Mathematical Sciences: Computational Error Estimation and Adaptive Error Control for Numerical Methods for Differential Equations

研究者使用后验误差估计和自适应误差控制开发和实现微分方程的精确近似方法。主要目标是反应扩散方程组。这些问题在实践中很重要，因为它们在应用科学和工程中作为数学模型出现，包括遗传学、材料科学、化学和生物学等方面的应用。挑战在于计算解的精确近似值，这些解通常包括其空间和时间行为的多个尺度，其行为在很大程度上取决于作为模型一部分规定的参数。此外，将计算作为一种科学工具，需要对近似值的准确性进行估计。解决这些问题的方法是建立在发展后验误差估计的基础上的，该估计根据计算完成后从近似中获得的可计算信息来约束误差。该分析既考虑了在小区间内求解微分方程的困难，又考虑了误差的全局累积。特别地，被近似解的稳定性是通过在近似过程中进行的辅助计算来测量的。结果是鲁棒和可靠的计算误差估计。此外，研究者还研究了数值格式的动态特性，以获得具有更高精度的特定问题的格式，并获得更准确的此类格式的误差估计。该项目的第三个组成部分是基于后验误差估计的自适应误差控制算法的开发和代码实现。该项目的最终目标是公开发布一个能够可靠有效地求解二维和三维反应扩散方程组的并行代码。应用科学中的数学模型，包括遗传学、材料科学、化学和生物学，通常表示为非线性反应扩散微分方程，其中包含源项与扩散能量项的平衡。这种建模的目的是用微分方程的解来描述物理情况。然而，大多数模型的非线性性质使得不可能显式地求解方程；因此，数值近似是科学研究中的一个重要工具。这种方法有其自身的困难。反应和扩散之间的平衡通常是微妙的，很难准确处理。此外，这类问题的解决方案通常在几个尺度上演变，即一些有趣的行为发生在空间和时间的非常局部的区域，而其他行为则在很长时间或更大的空间区域中演变。在实际应用中使用统一的数值离散化会导致大量的计算，即使是最大的计算机也会感到吃力。研究者的目标是产生能够适应目标解的局部行为的数值方案，以便使计算既尽可能准确又尽可能高效。另一个好处是可以报告对精度的估计，这提高了数值分析的科学水平。数学方法是利用从近似中得到的信息对误差进行估计，然后利用这些估计来适应离散化，即使计算自适应。研究者还在并行计算机的代码中实现了这一理论，该代码可以用最少的用户输入来解决非常普遍的问题。其目的是使代码公开可用，从而产生有利于工程和科学基础设施的科学工具。

Estep The investigator develops and implements accurate approximation methods for differential equations using a posteriori error estimates and adaptive error control. The main target is systems of reaction-diffusion equations. Such problems are important in practical terms because they occur as mathematical models in applied science and engineering, including applications in genetics, material science, chemistry, and biology, among others. The challenge is to compute accurate approximations of solutions that generically include multiple scales in their space and time behavior and whose behavior depends strongly on parameters prescribed as part of the model. Moreover, using computation as a scientific tool requires an estimate of the accuracy of the approximation. The approach to these problems is based on developing a posteriori error estimates that bound the error in terms of computable information obtained from the approximation once a computation is completed. The analysis takes into account both the difficulty of solving the differential equation over a small interval and the global accumulation of errors. In particular, the stability properties of the solution being approximated are measured by auxilary computations performed during the approximation. The result is robust and reliable computational error estimates. In addition, the investigator examines the dynamical properties of numerical schemes in the context of obtaining schemes with improved accuracy for a specified problem and obtaining more accurate error estimates for such schemes. The third component of the project is the development and implementation into code of adaptive error control algorithms based on the a posteriori error estimates. The ultimate goal of this project is the public release of a parallel code that can solve systems of reaction-diffusion equations in two and three dimensions reliably and efficiently. Mathematical models in applied science, including genetic s, material science, chemistry, and biology, are often expressed as nonlinear reaction-diffusion differential equations that contain source terms balanced against terms that diffuse energy. The goal of such modelling is to describe the physical situation in terms of the solution of the differential equation. However, the nonlinear nature of most models makes it impossible to solve the equations explicitly; consequently numerical approximation is an important tool in science. This approach has its own difficulties. The balance between reaction and diffusion is usually delicate and difficult to handle accurately. Moreover, solutions of such problems typically evolve on several scales, i.e. some interesting behavior occurs in very localized regions in space and time while other behavior evolves over long times or over larger regions in space. The use of a uniform numerical discretization for a real application results in huge computations that tax even the largest computers. The investigator aims to produce numerical schemes that adapt themselves to the localized behavior of the target solution so as to make the computations both as accurate as desired and as efficient as possible. Another benefit is that the estimate of the accuracy can then be reported, which increases the scientific level of numerical analysis. The mathematical approach is develop estimates of the error that use information obtained from the approximation, which can then be used to adapt the discretization, that is make the computations self-governing. The investigator also is implementing this theory in a code for parallel computers that can solve very general problems with minimum user input. The intent is to make the code publicly available, yielding a scientific tool that benefits the engineering and scientific infrastructure.