Fast Algorithms for Solving Big Data PDE Parameter Estimation Problems on Cloud Computing Platforms

参数估计问题出现在许多科学和经济学科中，例如，在医学成像、地球物理勘探、无损检测和经济结构估计中。尽管为设计有效的方法付出了巨大的努力，但解决参数估计问题仍然非常具有挑战性，因为参数化方程必须反复求解，直到参数估计具有满意的精度。本研究项目旨在开发和实现有效的数值方法来解决涉及大量测量和偏微分方程的参数估计问题。将开发可重用的开源软件，并将其提供给科学界。该项目正在开发的技术将适用于地球物理，以减少具有高度经济影响的大型调查的计算费用，例如石油和天然气勘探和地下水调查。该项目的结果也将适用于医学成像，以降低医疗保健筛查成本，提高某些疾病的诊断。参数估计可以被表述为带有参数化偏微分方程给出的约束的优化问题。未知数是偏微分方程的参数，它对应于被测物体的物理性质。目标是最小化PDE模拟和测量数据加上一些正则化项之间的不拟合。云计算平台以适中的成本提供对大量计算资源的访问，因此对于解决PDE参数估计问题非常有吸引力。这尤其适用于大数据问题，因为估计的计算成本主要是由PDE模拟的计算成本决定的。在许多情况下，后者随数据数量线性增长。然而，将目前最可靠的参数估计算法直接扩展到大规模并行平台会导致巨大的通信开销和内存需求。本项目旨在设计可替代的定制算法，有效利用云平台，并能够在合理的时间内解决大量数据的参数估计问题。该项目采用的方法基于三个基石。首先，将研究两种降阶建模技术及其组合。偏微分方程将在与问题域对齐的相当粗糙的矩形网格上离散化。在这些网格上，将使用带有自适应多尺度基的降阶模型。这两种技术都将显著降低与PDE模拟相关的计算成本。其次，随机优化方法将被设计为利用大数据集中通常存在的冗余。目标是减少所需的PDE模拟次数，推导参数选择规则，并量化解决方案的不确定性。第三，上述步骤将在大规模并行云计算平台上进行组合和实施。

Parameter estimation problems arise in many scientific and economic disciplines, for example, in medical imaging, geophysical explorations, nondestructive testing, and economic structural estimation. Despite enormous effort put into designing efficient methods, solving parameter estimation problems is still very challenging, since the parametrized equations have to be solved repeatedly until the parameters are estimated with satisfactory accuracy. This research project aims to develop and implement efficient numerical methods for solving parameter estimation problems that involve a large number of measurements and partial differential equations. Reusable, open source software will be developed and made available to the scientific community. The techniques under development in the project will be applicable in geophysics to reduce the computational costs of large surveys that are of high economic impact, for example, in oil and gas exploration and groundwater surveys. The results from this project will also be applicable in medical imaging to reduce health care screening costs and improve diagnosis of certain diseases.Parameter estimation can be formulated as an optimization problem with constraints that are given by the parametrized partial differential equations (PDEs). The unknowns are parameters of the PDEs, which correspond to physical properties of the object to be measured. The objective is to minimize the misfit between PDE simulations and measured data plus some regularization term. Cloud computing platforms provide access to immense computational resources at moderate costs and are thus highly attractive for solving PDE parameter estimation problems. This holds particularly for big data problems since the computational costs of the estimation are dominated by the computational costs for PDE simulations. The latter, in many cases, grows linearly with the number of data. Straightforward extensions of the currently most reliable parameter estimation algorithms to massively parallel platforms, however, lead to huge communication overhead and memory requirement. This project seeks to design alternative tailored algorithms that make efficient use of cloud platforms and are able to solve parameter estimation problem with massive amounts of data in reasonable time. The approach undertaken in this project is based on three cornerstones. First, two reduced-order modeling techniques and their combination will be investigated. The PDEs will be discretized on rather coarse rectangular meshes that are aligned to the problem domain. On these meshes, reduced order models with adaptive multiscale bases will be used. Both techniques will dramatically reduce the computational cost associated with the PDE simulations. Second, stochastic optimization methods will be designed to exploit redundancy typically present in big data sets. The goal is to reduce the required number of PDE simulations, derive parameter selection rules, and quantify uncertainty of the solution. Third, the above steps will be combined and implemented on massively parallel cloud computing platforms.