Derivation and Simulation in Radiative Transfer Theory

NSF奖摘要- DMS-0072008数学科学：辐射传输理论的推导和模拟摘要0072008巴尔这个项目研究高频波在高度不均匀介质中的传播。辐射输运理论提供了一个精确的宏观描述传播波和快速波动的底层介质之间的微观相互作用。研究的目标有三个方面。 首先，扩展辐射传输理论，包括边界条件和表面波和体波的相互作用。其次，使用基于偏振辐射传输概率表示的Monte Carlo方法数值求解体积和表面方程。 最后，通过建立合适的反问题来评估辐射传输的有效性，这些反问题允许从边界测量确定表征底层介质的物理参数。辐射传输理论最近的一个主要成功是模拟了地震波在地壳中的传播。 地震波传播超过数百公里的数值研究仍然禁止与微观模型，但模拟现在可以使用辐射传输理论和适当的统计方法。 研究结果有望更好地理解和预测地震。 辐射输运及其扩散近似的另一个应用是近红外光谱。 这种新的方法越来越多地用于医学成像，以监测人体组织的特性。正、逆输运问题的数值模拟仍然是一个活跃的研究领域。该项目提供了数学和数值分析来解决这些问题。

NSF Award Abstract - DMS-0072008Mathematical Sciences: Derivation and Simulation in Radiative Transfer TheoryAbstract0072008 BalThis project studies the propagation of high frequency waves in highly heterogeneous media. Radiative transport theory provides an accurate macroscopic description of the microscopicinteractions between the propagating waves and the rapidly-fluctuating underlying medium. The objectives of the research are threefold. First, extend radiative transport theory to include boundary conditions and surface and volume wave interaction. Second, solve the volume and surface equations numerically using a Monte Carlo method based on a probabilistic representation of radiative transfer with polarization. Finally, assess the domain of validity of radiative transfer by setting up suitable inverse problems that allow determination of statisticalparameters characterizing the underlying medium from boundary measurements.A major recent success of radiative transport theory is the modeling of the propagation of seismic waves in the earth's crust. Numerical study of seismic wave propagation over hundreds of kilometers remains prohibitive with a microscopic model, but simulations are now accessibleusing radiative transfer theory and suitable statistical methods. Results promise to yield better understanding and prediction of earthquakes. Another application of radiative transport and its diffusion approximation is near-infrared spectroscopy. This novel method is increasinglyused in medical imaging for monitoring properties of human tissues. Numerical simulation of forward and inverse transport problems remains an active field of research. The project provides the mathematical and numerical analysis to address these issues.