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Dimension Reduction for Nonlinear Stochastic Systems

非线性随机系统的降维

基本信息

批准号：
1953271
负责人：
Pierre Gremaud
金额：
$ 20万
依托单位：
North Carolina State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953271&HistoricalAwards=false
关键词：
Dimension Reduction Nonlinear Stochastic Systems

项目摘要

Predictive computational models play a central role not only in engineering and the sciences, but also in society at large. This project addresses fundamental issues common to virtually all computational models. If a model is too simple, the underlying phenomenon will not be faithfully described; if a model is too complex, its predictive power will be minimal. There is, somewhere, a happy middle; this project is about how to find the right balance in terms of model complexity. How complex should a computational model be to be useful? Both the mathematical community and domain scientists have long been worried about the lower bound – the minimum complexity. Models have be faithful to the underlying biology, chemistry or physics but how much of the science should one include? Quantum and relativistic effects can safely be ignored on many problems but what should be included and what can be left out to describe complex chemical reactions or physiological models? This project is about finding the lowest complexity sufficient for the tasks at hand. Models of complex systems, however, present features that render the task of making prediction with quantified uncertainties challenging. Such features include high-dimensional uncertain input parameters, time and/or space dependent quantities of interest, inherent stochasticity, as well as computational expense of simulating complex models. Research in this project will bring about key advances in modeling under uncertainty by developing mathematical techniques and computational methods to address such challenges in models of complex systems. The overarching goal is to develop methods that allow domain scientists to simplify their models so as to facilitate forward uncertainty quantification and parameter estimation at reasonable costs. Students will be trained and mentored in the interdisciplinary aspects of this project. In addition, his project involves the development of new course material that reflects and addresses challenges in present-day scientific computing.The research in this project makes important contributions to computational modeling under uncertainty by developing mathematical theory and algorithms for (i) multiscale sensitivity analysis of stochastic compartment models, (ii) derivative- and variance-based sensitivity analysis methods for time-dependent stochastic systems, (iii) model complexity reduction in compartment models, and (iv) goal oriented multilevel dimension reduction for fast parameter estimation. The algorithms will be applied to models from biochemistry and physiology. A family of complex models of neurovascular coupling will be analyzed with the methods to be developed during the project, including dimension reduction, model complexity reduction and parameter estimation based on existing murine data. The model complexity reduction framework can be used in a broad range of models, where modelers can benefit from removing certain model components. Overall, by enabling dimension reduction in broad classes of models, the project bridges gaps in the processes of modeling, prediction under uncertainty, and parameter estimation: models with most essential components will be discovered and computational budget can be focused on quantifying uncertainty in only those model parameters that are most important to model output.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.

预测计算模型不仅在工程和科学中发挥着核心作用，而且在整个社会中也发挥着核心作用。该项目解决了几乎所有计算模型所共有的基本问题。如果模型太简单，就无法忠实地描述底层现象；如果模型太复杂，其预测能力就会很小。在某个地方，有一个幸福的中间；该项目是关于如何在模型复杂性方面找到适当的平衡。计算模型应该有多复杂才能有用？数学界和领域科学家长期以来一直担心下限——最小复杂度。模型必须忠实于基础生物学、化学或物理学，但应该包含多少科学知识？在许多问题上可以安全地忽略量子和相对论效应，但是在描述复杂的化学反应或生理模型时应该包括哪些内容以及可以省略哪些内容？该项目旨在找到足以完成手头任务的最低复杂度。然而，复杂系统的模型所呈现的特征使得进行具有量化不确定性的预测任务变得具有挑战性。这些特征包括高维不确定输入参数、感兴趣的时间和/或空间相关量、固有随机性以及模拟复杂模型的计算费用。该项目的研究将通过开发数学技术和计算方法来解决复杂系统模型中的此类挑战，从而在不确定性建模方面取得重大进展。总体目标是开发允许领域科学家简化模型的方法，以便以合理的成本促进前向不确定性量化和参数估计。学生将在该项目的跨学科方面接受培训和指导。此外，他的项目还涉及开发反映和解决当今科学计算挑战的新课程材料。该项目的研究通过开发数学理论和算法，为不确定性下的计算建模做出了重要贡献：（i）随机室模型的多尺度敏感性分析，（ii）基于导数和方差的瞬态随机系统敏感性分析方法，（iii）降低模型复杂性隔室模型，以及（iv）面向目标的多级降维，用于快速参数估计。该算法将应用于生物化学和生理学模型。将使用项目期间开发的方法来分析一系列复杂的神经血管耦合模型，包括降维、降低模型复杂性和基于现有小鼠数据的参数估计。模型复杂性降低框架可用于广泛的模型，其中建模者可以从删除某些模型组件中受益。总体而言，通过在广泛的模型类别中实现降维，该项目弥合了建模、不确定性预测和参数估计过程中的差距：将发现具有最基本组成部分的模型，并且计算预算可以集中于量化那些对模型输出最重要的模型参数中的不确定性。该奖项反映了 NSF 的法定使命，并被认为值得通过使用基金会的知识进行评估来支持优点和更广泛的影响审查标准。