Mathematical Sciences: Some Approximation Problems in Differential Equations

9625813 Dieci研究者和他的合作者分析和实施技术来解决微分方程中突出的计算问题。本文主要研究了以下几个问题：(1)不变环面的计算与分岔，(2)Lyapunov指数的计算与应用，(3)时变矩阵的正交分解及其应用，(4)Riccati方程，(5)矩阵的对数与矩阵插值。在每种情况下，项目都结合了理论分析和广泛的计算测试，并开发了工作代码。这项研究的几个方面正在与Timo Eirola、Jens Lorenz、Erik Van Vleck、Alessandra Papini、Aldo Pasquali以及博士生Michael Keeve和Benedetta Morini合作进行。研究者分析并实现了计算参数依赖动力系统的不变环面的算法，并研究了它们的分岔。他研究了在环面上监测Lyapunov型数的方法，以检测它们的平滑性，理解和分类它们的分岔和分解机制。计算李雅普诺夫类型数（或李雅普诺夫指数）的整个问题也被考虑。为了获得时变线性系统和非线性系统的可靠的稳定性图像，需要这些数字的定量信息。他研究了该工具在错误控制方面的应用。有效可靠的李雅普诺夫指数计算技术要求研究时变矩阵的正交分解。研究Riccati方程的数值解，并开发了基于高斯龙格-库塔格式的积分代码。他研究矩阵的对数计算技术，强调慢变化矩阵序列的情况及其在矩阵插值中的应用。最后，他正在写一本关于数值动力系统的书，重点是直接近似技术。微分方程是描述物理现象最有力的模型之一。它们通过连续描述系统在特定时间（相）空间中的给定位置如何在不久的将来影响相邻位置来提供系统的紧凑描述。这种紧凑描述的代价是，我们需要解微分方程以获得关于系统的信息。微分方程的解，除了一些微不足道的情况外，是不可能精确地解出来的，我们必须借助于数值技术。研究者设计了微分方程的数值技术，动机是以下问题：我们应该近似什么，我们应该监测什么？现有的技术在短时间间隔内跟踪方程的特定解（轨迹）方面通常非常可靠，但是它们不足以提供系统的完整描述，这将需要在长时间间隔内了解许多解。这种需要对于复杂现象的模型尤其正确，这些模型可能采用耦合振荡器系统和耦合非线性微分方程，或高维时变矩阵方程的形式。这些案例是本项目感兴趣的，并在几乎所有应用科学中都有应用：从复杂的生物和化学相互作用的研究，到依赖时间的制造过程的最佳控制，再到一般依赖时间的现象的稳定性评估。具体研究了以下几个问题。(1)对于许多系统，所有的解最终都接近相空间的一个有限区域，之后一直停留在该区域。研究者设计了直接针对这些最终有限区域的技术，以避免瞬态行为。微分方程，因为它模拟了一个给定的物理系统，通常包含了一些关键的性质。例如，在对刚性旋转进行建模时，微分方程的解应该是刚性旋转。不幸的是，大多数数值技术甚至无法保留这样一个看似简单的特征。数值方法不能保持基本特征可能会产生非常不愉快的影响。研究者致力于保持溶液相关几何性质的技术。(3)对于给定的模型，其基本问题是稳定性问题。这个模型有多健壮？解决方案对模型中的微小变化有多敏感？或者，用一种看似不同但在数学上等效的方式，一个系统接近其最终状态的速度有多快？为了回答这些问题，研究者研究了监测指标的方法，并将这项研究应用于评估离散化过程中的误差传播。

9625813 Dieci The investigator and his collaborators analyze and implement techniques to resolve outstanding computational issues in differential equations. The following topics are studied: (1) computation and bifurcations of invariant tori, (2) computation of Lyapunov exponents and applications, (3) orthogonal factorizations for time-dependent matrices and applications, (4) Riccati equations, (5) logarithms of matrices and matrix interpolation. In each case the project combines theoretical analysis and extensive computational testing, and develops working codes. Several aspects of this research are being carried out in collaboration with Timo Eirola, Jens Lorenz, Erik Van Vleck, Alessandra Papini, Aldo Pasquali, and Ph.D. students Michael Keeve and Benedetta Morini. The investigator analyzes and implements algorithms to compute invariant tori of parameter-dependent dynamical systems and to study their bifurcations. He studies ways to monitor Lyapunov type numbers on the tori in order to detect their smoothness and understand and classify their bifurcations and breakdown mechanisms. The whole issue of computation of Lyapunov type numbers (or Lyapunov exponents) is also considered. Quantitative information on these numbers is needed in order to obtain a reliable stability picture for time-dependent linear systems and nonlinear systems. He examines applications of this tool to error control. Efficient and reliable techniques for computation of Lyapunov exponents require study of orthogonal factorizations of time-dependent matrices. The investigator works on numerical solution of Riccati equations, and develops an integration code based on Gauss Runge-Kutta schemes. He studies techniques for computation of logarithms of matrices, emphasisizing the case of a sequence of slowly varying matrices and its application to interpolants of matrices. Finally, he is writing a book on numerical dynamical systems with emphasis on direct approximation techniques . Differential equations are one of the most powerful models to describe physical phenomena. They provide a compact description of a system by giving a continuous description of how a given position in (phase) space of a system at a particular time influences neighboring positions in the immediate future. The price for such a compact description is that we then need to solve the differential equation in order to obtain information about the system. Solving the differential equation, except in trivial cases, cannot be done exactly, and we must resort to numerical techniques. The investigator devises numerical techniques for differential equations, motivated by the following question: What should we approximate, what should we monitor? Existing techniques are typically very reliable at tracking a specific solution (a trajectory) of the equation over a short time interval, but they are inadequate to provide a complete description of the system, which would need knowledge about many solutions over long time intervals. This need is particularly true for models of complicated phenomena, which may take the form of systems of coupled oscillators and coupled nonlinear differential equations, or high-dimensional time-varying matrix equations. These cases are of interest in this project and find applications in almost all applied sciences: from the study of complicated biological and chemical interactions, to the optimal control of time-dependent manufacturing processes, to stability assessment for time-dependent phenomena in general. In concrete, the following issues are studied. (1) For many systems, all solutions eventually approach a finite region of phase space, and then stay there afterwards. The investigator devises techniques that target directly such eventual finite regions, to avoid transient behavior. (2) A differential equation, because it mimics a given physical system, usually has built in some key properties. For example, in modeling a rigid rotat ion, the solutions of the differential equation should be rigid rotations. Unfortunately, most numerical techniques will fail at preserving even such a seemingly simple characteristic. The failure of a numerical method to preserve essential characteristics might have very unpleasant effects. The investigator works on techniques that maintain the relevant geometrical properties of the solutions. (3) For a given model, the fundamental question is the one of stability. How robust is the model? How sensitive are solutions to small changes in the model? Or, in a deceivingly different but mathematically equivalent way, how fast is a system approaching its eventual state? The investigator studies ways to monitor indicators in order to answer these questions, and also applies this study to assess error propagation during discretization.