贝叶斯最优连续离散滤波的网格方法和利用函数张量列表示

IF 1.1 4区 工程技术 Q3 ENGINEERING, MULTIDISCIPLINARY Inverse Problems in Science and Engineering Pub Date : 2021-01-06 DOI:10.1080/17415977.2020.1862109
C. Fox, S. Dolgov, Malcolm Morrison, T. Molteno
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引用次数: 1

摘要

非线性系统的最优连续离散滤波需要与Bayes的条件更新交替演化前向Kolmogorov方程(即Fokker-Planck方程)。我们提出了两种数值网格方法来表示网格或网格上的密度函数。对于低维光滑系统,有限体积方法是一种有效的求解方法,它给出的估计收敛于最优连续时间值。我们给出了数值例子,表明该有限体积滤波器能够处理由秩缺失观测结果引起的多模态滤波分布,并且可以在滤波过程中进行贝叶斯最优参数估计。有限体积滤波器中使用的密度函数naïve离散化导致计算成本和存储空间随着维数的增加呈指数增长,这使得有限体积滤波器对高维问题不可行。我们通过使用密度函数的张量列表示(或近似)来规避这种“维度诅咒”,并使用一个有效的隐式PDE求解器来操作张量列表示。我们给出了连续时间跟踪n个弱耦合摆的数值例子,以证明在80维范围内用复密度函数进行滤波。
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Grid methods for Bayes-optimal continuous-discrete filtering and utilizing a functional tensor train representation
Optimal continuous-discrete filtering for a nonlinear system requires evolving the forward Kolmogorov equation, that is a Fokker–Planck equation, in alternation with Bayes' conditional updating. We present two numerical grid-methods that represent density functions on a mesh, or grid. For low-dimensional, smooth systems the finite-volume method is an effective solver that gives estimates that converge to the optimal continuous-time values. We give numerical examples to show that this finite-volume filter is able to handle multi-modal filtering distributions that result from rank-deficient observations, and that Bayes-optimal parameter estimation may be performed within the filtering process. The naïve discretization of density functions used in the finite-volume filter leads to an exponential increase of computational cost and storage with increasing dimension, that makes the finite-volume filter unfeasible for higher-dimensional problems. We circumvent this ‘curse of dimensionality’ by using a tensor train representation (or approximation) of density functions and employ an efficient implicit PDE solver that operates on the tensor train representation. We present numerical examples of tracking n weakly coupled pendulums in continuous time to demonstrate filtering with complex density functions in up to 80 dimensions.
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来源期刊
Inverse Problems in Science and Engineering
Inverse Problems in Science and Engineering 工程技术-工程:综合
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审稿时长
6 months
期刊介绍: Inverse Problems in Science and Engineering provides an international forum for the discussion of conceptual ideas and methods for the practical solution of applied inverse problems. The Journal aims to address the needs of practising engineers, mathematicians and researchers and to serve as a focal point for the quick communication of ideas. Papers must provide several non-trivial examples of practical applications. Multidisciplinary applied papers are particularly welcome. Topics include: -Shape design: determination of shape, size and location of domains (shape identification or optimization in acoustics, aerodynamics, electromagnets, etc; detection of voids and cracks). -Material properties: determination of physical properties of media. -Boundary values/initial values: identification of the proper boundary conditions and/or initial conditions (tomographic problems involving X-rays, ultrasonics, optics, thermal sources etc; determination of thermal, stress/strain, electromagnetic, fluid flow etc. boundary conditions on inaccessible boundaries; determination of initial chemical composition, etc.). -Forces and sources: determination of the unknown external forces or inputs acting on a domain (structural dynamic modification and reconstruction) and internal concentrated and distributed sources/sinks (sources of heat, noise, electromagnetic radiation, etc.). -Governing equations: inference of analytic forms of partial and/or integral equations governing the variation of measured field quantities.
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