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。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，为了将感兴趣的问题缩小到适合可用计算平台的可处理的大小，模型顺序缩减是必要的。模型降阶技术允许构建一个降阶模型（ROM），它保留了高保真计算模型的关键特征，同时更便宜地模拟。通常，一个ROM代表系统的一个特定实例，如果系统的属性发生重大变化，则需要从头开始重新计算。因此，构造一个能够捕获系统对其参数依赖性的ROM的问题就出现了。这就是所谓的参数化模型简化的主要目标。它的主要挑战是开发有效的rom，能够准确地预测参数化高保真模型的解，这些参数值位于“训练”集之外。该项目将包括研究生的培训。该项目旨在通过开发一个ROM来解决上述挑战，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思想扩展到参数系统。应用的主要技术有张量分解、低秩张量逼近和补全。特别地，采用正则多进、Tucker（又称高阶奇异值分解，HOSVD）和张量序列等低秩张量近似代替截断奇异值分解这一传统降维的关键技术。得到的简化模型被称为张量ROM （TROM）。该项目的三个主要目标是：(1)为非线性动力系统开发两阶段（训练/评估）TROM，包括离散经验插值方法的张量版本；(2)开发新的低秩张量补全方法，通过使用参数空间的稀疏采样来减轻训练阶段的负担；(3)将TROM集成到逆建模工作流程中，特别是基于表面Cahn-Hilliard方程的多组分脂质膜相场模型的参数估计，以及用于医学成像和地球物理监测的波定量成像。该奖项反映了美国国家科学基金会的法定使命，并通过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，被认为值得支持。

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.