Advances in sampling methods with a dependence structure

点击翻译按钮获取中文摘要

To remain competitive in today's world, the Canadian economy rests on advances in science and industry, which increasingly depend on the availability of efficient computational tools. These tools help scientists and analysts evaluate quantities of interest for a system under study. Many of these tools rely on some form of random sampling to approximate quantities for which no explicit formula exists. Random sampling is typically used to "simulate" scenarios of the system. For each scenario, the corresponding value of the quantity of interest is evaluated. By repeating this process several times, a sample of possible values for this quantity is created, which can then be used for inference. This approach is often referred to as the "Monte Carlo method". A drawback of this method is that by nature, random sampling can produce irregularities. Indeed, since scenarios are sampled independently from one another, we may get too many that are similar and/or not enough of a certain type. Quasi-Monte Carlo methods aim at addressing this issue by replacing random sampling by more structured sampling. More precisely, new scenarios are sampled by implicitly taking into account the scenarios sampled so far. This is achieved through the use of low-discrepancy sequences, which are constructions that attempt to place points in a very uniform way in the space over which they are defined. Sophisticated techniques are then used to transform each point into a scenario of the system. These methods have gained considerable attention over the last 20 to 25 years, as they have proven to be useful for solving high-dimensional problems in finance, e.g., involving the simulation of several financial assets over long periods of time. That is, with the same computational effort, they provide estimators with a smaller error than Monte Carlo-based ones. The main goal of this research program is to advance our understanding of quasi-Monte Carlo methods by focusing on the dependence being induced in their underlying sampling schemes. This new approach has the potential to improve the effectiveness of these methods. In addition, we aim to make significant progress in the design and analysis of algorithms that use low-discrepancy sequences to construct approximations adaptively, i.e., learning along the way some of the features of the system to further direct sampling into important regions. Finally, when using low-discrepancy sequences it is more difficult to apply the techniques by which "points" are transformed into "scenarios". This has limited the kinds of models that can be tackled by quasi-Monte Carlo methods. Our research will attempt to address these limitations. This research program will involve at least 10 students from all levels, who will gain valuable expertise on Monte Carlo and quasi-Monte Carlo methods. This research blends theoretical and practical work, so students will be well equipped to transfer the acquired knowledge to either industry or academia.