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

跨尺度信息交换是多尺度建模和仿真中最重要的挑战之一。 在多尺度环境中，信息在呈现为较粗尺度时必然被截断，而在穿过相反路径时被丰富，信息在放大和缩小时分别丢失和损坏。 减轻这些错误可以设置在严格的地面上通过概率描述的信息，其中有限维近似的措施提供了一个分析的路径来描述粗化和细化的信息。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。 本次研讨会的“随机多尺度方法：数学分析和算法”将有助于定义在科学和工程中的各种问题的随机多尺度方法的发展的挑战和机遇。 不确定性量化、模型验证和不确定性下的优化问题已经成为科学和工程领域的中心议题。 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，科学界有必要在这个节骨眼上阐明随机多尺度概念的数学基础，以确保科学能力作为经济增长和社会福祉的引擎的稳步发展。 本次研讨会将在数学家、机械学家和计算科学家之间展开对话，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描述的问题，以及用越来越复杂的数学模型来描述这种现实的可能性。物理问题的多尺度描述可以被视为利用这些新机会的尝试，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社区已经从本质上发展起来，沿着不同的路径。然而，要想朝着颠覆性科学影响的方向前进，需要大量的交流和合作。 “随机多尺度方法：数学分析和数学方法”讲习班的目的是将这两个领域的主要研究人员聚集在一起，以描绘新的视野，形成新的协同作用，加速多尺度能力的发展，使其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.