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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=742977
• 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.

部分微分方程（PDE）被广泛用作物理，生物学或工程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，产生所谓的参数偏微分方程（PDE）。如今，通常在数学模型中包括影响这些复杂现象的固有不确定性。建模不确定性的一种方法是使用随机变量或随机字段。具有随机输入的这种PDE具有等效的参数确定性公式，其中参数空间具有概率度量。该程序的长期目标是设计，分析和实施数值方法，以近似于参数/随机PDE的解决方案。主要的重点将放在所谓的正向问题的模型订单降低技术和自适应策略上：给定参数值（在某些参数空间中），找到相应的解决方案的近似值。为了具有免疫所谓的维度诅咒的方法，将重点放在高维问题上，即当参数空间的维度很大甚至无限时。主要目标是：（i）比较线性简化模型并将其应用于实际兴趣的问题； （ii）设计和分析具有可证明性能保证的非线性减少模型； （iii）引入自适应策略来求解随机PDE并将其与现有方法进行比较。降低的基础方法是突出的有效线性还原模型。该方法取决于解决方案相对于参数的潜在平滑度，以构建近似溶液的线性空间。线性空间是所谓的快照的跨度，即适当选择的参数值的问题解决方案。在某些情况下，构造一个线性空间以将参数近似于解决方案映射是不可行的。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还原方法，并通过数值测试。与线性简化的模型相反，该理论对此理论充分理解，在非线性策略的精确绩效保证方面几乎没有知道。因此，该程序的主要目标之一是开发用于构建非线性减少模型并对其性能进行精确分析的新算法。该研究计划将培训3博士学位，3个MSC和2名BSC学生，他们将获得数值分析方面的专业知识，并在用于解决参数/随机PDE的数值方法方面取得了重大进展。具有可证明性能保证的快速有效的前向求解器是许多应用中的必要工具，例如最佳工程设计，天气预测或医学诊断。