Numerical solutions for the inverse heat problems in \({\mathbb{R}{}}^ N\) (Q1803352)

Consider the heat equation \(\partial_ t u(x,t)=\partial_ x^ 2 u(x,t)\), \(x\in\mathbb{R}\), \(t>0\). Using the sinc approximation of \(u(x,h/2\pi)\) with the knots \(x_ k=kh\), \(k\in\mathbb{Z}\), the author proposes an effective algorithm to find the sinc approximation of \(u(x,0)\). The knot values of \(u(x,0)\) satisfy a Hermitian Toeplitz system with a matrix \(B_ n\), \(\sigma(B_ n)\subset[e^{-1/4},1]\). The precondition of conjugate gradient method is discussed. The tensor product idea is used to treat the multidimensional problem with \(x\in\mathbb{R}^ N\). Interesting numerical examples are presented. Reviewer's remark. To compute \(u(x,0)\) knowing \(u(x,1)\) one has to apply the algorithm \(2\pi/h\) times. The norm of the inverse evolution operator of the method for a \(h/2\pi\)-time step is estimated by \(e^{1/4}>1\). Consequently, on long time intervals, the proposed method is unstable. Of course, this is caused by the ill-posedness of the inverse problem.