Core Concepts
The authors introduce a stable computational method for solving ill-posed partial differential equation (PDE) problems by mapping them to higher-dimensional Schrödinger-type equations, which can be solved in a computationally stable way both forward and backward in time.
Abstract
The content discusses a computational strategy for numerically solving ill-posed PDE problems based on Schrödinger transformation. The key points are:
Ill-posed or unstable PDE problems, such as the backward heat equation and linear convection equations with imaginary wave speed, are challenging to solve numerically due to exponential growth of errors.
The authors propose mapping the original ill-posed PDE to a higher-dimensional Schrödinger-type equation, which is a Hamiltonian system that is time-reversible and can be solved stably both forward and backward in time.
For the backward heat equation, the Schrödinger transformation lifts the problem to a well-posed Schrödinger equation in one higher dimension. The original variable can be recovered by integrating or evaluating the solution over a suitably chosen domain in the extended dimension.
Similar strategies are applied to solve the unstable linear convection equation with imaginary wave speed.
Error analysis is provided for the numerical discretization of the Schrödinger-transformed equations, showing convergence rates.
The methods can be implemented on both classical and quantum computers, with the quantum algorithms also outlined.