Schr\"odingerisation based computationally stable algorithms for ill-posed problems in partial differential equations

We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schr\"odingerisation, introduced in [S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603], which maps all linear PDEs into Schr\"odinger-type equations in one higher dimension, for quantum simulations of these PDEs. Although the original problem is ill-posed, the Schr\"odingerized equations are Hamiltonian systems and time-reversible, allowing stable computation both forward and backward in time. The original variable can be recovered by data from suitably chosen domain in the extended dimension. We will use the backward heat equation and the linear convection equation with imaginary wave speed as examples. Error analysis of these algorithms are conducted and verified numerically. The methods apply to both classical and quantum computers, and we also layout the quantum algorithms for these methods.