Non-iterative Numerical Methods for Boundary Value Problems

静态非线性偏微分方程（PDE）描述了物理和工程中的各种问题。数值格式通常用于近似满足特定边界条件的解。这样的计划通常需要解决一个大的系统耦合非线性离散方程。这种系统的迭代方法在计算上可能非常昂贵，通常导致从业者使用替代问题描述来避免解决完整的边界值问题。调查员提出了一个家庭的快速（非迭代）的方法为广泛的一类静态偏微分方程，“信息流”的方向定义了一个自然的顺序离散方程。在与J.A.的合作中，Sethian，非迭代有序迎风方法（OUM）被引入到各向异性（混合）控制和前向传播中产生的Hamilton-Jacobi偏微分方程。研究者建议将OUM扩展到描述微分博弈和非自治最优控制问题的边值问题。 在与J. Guckenheimer的一项联合工作中，OUM先前被应用于一个特殊的拟线性偏微分方程系统，以近似向量场的不变流形。研究者提出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排序。 这种方法是相关的机器人导航和光刻，地震成像和计算几何，光学和瞬态弹性成像，差分游戏和图像分割等不同的应用。 火星车在火星表面行驶的最佳轨迹是什么？ 地面上某个特定点的传感器会在多长时间内感觉到地下爆炸，感觉到的强度有多大？ 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“故障”后会自动恢复吗？要在真实的时间内解决这些重要的实际问题，需要高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.