Mathematical Sciences: Some Approximation Problems in Differential Equations

9625813 Dieci，这位研究员和他的合作者分析并实现了解决微分方程式中悬而未决的计算问题的技术。主要研究内容如下：(1)不变环面的计算和分支；(2)Lyapunov指数的计算及其应用；(3)含时矩阵的正交分解及其应用；(4)Riccati方程；(5)矩阵的对数和矩阵内插。在每种情况下，该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是与蒂莫·埃罗拉、延斯·洛伦茨、埃里克·范·弗莱克、亚历山德拉·帕皮尼、阿尔多·帕斯夸里以及博士生迈克尔·基夫和贝内德塔·莫里尼合作进行的。研究人员分析并实现了计算参数依赖动力系统不变环面的算法，并研究了它们的分叉。他研究了监测环面上Lyapunov型数的方法，以检测其光滑性，并了解和分类其分叉和破裂机制。还考虑了Lyapunov型数(或Lyapunov指数)的整个计算问题。为了获得依赖于时间的线性系统和非线性系统的可靠稳定性图，需要关于这些数字的定量信息。他研究了该工具在差错控制方面的应用。计算Lyapunov指数的有效而可靠的技术需要研究依赖于时间的矩阵的正交分解。研究人员致力于Riccati方程的数值求解，并开发了基于Gauss Runge-Kutta格式的积分程序。他研究了计算矩阵对数的技术，强调了缓慢变化的矩阵序列的情况及其在矩阵内插中的应用。最后，他正在写一本关于数值动力系统的书，重点是直接逼近技术。微分方程是描述物理现象的最强大的模型之一。它们通过连续描述系统在特定时间在(相)空间中的给定位置如何影响在不久的将来的相邻位置来提供对系统的紧凑描述。如此紧凑的描述的代价是，我们需要求解微分方程，才能获得关于系统的信息。除非是在微不足道的情况下，否则解微分方程是不可能精确的，我们必须求助于数值技术。这位研究人员设计了微分方程式的数值技术，其动机是以下问题：我们应该近似什么，我们应该监控什么？现有技术通常非常可靠地在短时间间隔内跟踪方程的特定解(轨迹)，但它们不足以提供系统的完整描述，这将需要关于长时间间隔上的许多解的知识。对于复杂现象的模型尤其如此，其形式可能是耦合振子系统和耦合非线性微分方程组，或者高维时变矩阵方程。这些案例对这一项目很感兴趣，几乎应用于所有应用科学：从研究复杂的生物和化学相互作用，到时间相关制造过程的最优控制，到一般时间相关现象的稳定性评估。具体而言，主要研究了以下几个问题。(1)对于许多系统，所有的解最终都会逼近相空间的一个有限区域，然后停留在那里。研究人员设计了直接针对这样的最终有限区域的技术，以避免瞬时行为。(2)一个微分方程，因为它模拟一个给定的物理系统，通常具有一些关键性质。例如，在建立刚体旋转模型时，微分方程解应该是刚体旋转。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。如果一种数值方法不能保留本质特征，可能会产生非常令人不快的后果。研究人员致力于保持解的相关几何属性的技术。(3)对于给定的模型，根本问题是稳定性问题。该模型的健壮性有多强？解决方案对模型中的微小变化有多敏感？或者，以一种具有欺骗性的不同但在数学上等价的方式，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错误传播。

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.