反射边界条件下MonteCarlo输运问题的重要抽样和伴随混合方法
Importance Sampling and Adjoint Hybrid Methods in Monte Carlo Transport with Reflecting Boundaries
摘要：
伴随方法是一类重要抽样方法，用于加速输运方程的蒙特卡罗（MC）模拟。理想情况下，伴随方法允许零方差MC估计提供了一个伴随输运方程的解决方案是已知的。混合方法的目的是（i）用确定性方法近似求解伴随输运方程;（ii）使用该解构造具有低方差的无偏MC采样算法。这种方法的问题是，这两个步骤都可能非常昂贵。在本文中，我们简化了步骤（i）和（ii），只计算部分伴随解。更具体地说，在有限的体积散射和复杂的反射在边界的几何形状，我们考虑的情况下，伴随解决方案“忽略”体积散射，从而显着减少步骤（i）和（ii）的自由度。这种几何结构的一个主要应用是使用基于物理的信号模型对环境进行遥感。体积散射，然后采用模拟采样算法（或更确切地说，模拟采样的一个简单的修改称为启发式采样算法），以获得无偏估计。在几何形状与弱体积散射（与域的利益的大小可比的运输平均自由程），我们证明了数值显着的方差减少和加速（品质因数）。
英文摘要：
Adjoint methods form a class of importance sampling methods that are used to accelerate Monte Carlo (MC) simulations of transport equations. Ideally, adjoint methods allow for zero-variance MC estimators provided that the solution to an adjoint transport equation is known. Hybrid methods aim at (i) approximately solving the adjoint transport equation with a deterministic method; and (ii) use the solution to construct an unbiased MC sampling algorithm with low variance. The problem with this approach is that both steps can be prohibitively expensive. In this paper, we simplify steps (i) and (ii) by calculating only parts of the adjoint solution. More specifically, in a geometry with limited volume scattering and complicated reflection at the boundary, we consider the situation where the adjoint solution "neglects" volume scattering, whereby significantly reducing the degrees of freedom in steps (i) and (ii). A main application for such a geometry is in remote sensing of the environment using physics-based signal models. Volume scattering is then incorporated using an analog sampling algorithm (or more precisely a simple modification of analog sampling called a heuristic sampling algorithm) in order to obtain unbiased estimators. In geometries with weak volume scattering (with a domain of interest of size comparable to the transport mean free path), we demonstrate numerically significant variance reductions and speed-ups (figures of merit).
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