Dimension Reduction for Nonlinear Stochastic Systems

预测计算模型不仅在工程和科学领域，而且在整个社会中发挥着核心作用。该项目解决了几乎所有计算模型的基本问题。如果一个模型太简单，那么它所隐含的现象就不能被忠实地描述出来;如果一个模型太复杂，那么它的预测能力就很小。在某个地方，有一个快乐的中间;这个项目是关于如何在模型复杂性方面找到正确的平衡。一个计算模型应该有多复杂才有用？数学界和领域科学家长期以来一直担心下限-最小复杂性。模型必须忠实于基本的生物学、化学或物理学，但一个模型应该包括多少科学？在许多问题上，量子和相对论效应可以被安全地忽略，但是在描述复杂的化学反应或生理模型时，应该包括什么，可以省略什么？这个项目是关于找到最低的复杂度足以完成手头的任务。复杂系统的模型，但是，目前的功能，使量化的不确定性预测的任务具有挑战性。这些特征包括高维不确定的输入参数、时间和/或空间相关的感兴趣的量、固有的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，从而在不确定性下的建模方面取得关键进展。总体目标是开发方法，使域科学家简化他们的模型，以促进向前的不确定性量化和参数估计在合理的成本。学生将在该项目的跨学科方面接受培训和指导。此外，他的项目还包括开发新的课程材料，以反映和应对当今科学计算的挑战。该项目的研究通过开发以下数学理论和算法对不确定性下的计算建模做出了重要贡献：（i）随机隔室模型的多尺度灵敏度分析，（ii）基于导数和方差的时变随机系统灵敏度分析方法，（iii）基于导数和方差的时变随机系统灵敏度分析方法，（iv）基于导数和方差的时变随机系统灵敏度分析方法。（iii）房室模型中的模型复杂性降低，以及（iv）用于快速参数估计的面向目标的多级维数降低。该算法将应用于生物化学和生理学模型。一个家庭的复杂模型的神经血管耦合将进行分析的方法，在该项目中，包括降维，模型的复杂性降低和参数估计的基础上，现有的小鼠数据。模型复杂性降低框架可以用于广泛的模型中，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：具有最基本组成部分的模型将被发现，计算预算可以集中在量化那些对模型输出最重要的模型参数的不确定性上。这个奖项反映了NSF的法定使命，并且被认为是值得支持的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。

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.