Non-iterative Numerical Methods for Boundary Value Problems

静态非线性偏微分方程（PDEs）描述了物理和工程中的各种问题。通常采用数值格式来逼近满足特定边界条件的解。这种方案通常需要求解一个大型的耦合非线性离散方程系统。这种系统的迭代方法在计算上可能非常昂贵，通常导致从业者使用替代问题描述来避免解决完整的边值问题。研究人员提出了一类快速（非迭代）的静态偏微分方程的方法，其中“信息流”的方向定义了离散方程的自然顺序。在与J.A. Sethian的联合工作中，引入了Hamilton-Jacobi偏微分方程在各向异性控制和前传播中的非迭代有序逆风方法（oum）。研究者建议将oum扩展到描述微分对策和非自治最优控制问题的边值问题。在与J. Guckenheimer的联合工作中，oum先前被应用于拟线性偏微分方程的一个特殊系统，以近似向量场的不变流形。研究人员提出了一种扩展不变流形方法来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。这种方法适用于各种应用，如机器人导航和光刻、地震成像和计算几何、光学和瞬态弹性成像、微分游戏和图像分割。对于在火星表面运行的探测车来说，哪条轨道是最优的？地面上某一特定地点的传感器能感觉到地下爆炸的延迟时间和强度有多大？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？电力系统是否会在“故障”后自动恢复？要实时回答这些重要的实际问题，就需要有效而稳健的数值方法来求解相应的偏微分方程。

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.