Non-iterative Numerical Methods for Boundary Value Problems

静态非线性偏微分方程组描述了物理和工程中的各种问题。数值格式通常用来逼近满足特定边界条件的解。这类格式通常需要求解一大组耦合的非线性离散方程。这种系统的迭代方法在计算上可能非常昂贵，通常会导致实践者使用替代的问题描述来避免求解完整的边值问题。研究人员针对一类静态偏微分方程组提出了一类快速(非迭代)方法，其中“信息流”的方向定义了离散化方程的自然顺序。在与J.A.Sethian的合作中，引入了非迭代有序迎风方法(OUM)来求解各向异性(&amp；杂交)控制和前向传播中的Hamilton-Jacobi偏微分方程组。研究人员建议将OUMS扩展到描述微分对策的边值问题和非自治最优控制问题。在与J.Guckenheimer的一项联合工作中，OUM先前被应用于一种特殊的拟线性偏微分方程组，以逼近矢量场的不变流形。研究人员建议将不变流形方法扩展到计算边值问题的多值解。所提出的方法的效率源于“因果关系”的概念--计算元素的不明显但自然的顺序。这种方法适用于机器人导航和光刻、地震成像和计算几何、光学和瞬时弹性成像、微分游戏和图像分割等各种应用。火星车在火星表面旅行的最佳轨迹是什么？在地面上的某一给定点，传感器会以多大的延迟和多大的强度感受到地下爆炸？在集成电路制造中，蚀刻和沉积的最佳参数值是什么？飞机避碰的最小安全距离是多少？电力系统会在发生故障后自动恢复吗？要实时地回答这些重要的实际问题，就需要高效而稳健的数值方法来求解相应的偏微分方程组。

Static non-linear Partial Differential Equations (PDEs) describe a variety of problems in physics and engineering. Numerical schemes are commonly used to approximate the solution satisfying particular boundary conditions. Such schemes usually require solving a large system of coupled non-linear discretized equations. Iterative methods for such systems can be very expensive computationally, often leading practitioners to use alternative problem descriptions to avoid solving the full boundary value problem. The investigator proposes a family of fast (non-iterative) methods for a wide class of static PDEs, for which the direction of "information flow" defines a natural ordering on the discretized equations. In a joint work with J.A. Sethian, non-iterative Ordered Upwind Methods (OUMs) were introduced for Hamilton-Jacobi PDEs arising in anisotropic (& hybrid) control and in front propagation. The investigator proposes to extend OUMs to boundary value problems describing differential games and non-autonomous optimal control problems. In a joint work with J. Guckenheimer, the OUMs were previously applied to a special system of quasilinear PDEs to approximate invariant manifolds of vector fields. The investigator proposes to extend the invariant manifold approach to compute multi-valued solutions of boundary value problems.The efficiency of the proposed methods stems from the notion of "causality" -- unobvious yet natural ordering of the elements of computation. This approach is relevant for the applications as diverse as robotic navigation and photolithography, seismic imaging and computational geometry, optics and transient elastography, differential games and segmentation of images. Which trajectory is optimal for a rover traveling on the surface of Mars? With what delay and how strongly will an underground explosion be felt by a sensor at a given point on the surface? What are the optimal parameter values for etching and deposition in the integrated circuit manufacturing? What is the minimum safe distance for the aircraft collision avoidance? Will the electrical power system automatically recover after a "fault"? Answering these important practical questions in real time requires efficient and robust numerical methods for solving the corresponding partial differential equations.