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

偏微分方程（PDEs）被广泛地用作物理、生物或工程问题的数学模型。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（PDEs）。目前，通常在数学模型中包含影响这些复杂现象的固有不确定性。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一个概率测度。该计划的长期目标是设计、分析和实现近似参数/随机偏微分方程解的数值方法。主要焦点将放在模型降阶技术和所谓的前向问题的自适应策略上：给定参数值（在某些参数空间中），找到相应解的近似值。为了获得不受所谓维数诅咒影响的方法，将重点放在高维问题上，即当参数空间的维数很大甚至无限时。主要目标是：(1)比较线性简化模型并将其应用于实际问题；（二）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，并与现有方法进行比较。简化基法是一种突出的高效线性简化模型。该方法依赖于解相对于参数的潜在平滑性来建立一个近似解的线性空间。线性空间是所谓快照的跨度，即对参数的适当选择值的问题的解。在某些情况下，构造一个线性空间来逼近参数到解映射在数值上是不可行的。众所周知，非线性方法可以提高计算效率。近年来，人们开发了几种非线性约简方法，并进行了数值试验。与理论被充分理解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、3名硕士和2名理学士，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发展方面取得重大进展。快速高效的前向求解器具有可证明的性能保证，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。

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.