Mathematical Sciences: Some Approximation Problems in Differential Equations

小行星9625813 研究人员和他的合作者分析和实施技术，以解决微分方程中的突出计算问题。 研究了以下几个问题：（1）不变环面的计算与分支，（2）李雅普诺夫指数的计算及其应用，（3）时变矩阵的正交分解及其应用，（4）Riccati方程，（5）矩阵的分解与矩阵插值。 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正在与Timo Eirola，Jens Lorenz，Erik货车Vleck，Alessandra Papini，Aldo Pasquali和Ph.D.学生Michael Keeve和Benedetta Morini。 调查分析和实现算法来计算参数依赖动力系统的不变环面，并研究其分叉。 他研究的方法来监测李雅普诺夫型数的环面，以检测其平滑度和理解和分类其分叉和故障机制。 整个问题的计算李雅普诺夫型数（或李雅普诺夫指数）也被认为是。 这些数字的定量信息是必要的，以获得一个可靠的稳定性图片依赖于时间的线性系统和非线性系统。 他研究了这个工具在错误控制中的应用。 计算李雅普诺夫指数的有效和可靠的技术需要研究时变矩阵的正交分解。 研究了Riccati方程的数值解，并开发了基于Gauss Runge-Kutta格式的积分程序。 他研究的技术计算的矩阵矩阵，强调的情况下，一系列的缓慢变化的矩阵及其应用插值的矩阵。 最后，他正在写一本书的数值动力系统，重点是直接逼近技术。 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的（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，以获得有关系统的信息。 解微分方程，除了在平凡的情况下，不可能精确地完成，我们必须求助于数值技术。 研究人员设计微分方程的数值技术，其动机是以下问题：我们应该近似什么，我们应该监控什么？ 现有技术通常在短时间间隔内跟踪方程的特定解（轨迹）方面非常可靠，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内的许多解的知识。 这一需要对于复杂现象的模型尤其如此，这些复杂现象可能采取耦合振荡器和耦合非线性微分方程系统或高维时变矩阵方程的形式。 这些案例在这个项目中很有意义，几乎在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，到时间依赖性制造过程的最佳控制，再到一般时间依赖性现象的稳定性评估。 具体而言，本文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有限区域，然后停留在那里。 研究人员设计了直接针对这种最终有限区域的技术，以避免瞬态行为。(2)微分方程，因为它模拟了一个给定的物理系统，通常具有一些关键性质。 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不能保持这样一个看似简单的特征。 如果一种数值方法不能保持基本特征，可能会产生非常不愉快的后果。 调查工作的技术，保持相关的几何性质的解决方案。(3)对于一个给定的模型，基本问题是稳定性问题。 该模型的稳健性如何？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价的方式，一个系统接近其最终状态的速度有多快？研究人员研究监测指标的方法，以回答这些问题，并应用这项研究来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.