Derivation and Simulation in Radiative Transfer Theory

数学科学：辐射传递理论的推导与模拟[摘要]本项目研究高非均质介质中高频波的传播。辐射输运理论对传播波与快速波动的底层介质之间的微观相互作用提供了精确的宏观描述。研究的目的有三个方面。首先，扩展辐射输运理论，使其包括边界条件和表面波与体积波的相互作用。其次，利用基于极化辐射传递概率表示的蒙特卡罗方法，对体积和表面方程进行数值求解。最后，通过设置合适的反问题来评估辐射传输的有效性域，这些反问题允许从边界测量中确定表征底层介质的统计参数。辐射输运理论最近取得的一个重大成功是对地震波在地壳中的传播进行了建模。用微观模型对数百公里范围内的地震波传播进行数值研究仍然是禁止的，但是现在可以利用辐射传递理论和适当的统计方法进行模拟。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一直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.