Schrödingerisation based computationally stable algorithms for ill-posed problems in partial differential equations
CoRR(2024)
摘要
We introduce a simple and stable computational method for ill-posed partial
differential equation (PDE) problems. The method is based on
Schrödingerisation, introduced in [S. Jin, N. Liu and Y. Yu, Phys. Rev. A,
108 (2023), 032603], which maps all linear PDEs into Schrödinger-type
equations in one higher dimension, for quantum simulations of these PDEs.
Although the original problem is ill-posed, the Schrödingerized 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.
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