线性多步法粒子滤波的计算问题与数值实验
Computational issues and numerical experiments for Linear Multistep Method Particle Filtering
AI总结：
本文研究了线性多步法粒子滤波在估计进化系统参数中的有效性，通过骨骼肌代谢问题验证了该方法的计算效率及误差来源，分析了线性多步法与Runge-Kutta方法的对比效果。
摘要：
线性多步法粒子滤波（LMM PF）是一种用于预测由微分方程系统支配的进化系统时间演化的算法。如果某些参数未知，可以通过计算来估计这些参数。本文假设所有未知参数均被视为随机变量，其随机性表示其值的不确定性而非内在属性。因此，系统状态和参数通过概率密度描述，通常以代表性样本形式呈现。该方法在参数估计反问题中特别有吸引力，因为统计方法自然提供了通过分布扩散来评估解不确定性的手段。底层采样技术的计算效率对方法的成功至关重要，因为解的准确性取决于能否从未知参数分布中生成代表性样本。本文在骨骼肌代谢问题上测试了LMM PF，该问题此前在Ensemble Kalman过滤框架中处理。本文利用数值证据突显了主要误差源与所采用线性多步法的影响。最后，我们分析了将LMM替换为Runge-Kutta类积分方法对支持粒子滤波技术的影响。
英文摘要：
The Linear Multistep Method Particle Filter (LMM PF) is a method for predicting the evolution in time of a evolutionary system governed by a system of differential equations. If some of the parameters of the governing equations are unknowns, it is possible to organize the calculations so as to estimate them while following the evolution of the system in time. The underlying assumption in the approach that we present is that all unknowns are modelled as random variables, where the randomness is an indication of the uncertainty of their values rather than an intrinsic property of the quantities. Consequently, the states of the system and the parameters are described in probabilistic terms by their density, often in the form of representative samples. This approach is particularly attractive in the context of parameter estimation inverse problems, because the statistical formulation naturally provides a means of assessing the uncertainty in the solution via the spread of the distribution. The computational efficiency of the underlying sampling technique is crucial for the success of the method, because the accuracy of the solution depends on the ability to produce representative samples from the distribution of the unknown parameters. In this paper LMM PF is tested on a skeletal muscle metabolism problem, which was previously treated within the Ensemble Kalman filtering framework. Here numerical evidences are used to highlight the correlation between the main sources of errors and the influence of the linera multistep method adopted. Finally, we analyzed the effect of replacing LMM with Runge-Kutta class integration methods for supporting the PF technique.
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