Linear and nonlinear reduced models for the numerical approximation of high-dimensional functions

高维函数数值逼近的线性和非线性简化模型

• 批准号: RGPIN-2021-04311
• 负责人: Guignard, Diane
• 金额: $ 1.68万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759207
• 关键词: Linear; nonlinear; reduced; models; numerical

Partial differential equations (PDEs) are widely used as the mathematical model for problems arising in physics, biology or engineering. In most cases, these problems depend on many parameters, for instance the geometry of the physical domain, the boundary/initial conditions or the coefficients, yielding so-called parametric partial differential equations (PDEs). Nowadays, it is common to include the inherent uncertainty affecting these complex phenomena in the mathematical model. A way to model the uncertainty is to use random variables or random fields. Such PDE with random input has an equivalent parametric deterministic formulation, where the parameter space is endowed with a probability measure. The long-term goal of this program is to design, analyze and implement numerical methods for approximating the solutions to parametric/random PDEs. The main focus will be on model order reduction techniques and adaptive strategies for the so-called forward problem: given a value of the parameter (in some parameter space), find an approximation of the corresponding solution. In order to have methods that are immune to the so-called curse of dimensionality, emphasis will be given to high-dimensional problems, namely when the dimension of the parameter space is large or even infinite. The main objectives are: (I) to compare linear reduced models and apply them to problems of practical interest; (II) to design and analyze nonlinear reduced models with provable performance guarantees; (III) to introduce adaptive strategies for solving random PDEs and compare them to existing methods. A prominent efficient linear reduced model is the reduced basis method. This method hinges on the potential smoothness of the solution with respect to the parameters to build a linear space in which the solution is approximated. The linear space is the span of so-called snapshots, namely the solution of the problem for suitably selected values of the parameters. In some cases, the construction of one linear space for approximating the parameter to solution map is not feasible numerically. It is well-known that nonlinear methods can provide improved efficiency. Recently, several nonlinear reduced methods have been developed and tested numerically. Contrary to linear reduced models, for which the theory is well-understood, little is known in terms of precise performance guarantees for nonlinear strategies. One of the main goals of this program is thus to develop new algorithms for constructing nonlinear reduced models and perform a precise analysis of their performances. This research program will train 3 PhD, 3 MSc and 2 BSc students, who will gain expertise in numerical analysis, and make significant progress in the development of numerical methods for solving parametric/random PDEs. Fast and efficient forward solvers with provable performance guarantees are essential tools in many applications, such as optimal engineering design, weather prediction or medical diagnosis.

偏微分方程 (PDE) 被广泛用作物理、生物学或工程学中出现的问题的数学模型。在大多数情况下，这些问题取决于许多参数，例如物理域的几何形状、边界/初始条件或系数，从而产生所谓的参数偏微分方程（PDE）。如今，在数学模型中包含影响这些复杂现象的固有不确定性是很常见的。对不确定性进行建模的一种方法是使用随机变量或随机场。这种具有随机输入的偏微分方程具有等效的参数确定性公式，其中参数空间被赋予概率测度。该项目的长期目标是设计、分析和实施数值方法来逼近参数/随机偏微分方程的解。主要关注点是模型降阶技术和所谓前向问题的自适应策略：给定参数值（在某些参数空间中），找到相应解决方案的近似值。为了拥有不受所谓维数灾难影响的方法，将重点关注高维问题，即当参数空间的维数很大甚至无穷大时。主要目标是：（I）比较线性简化模型并将其应用于实际感兴趣的问题； (II) 设计和分析具有可证明性能保证的非线性简化模型； (III) 引入求解随机偏微分方程的自适应策略并将其与现有方法进行比较。一个突出的高效线性简化模型是简化基方法。该方法取决于解相对于参数的潜在平滑度，以构建近似解的线性空间。线性空间是所谓快照的跨度，即适当选择参数值的问题的解决方案。在某些情况下，构建一个线性空间来将参数逼近解图在数值上是不可行的。众所周知，非线性方法可以提高效率。最近，已经开发了几种非线性简化方法并进行了数值测试。与理论众所周知的线性简化模型相反，人们对非线性策略的精确性能保证知之甚少。因此，该计划的主要目标之一是开发用于构建非线性简化模型的新算法并对其性能进行精确分析。该研究项目将培养 3 名博士生、3 名硕士生和 2 名理学士学生，他们将获得数值分析方面的专业知识，并在求解参数/随机偏微分方程的数值方法的开发方面取得重大进展。具有可证明性能保证的快速高效的正向求解器是许多应用中必不可少的工具，例如优化工程设计、天气预报或医疗诊断。