Workshop on Stochastic Multiscale Methods: Mathematical Analysis and Algorithms; August 2009, Los Angeles, CA

在多尺度建模和仿真中，跨尺度交换信息是最重要的挑战之一。在一个多尺度的背景下，信息必然地被截断，当它以更粗的尺度呈现时，它被截断，而当它穿过相反的路径时，它被丰富。信息丢失和损坏，就像它们分别被放大和缩小一样。减少这些误差可以通过信息的概率描述建立在严格的基础上，其中测量的有限维近似为描述信息的粗化和提炼提供了一条分析路径。因此，随机分析为多尺度方法的分析提供了一个合理的背景。这次关于“随机多尺度方法：数学分析和算法”的研讨会将有助于确定在为科学和工程中的各种问题开发随机多尺度方法方面的挑战和机会。不确定性量化、模型验证和不确定性下的优化问题已成为许多科学和工程领域的中心问题。同样，多尺度建模和计算能力正在成为衡量基于模型的预测的标准。因此，科学界在这一时刻理应阐明随机多尺度概念的数学基础，以确保作为经济增长和社会福祉引擎的科学能力稳步发展。这次研讨会将开启数学家、机械师和计算科学家之间的对话，为随机多尺度方法的加速发展奠定基础。计算资源的快速增长提高了人们的期望，即科学知识确实可以成为社会福祉和改善的驱动力。与此同时，在传感器、互联网和其他通信方式的技术发展的帮助下，我们测量周围自然和社会世界的能力显著提高。因此，科学同时面临着以前所未有的分辨率对现实进行复杂的描述，以及用日益复杂的数学模型来描述这种现实的可能性。对物理问题的多尺度描述可以被视为试图利用这些新的机会，同时解决它们不可避免地带来的概念挑战。随机分析和计算科学的共同体基本上沿着不同的道路发展。然而，朝着颠覆性科学影响的方向前进的道路需要重大的交流和合作。本次讲习班“随机多尺度方法：数学分析和算法”的目的是将这两个领域的主要研究人员聚集在一起，以期划定新的视野并形成新的协同作用，以加速多尺度能力的演变，使之成为科学和经济进步的推动者。

Exchanging information across scales is one of the most significant challenges in multiscale modeling and simulation. By necessity, and naturally within a multiscale context, information is truncated as it is presented to a coarser scale, and is enriched as it traverses the opposite path. Information is lost and corrupted as it is, respectively, upscaled and downscaled. Mitigating these errors can be set on rigorous ground through a probabilistic description of information, whence finite-dimensional approximations of measures provides an analytical path for describing the coarsening and refining of information. Stochastic analysis, therefore, provides a rational context for the analysis of multiscale methods. This workshop on "Stochastic Multiscale Methods: Mathematical Analysis and Algorithms"will serve to define challenges and opportunities in the development of stochastic multiscale methods for various problems in science and engineering. Issues of uncertainty quantification, model validation, and optimization under uncertainty have taken center stage in many areas of science and engineering. Likewise, multiscale modeling and computing capabilities are becoming the standard against which model-based predictions are gauged. It thus behooves the scientific community, at this juncture, to elucidate the mathematical foundation of stochastic multiscale concepts so as to ensure a steady evolution of scientific capabilities as engines of economical growth societal well-being. This workshop will initiate a dialog between mathematicians, mechanicians, and computational scientists that will lay the foundation for an accelerated growth in stochastic multiscale methods.Rapid growth in computational resources has heightened the expectation that scientific knowledge can indeed be a driver for societal well-being and betterment. At the same time, our ability to measure the natural and social world around has significantly increased, aided by technological development in sensors, the internet, and other modalities of communication. Science is thus faced, simultaneously, with a complex description of reality at an unprecendented resolution, and the possibility to describe this reality with mathematical models of increasing complexity.Multiscale descriptions of physical problems can be viewed as attempts to take advantage of these new oppotunities, while tackling the conceptual challenges they inevitably present.The communities of stochastic analysis and computational science have evolved essentially along separate paths. The path forward, however, in the direction of disruptive scientific impact, requires significant exchange andcollaboration. It is the intent of this Workshop ``Stochastic Multiscale Methods:Mathematical Analysis and Algorithms'' to bring together leading researchers in these two fields with view to delineate new horizons and forge new synergies that will accelerate the evolution of multiscale capabilities to become an enabler of scientific and economic progress.