Partial differential equation (PDE) models with multiple temporal/spatial scales are prevalent in several disciplines such as physics, engineering, and many others. These models are of great practical importance but notoriously difficult to solve due to prohibitively small mesh and time step sizes limited by the scaling parameter and CFL condition. Another challenge in scientific computing could come from curse-of-dimensionality. In this paper, we aim to provide a quantum algorithm, based on either direct approximations of the original PDEs or their homogenized models, for prototypical multiscale problems in partial differential equations (PDEs), including elliptic, parabolic and hyperbolic PDEs. To achieve this, we will lift these problems to higher dimensions and leverage the recently developed Schr\"{o}dingerization based quantum simulation algorithms to efficiently reduce the computational cost of the resulting high-dimensional and multiscale problems. We will examine the error contributions arising from discretization, homogenization, and relaxation, analyze and compare the complexities of these algorithms in order to identify the best algorithms in terms of complexities for different equations in different regimes.

翻译：具有多个时间/空间尺度的偏微分方程模型在物理学、工程学等多个学科中普遍存在。这些模型具有重要的实际意义，但由于尺度参数和CFL条件限制导致网格和时间步长过小，求解难度极大。科学计算中的另一挑战来自维度灾难。本文旨在针对偏微分方程中的典型多尺度问题（包括椭圆型、抛物型和双曲型偏微分方程），提出基于原始偏微分方程直接逼近或其均匀化模型的量子算法。为实现这一目标，我们将把这些问题提升至更高维度，利用近期发展的基于薛定谔化的量子模拟算法，有效降低由此产生的高维多尺度问题的计算成本。我们将分析离散化、均匀化和松弛化引入的误差贡献，研究并比较这些算法的复杂度，以识别在不同方程和不同参数区间中复杂度最优的算法。