Uncertainty Quantification for Systems Governed by Partial Differential Equations; May 2010; Edinburgh, Scotland

在确定性建模中，假设输入参数是完全已知的。这将导致简化、易于处理的计算，并产生与特定输入选择相对应的输出模拟。然而，大多数物理、生物、社会、经济、金融等过程都包含一定程度的不确定性。不确定性量化(UQ)是在只给出关于输入的统计(即不完整)信息的情况下，确定有关感兴趣过程的输出的统计信息的任务。研讨会的重点是由偏微分方程组(PDE)控制的过程。人们早就认识到，数学模型需要考虑到这种不确定性。然而，UQ在许多应用领域的研究还处于起步阶段。目前，在数学、统计学、科学和工程等不同领域，有许多与UQ相关的活动。然而，根本性和挑战性的数学问题仍然没有得到解决，特别是与解决高维度问题所伴随的“维度诅咒”作斗争仍然是一个悬而未决的问题。研讨会旨在帮助改善这种情况。讲习班汇集了与UQ相关的所有数学和统计领域的专家以及在应用领域工作的科学家和工程师。讲习班的目标如下：回顾这一迅速发展的领域的发展；聚集数学、统计和其他领域相关领域的国际领先专家，并使他们之间进行有效的对话；使工业研究人员了解该领域的最新发展，使数学科学家了解工业面临的重要问题；促进不同相关数学学科(如数值分析、概率论、统计学、高性能计算)之间的交流；鼓励初级研究人员在该领域工作；以及加强来自不同研究领域的研究人员之间的互动。研讨会以三个短期课程开始，旨在让每个人快速了解研讨会中考虑的UQ和随机PDE的不同方面。虽然短期课程对参加研讨会的每个人都很有价值，但对初级研究人员来说尤其有价值。研讨会结束时专门讨论了随机PDE和UQ研究的未来方向，特别强调了需要解决的悬而未决的问题，以便使基于随机PDE的UQ成为政府和行业中必须在涉及风险和不确定性的环境中做出决策的人容易、常规和容易获得的工具。不确定性量化(UQ)是对科学家和工程师对物理、生物、社会、经济、金融、军事等过程的预测中的不确定性进行准确评估的过程。例如，飓风路径的预测、桥梁或飞机的结构完整性、金融工具的未来价格，以及军事装备的寿命到故障，都受到不确定性的影响。因此，准确地量化不确定性对于设计过程中的工程师、政府官员在制定政策决策时(包括与国土安全和军事战略有关的决策)、对评估自然灾害和人为灾害情况下的危险和补救措施的响应小组以及在许多其他环境中都是至关重要的。研讨会的目标是促进UQ科学的最新发展。这一目标是通过将来自大学和行业的数学家、统计学家、工程师和科学家聚集在一起交流思想和开发新方法来实现的。向科学UQ用户进行重大和有效的知识转让也受到影响。研讨会的组织者承诺在研讨会中纳入不同级别、性别、年龄和种族的参与者。还制定了一项周密的计划，以便及时有效地传播关于研讨会上发生的事态发展的信息。研讨会结束时专门讨论了UQ研究的未来方向，特别强调了需要解决的悬而未决的问题，以便使基于随机PDE的UQ成为政府和行业中必须在涉及风险和不确定性的环境中做出决策的人容易、常规和容易获得的工具。

In deterministic modeling, complete knowledge of input parameters is assumed. This leads to simplified, tractable computations and produces simulations of outputs that correspond to specific choices of inputs. However, most physical, biological, social, economic, financial, etc. processes involve some degree of uncertainty. Uncertainty quantification (UQ) is the task of determining statistical information about the outputs of a process of interest, given only statistical (i.e., incomplete) information about the inputs. The particular focus of the workshop are processes governed by partial differential equations (PDEs). It has long been recognized that mathematical models need to account for such uncertainties. However, the science of UQ in many application areas is still in its infancy. There is much current activity in disparate areas of mathematics, statistics, science, and engineering that is relevant to UQ. However, fundamental and challenging mathematical issues remain unsolved, in particular combating the "curse of dimensionality" attendant to solving problems in high dimensions remains an unresolved issue. The workshop is meant to help ameliorate this situation. The workshop brings together experts in all areas of mathematics and statistics relevant to UQ as well as scientists and engineers working in application areas. The objectives of the workshop are as follows: to review developments in this rapidly developing field; to bring together internationally leading experts working in relevant fields of mathematics, statistics, and other areas and enable an effective dialogue between them; to expose industrial researchers to the recent developments in the field and mathematical scientists to the important problems facing industry; to promote communication between the various relevant mathematical disciplines (e.g., numerical analysis, probability theory, statistics, high-performance computing); to encourage junior researchers to work in the field; and to strengthen interactions between researchers coming from different areas of research. The workshop commences with three short courses that are meant to get everyone up to speed on the disparate aspects of UQ and stochastic PDEs considered in the workshop. Although the short courses are of value for everyone attending the workshop, they are especially valuable for junior researchers. The workshop closes with a session devoted to a discussion of future directions in stochastic PDE and UQ research with a special emphasis on the outstanding open problems that need to be solved in order to make stochastic PDE-based UQ a tool that is easily, routinely, and readily available to those in government and industry that have to make decisions in environments involving risk and uncertainty. Uncertainty quantification (UQ) is the process of accurately assessing the uncertainties in predictions made by scientists and engineers about physical, biological, social, economic, financial, military, etc. processes. For example, predicting hurricane paths, the structural integrity of a bridge or airplane, future prices of financial instruments, and the lifetime to failure of military equipment are all subject to uncertainty. Thus, accurately quantifying that uncertainty is of paramount importance to engineers in the design process, to government officials when making policy decisions including those related to homeland security and military strategies, to response teams assessing dangers and remedies in natural and man-made disaster situations, and in many other settings. The workshop objective is to advance the state of the art of the science of UQ. The objective is met by bringing together mathematicians, statisticians, engineers, and scientists from universities and industry to exchange ideas and to develop new methodologies. A significant and effective transfer of knowledge to the users of scientific UQ is also affected. The organizers of the workshop are committed to include a diverse, with respect to rank, gender, age, and ethnicity, set of participants in the workshop. There is also a well-formulated plan for the timely and effective dissemination of information about developments occurring at the workshop. The workshop closes with a session devoted to a discussion of future directions in UQ research with a special emphasis on the outstanding open problems that need to be solved in order to make stochastic PDE-based UQ a tool that is easily, routinely, and readily available to those in government and industry that have to make decisions in environments involving risk and uncertainty.