Tensorial Reduced Order Models: Development, Analysis, and Applications

动力系统的大规模数值模拟在计算科学和工程的各个领域都是普遍存在的。由于生物医学或地球科学中的许多计算任务涉及数十亿个自由度，因此有必要将模型降阶以将感兴趣的问题缩小到适合可用计算平台的易于处理的大小。模型降阶技术允许构建降阶模型（ROM），该降阶模型保留高保真计算模型的关键特征，同时仿真成本低得多。传统上，ROM表示系统的特定实例，并且如果系统经历其属性的显著改变，则需要从头开始重新计算。因此，构建一个ROM的问题，捕捉系统的依赖性，其参数出现。这就是所谓的参数模型简化的主要目标。它的主要挑战是开发有效的ROM，可以准确地预测解决方案的参数化高保真模型的参数值，位于“训练”集之外。该项目将包括培训研究生，旨在通过开发一个ROM来应对上述挑战，该ROM使用现代数值多线性代数的概念和工具，将传统的基于投影的模型降阶的思想扩展到参数系统。所使用的主要技术是张量分解，低秩张量近似和完成。特别是，低秩张量近似，如规范多矢，塔克（a.k.a.高阶SVD，HOSVD）和张量训练代替截断SVD，截断SVD是一种关键传统降维技术。由此产生的简化模型被称为张量ROM（TROM）。该项目的三个主要目标是：（1）开发两阶段（训练/评估）用于非线性动力系统的TROM，包括离散经验插值方法的张量版本;（2）开发与TROM一起使用的新颖的低秩张量完成方法，以通过与参数空间的稀疏采样一起工作来减轻训练阶段的负担;（3）将TROM集成到逆建模工作流中，特别是通过表面Cahn-Hilliard方程建模的多组分脂质膜的相场模型的参数估计，该奖项反映了NSF的法定使命，并被认为是值得支持的，使用基金会的知识价值和更广泛的影响审查标准进行评估。

Large-scale numerical simulation of dynamical systems is ubiquitous in all areas of computational science and engineering. With many computational tasks in biomedical or Earth sciences involving billions of degrees of freedom, model order reduction is necessary to scale the problem of interest down to a tractable size that fits onto the available computing platforms. Model order reduction techniques allow to construct a reduced order model (ROM) that retains the key features of a high-fidelity computational model while being much cheaper to simulate. Conventionally, a ROM represents a specific instance of the system and needs to be recomputed from scratch should the system experience significant changes to its properties. Thus, the question of constructing a ROM that captures the dependence of the system on its parameters arises. This is the main objective of the so-called parametric model reduction. Its main challenge is to develop efficient ROMs that can accurately predict solutions of parametrized high-fidelity models for parameter values that lie outside of the "training" set. The project will include training of graduate students.This project aims at addressing the above-mentioned challenges by developing a ROM that extends the ideas of conventional projection-based model order reduction to parametric systems using the concepts and tools of modern numerical multi-linear algebra. The main techniques utilized are tensor decompositions, low-rank tensor approximation and completion. In particular, low-rank tensor approximations such as canonical polyadic, Tucker (a.k.a. high order SVD, HOSVD) and tensor train are employed in place of truncated SVD, a key conventional dimension-reduction technique. The resulting reduced model is referred to as tensorial ROM (TROM). The three main objectives of the project are: (1) developing a two-stage (training/evaluation) TROM for non-linear dynamical systems, including a tensor version of the Discrete Empirical Interpolation Method; (2) developing novel low-rank tensor completion methods for use with TROM to ease the burden of the training stage by working with a sparse sampling of the parameter space; (3) integrating TROM into inverse modeling workflows, in particular, parameter estimation of the phase-field model for a multi-component lipid membrane modeled by surface Cahn-Hilliard equations, and quantitative imaging with waves for medical imaging and geophysical monitoring.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.