Multilevel methods in PDE constrained optimization

PDE 约束优化中的多级方法

• 批准号: 1016177
• 负责人: Andrei Draganescu
• 金额: $ 15万
• 依托单位: University of Maryland Baltimore County
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016177&HistoricalAwards=false
• 关键词: Multilevel; methods; PDE; constrained; optimization

The objective of this project is to develop efficient multilevelalgorithms for large-scale optimization problems constrained by partialdifferential equations (PDEs) with additional inequality constraints (ICs)on the controls and states. The computational revolution of the last twodecades has fostered not only high-resolution numerical computations basedon PDE models, but also a shift from model based simulation to model baseddesign. The latter translates into the question of solving optimizationproblems in order to identify initial and/or boundary values, materialproperties, sources, and other parameters for which the PDE models behavein a desired way. However, in general, by increasing resolution not onlydo optimization problems get larger, but they also become more difficultto solve, thus rendering an ever widening gap between the resolution ofPDEs and that of associated parameter identification problems that can besolved using state of the art resources; in order to take full advantageof these resources, highly efficient algorithms are critical. While suchefficient algorithms have been developed over the past few years, they aremostly restricted to problems without ICs. The addition of ICs on thecontrols and/or states normally increases the difficulty of the problemdue to the presence of Lagrange multipliers that have lower regularitythan the solution. Recent years have witnessed a sensible progress in theoptimization algorithms that target such problems, however, it is expectedthat significant efficiency can further be gained by improvements in thelinear algebra technology needed during the optimization process. In thisproject the PI specifically aims to develop optimal order multilevelpreconditioners for the linear systems arising in the interior pointmethod and semismooth Newton method solution processes of optimizationproblems constrained by linear and semilinear elliptic or parabolic PDEswith ICs on the controls and/or states. For the more difficult problem ofstate ICs, both Lavrentiev and Moreau-Yosida regularizations will beconsidered. The long term goal is to develop efficient multilevelalgorithms for large-scale control problems for fluid flows (Stokes, andNavier-Stokes systems).The results of this project are expected to enable end users of the software - engineers, applied scientists - to solvehigh-resolution, relevant optimization problems at a cost that iscomparable (a small multiple of) to that of performing a singlesimulation. Long-term targeted applications include data assimilation forweather prediction and air contamination modeling. Fast data assimilationfor high resolution models would enable, for example, gaining in a timelymanner a better quantitative understanding of the current state of theatmosphere around a hurricane, thus potentially improving the currentpredictive capabilities. From an educational perspective, the successfulproject will help the PI's efforts in promoting this field of research atUniversity of Maryland Baltimore County (UMBC), and it will allow graduate and undergraduate UMBC students to gainexperience in a research area of strategic interest, which is likely toincrease their opportunities of finding a good position in a researchuniversity or laboratory.

该项目的目标是为受偏微分方程（PDE）约束的大规模优化问题开发有效的多级算法，并在控制和状态上附加不等式约束（IC）。 过去几十年的计算革命不仅促进了基于偏微分方程模型的高分辨率数值计算，而且促进了从基于模型的模拟到基于模型的设计的转变。后者转化为解决优化问题的问题，以便识别初始值和/或边界值、材料属性、来源以及偏微分方程模型以所需方式运行的其他参数。然而，一般来说，通过增加分辨率，不仅优化问题变得更大，而且它们也变得更难以解决，从而导致偏微分方程的分辨率与可以使用最先进的资源解决的相关参数识别问题的分辨率之间的差距不断扩大；为了充分利用这些资源，高效的算法至关重要。虽然过去几年已经开发出如此高效的算法，但它们大多仅限于解决没有 IC 的问题。在控件和/或状态上添加 IC 通常会增加问题的难度，因为存在规律性低于解决方案的拉格朗日乘子。近年来，针对此类问题的优化算法取得了显着进展，然而，预计通过优化过程中所需的线性代数技术的改进，可以进一步获得显着的效率。 在这个项目中，PI 的具体目标是为线性系统开发最优阶多级预处理器，用于线性和半线性椭圆或抛物线 PDE 约束的优化问题的内点法和半光滑牛顿法求解过程中出现的线性系统，并在控制和/或状态上使用 IC。 对于更困难的状态 IC 问题，将考虑 Lavrentiev 和 Moreau-Yosida 正则化。长期目标是为流体流动的大规模控制问题（斯托克斯和纳维-斯托克斯系统）开发高效的多级算法。该项目的结果预计将使软件的最终用户（工程师、应用科学家）能够以与执行单个模拟相当（一小倍）的成本解决高分辨率、相关的优化问题。长期目标应用包括天气预报和空气污染建模的数据同化。 例如，高分辨率模型的快速数据同化将能够及时更好地定量了解飓风周围的当前大气状态，从而有可能提高当前的预测能力。 从教育角度来看，该项目的成功将有助于 PI 努力推动马里兰大学巴尔的摩县分校 (UMBC) 的这一研究领域的发展，并将让 UMBC 的研究生和本科生获得具有战略意义的研究领域的经验，这可能会增加他们在研究型大学或实验室找到好职位的机会。