CAREER: Optimal Approximation Algorithms in High Dimensions

现代计算硬件的不断增强的能力使基于计算机的复杂数学模型的模拟能够非常详细地解决重要的物理现象。随着这些计算能力的出现，在模型中包含更复杂的物理相互作用的需求增加了，从而增加了计算资源的压力。现代工程设计利用这样的模型，这些设计问题通常涉及(1)影响可靠性、成本和故障的众多可调参数，(2)外部影响的不确定性，表现为模型中的随机性，以及(3)涉及模型形式不确定性的认知无知。在实际应用中，这些效应的集合导致依赖于累积的高维参数的预测。该项目侧重于开发和部署新颖的、接近最优的实验设计和采样算法，以准确和有效地模拟由高维输入参数化的物理模型。该项目的工作包括在计算领域应用最近发展的近似理论结果，将理论数学扩展到计算目的的有针对性的进展，以及大规模计算算法的开发和实现。本课题的技术方面旨在为高维逼近任务提供可行的计算算法和具体的数学保证。完成这项任务的三个主要核心组成部分涉及利用(1)随机和确定性实验和抽样设计的最优性特征的算法的设计、实现和分析，(2)识别有效抽样方案的计算算法，以及(3)新兴近似范式（如稀疏近似和降维）的策略和技术。横切主题是将这些方法应用于现代科学计算中的问题。该项目通过开发具有可证明的近最优性的面向应用的抽样设计，对应用近似理论和计算近似方法领域做出了基本贡献。本项目的理论研究将经典的近似和线性代数技术与新兴的数据约简和降阶建模算法联系起来。这些算法的实现将显著提高面向目标的设计、参数研究和约简、稀疏和压缩表示、模型验证和校准以及数据驱动仿真的理论认识和计算可行性。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。

The increasing power of modern computational hardware has enabled computer-based simulation of sophisticated mathematical models that resolve important physical phenomena in great detail. With the advent of these computational abilities has come an increased demand to include more complex physical interactions in the models, and thus an increased strain on computational resources. Modern engineering design utilizes such models, and these design problems typically involve (1) numerous tunable parameters that affect reliability, cost, and failure, (2) uncertainty about external influences manifesting as randomness in the model, and (3) epistemic ignorance involving model form uncertainty. In realistic applications, the collection of these effects leads to predictions that depend on a cumulatively high-dimensional parameter. This project focuses on development and deployment of novel, near-optimal experimental design and sampling algorithms for the accurate and efficient simulation of physical models parameterized by high-dimensional inputs. The work of this project involves the application of recently developed approximation theory results in the computational arena, targeted advances that extend theoretical mathematics for computational purposes, and the development and implementation of algorithms for large-scale computations.The technical aspects of this project are designed to provide feasible computational algorithms and concrete mathematical guarantees for tasks in high-dimensional approximation. The three major core components for the completion of this task involve the design, implementation, and analysis of algorithms that leverage optimality characteristics of (1) random and deterministic experimental and sampling design, (2) computational algorithms for identifying efficient sampling schemes, and (3) strategies and techniques for emerging approximation paradigms such as sparse approximation and dimension reduction. A crosscutting theme is application of these methods to problems of modern interest in scientific computing. This project involves fundamental contributions to the fields of applied approximation theory and computational approximation methods through the development of applications-oriented sampling designs with provable near-optimality. Theoretical investigations of this project connect classical techniques in approximation and linear algebra with emerging algorithms in data reduction and reduced order modeling. The implementation of these algorithms will significantly enhance theoretical understanding and computational feasibility for goal-oriented design, parameter study and reduction, sparse and compressive representations, model verification and calibration, and data-driven simulations.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.