^{(10)} \stackrel{i.i.d}{\sim} \mathcal{N}([0,0]^\intercal, I_{2}), \] where \(I_2\) is the \(2\times2\) identity matrix, and the subscript is the epoch in \({0,1,\cdots,8,9}\).