Collaborative Research: Stochastic Methods for Complex Systems

该项目解决了材料科学、化学、不确定度量化和相关领域的计算挑战。我们感兴趣的数量，如化学反应速率、合金强度等，可以使用一个通用的数学建模框架来估计，该框架可以计算平均值，并提供关于平均值的方差的量化。当一个人希望研究的属性非常罕见，并且需要多次重复的计算机模拟来估计平均值和方差时，通过计算机模拟来估计这些数量尤其具有挑战性。虽然计算能力的增长在一定程度上缓解了这一挑战，但一些模拟，包括化学反应速率，无法通过蛮力计算来解决。研究人员打算开发新的计算机算法和近似方法，而不是依赖原始的计算能力，从而实现更有效、更准确的预测。这包括使用相互作用的数学模型副本，它们在彼此之间传递信息，从而产生更高质量的估计。这些算法和近似将允许对感兴趣的数量进行更可靠的预测，并访问更大的模型（例如更大、更复杂的分子）。从数学上讲，该项目将提供对计算机算法的严格理解，为各个领域的科学家提供信心。多尺度分布出现在各种应用中，包括材料科学、化学和不确定度量化。给定有效的采样策略，我们可以计算各种感兴趣的量，包括集合平均值、平均首次通过时间和罕见事件的概率。然而，高维度的多尺度分布对采样来说尤其具有挑战性。一个例子是由包含超级盆地的能量景观引起的玻尔兹曼分布。这样的景观特征是局部极小值的集群，对应于分布中模态的紧密分组。本项目将研究四种采样算法：加权集合采样、并行复制动态、局部熵平滑和分段确定性马尔可夫过程。加权集合抽样将状态空间划分为若干个仓，然后以最优的方式在这些仓内进行抽样。该项目将研究样本分配策略的选择，并考虑该方法的有限和无限系统大小限制。并行复制动力学也涉及使用样本集合，但是，与加权集合相反，它使用副本有效地找到从一个亚稳区域进入另一个亚稳区域的第一个出口。局部熵平滑通过局部集合采样和平均去除能量景观的超盆地特征。最后，研究人员将使用分段确定性马尔可夫过程来执行拒绝自由抽样，而不需要估计梯度。这些算法将被严格分析，并将在各种现实的高维问题上进行测试，包括化学反应网络和随机分子动力学。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project addresses computational challenges in materials science, chemistry, uncertainty quantification, and related fields. Quantities of interest such as chemical reaction rates, strength of alloys, and more, can be estimated using a common mathematical modeling framework that computes mean values and provides quantification of the variance about the means. Estimating these quantities by computer simulation can be particularly challenging when the property that one wishes to study is rare and many repeated computer simulations would be required to estimate the mean value and the variance. While this challenge is somewhat alleviated by growth in computing power, some simulations, including chemical reaction rates, cannot be addressed via brute force computation. Rather than rely on raw computing power, the investigators intend to develop novel computer algorithms and approximations that will allow for more efficient and more accurate predictions. This includes the use of interacting copies of mathematical models, which communicate information between one another, resulting in higher quality estimates. These algorithms and approximations will allow more faithful prediction of quantities of interest and access to bigger models (such as larger, more complicated molecules). Mathematically the project will provide a rigorous understanding of the computer algorithms, providing confidence to scientists in a variety of fields. Multiscale distributions appear in a variety of applications, including materials science, chemistry, and uncertainty quantification. Given efficient sampling strategies, one can compute a variety of quantities of interest, including ensemble averages, mean first passage times, and probabilities of rare events. However, multiscale distributions in high number of dimensions are particularly challenging to sample. One example is the Boltzmann distribution induced by an energy landscape containing superbasins. Such a landscape features clusters of local minima that correspond to close groupings of modes in the distribution. This project will investigate four sampling algorithms: weighted ensemble sampling, parallel replica dynamics, local entropy smoothing, and piecewise deterministic Markov processes. Weighted ensemble sampling partitions state space into bins and then elects to sample within those bins in an optimal way. The project will investigate the choice of the sample allocation strategy and consider both finite and infinite system size limits for the method. Parallel replica dynamics also involves using an ensemble of samples, but, in contrast to weighted ensemble, it uses the replicas to efficiently find first exits out of one metastable region and into another. Local entropy smoothing removes the superbasin features of the energy landscape by performing local ensemble sampling and averaging. Finally, the investigators will use piecewise deterministic Markov processes to perform rejection free sampling without requiring estimates of gradients. These algorithms will be rigorously analyzed, and they will be tested on a variety of realistic high-dimensional problems including chemical reaction networks and stochastic molecular dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.