CAREER: Optimal Approximation Algorithms in High Dimensions

职业：高维最优逼近算法

• 批准号: 1848508
• 负责人: Akil Narayan
• 金额: $ 40万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1848508&HistoricalAwards=false
• 关键词: CAREER; Optimal; Approximation; Algorithms; Dimensions

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.

现代计算硬件的能力不断增强，使得基于计算机的复杂数学模型模拟成为可能，这些模型可以详细解决重要的物理现象。随着这些计算能力的出现，对模型中包含更复杂的物理交互的需求不断增加，从而对计算资源的压力也随之增加。现代工程设计利用此类模型，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）涉及模型的认知无知形成不确定性。在实际应用中，这些效应的集合会导致依赖于累积高维参数的预测。该项目的重点是开发和部署新颖的、接近最优的实验设计和采样算法，以准确有效地模拟由高维输入参数化的物理模型。该项目的工作涉及最近开发的近似理论结果在计算领域的应用、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。该项目的技术方面是设计的为高维逼近任务提供可行的计算算法和具体的数学保证。完成此任务的三个主要核心组成部分涉及算法的设计、实现和分析，这些算法利用（1）随机和确定性实验和抽样设计，（2）用于识别有效抽样方案的计算算法，以及（ 3）新兴近似范式的策略和技术，例如稀疏近似和降维。一个横切主题是将这些方法应用于现代科学计算中感兴趣的问题。该项目通过开发具有可证明的接近最优性的面向应用的采样设计，对应用逼近理论和计算逼近方法领域做出了基础贡献。该项目的理论研究将近似和线性代数中的经典技术与数据缩减和降阶建模中的新兴算法联系起来。这些算法的实施将显着增强面向目标的设计、参数研究和简化、稀疏和压缩表示、模型验证和校准以及数据驱动模拟的理论理解和计算可行性。该奖项反映了 NSF 的法定使命，并被视为值得通过使用基金会的智力优点和更广泛的影响审查标准进行评估来支持。

项目成果

Fast Barycentric-Based Evaluation Over Spectral/hp Elements

对光谱/hp 元素进行基于重心的快速评估

• DOI: 10.1007/s10915-021-01750-2
• 发表时间: 2022-01
• 期刊: Journal of Scientific Computing
• 影响因子: 2.5
• 作者: Laughton, Edward;Zala, Vidhi;Narayan, Akil;Kirby, Robert M.;Moxey, David
• 通讯作者: Moxey, David

Learning Proper Orthogonal Decomposition of Complex Dynamics Using Heavy-ball Neural ODEs

使用重球神经常微分方程学习复杂动力学的正确正交分解

• DOI: 10.1007/s10915-023-02176-8
• 发表时间: 2023-05
• 期刊: Journal of Scientific Computing
• 影响因子: 2.5
• 作者: Baker, Justin;Cherkaev, Elena;Narayan, Akil;Wang, Bao
• 通讯作者: Wang, Bao

A Bandit-Learning Approach to Multifidelity Approximation

多保真度逼近的强盗学习方法

• DOI: 10.1137/21m1408312
• 发表时间: 2022-02
• 期刊: SIAM Journal on Scientific Computing
• 影响因子: 3.1
• 作者: Xu, Yiming;Keshavarzzadeh, Vahid;Kirby, Robert M.;Narayan, Akil
• 通讯作者: Narayan, Akil

Structure-Preserving Nonlinear Filtering for Continuous and Discontinuous Galerkin Spectral/hp Element Methods

连续和不连续伽辽金谱/hp 元素方法的结构保持非线性滤波

• DOI: 10.1137/20m1337223
• 发表时间: 2021-01
• 期刊: SIAM Journal on Scientific Computing
• 影响因子: 3.1
• 作者: Zala, Vidhi;Kirby, Robert M.;Narayan, Akil
• 通讯作者: Narayan, Akil

Meta-learning with adjoint methods

使用伴随方法的元学习

• 发表时间: 2023-01
• 期刊: Proceedings of the 26th international conference on artificial intelligence and statistics
• 作者: Li, Shibo;Wang, Zeng;Narayan, Akil;Kirby, Robert;Zhe, Shandian
• 通讯作者: Zhe, Shandian