CAREER: Explicit Adaptive Methods for Coupled Problems

椭圆型问题上具有可证明的最优误差衰减率的自适应算法的设计得到了很好的理解，抛物型方程得到了令人鼓舞的结果，而双曲型方程得到的结果很少。尽管耦合问题在科学和工程中无处不在，但它们的适应性处理还处于起步阶段。事实上，没有严格论证的特别自适应非常流行，但其效率受到坚实的数学基础的影响。然而，耦合系统中涉及的资源越来越多，这使得自适应算法变得更加重要。提出的研究目的是设计、分析和实现针对耦合问题的自适应算法。提出了以下目标：(i)开发一个系统框架，用于设计显式自适应算法，在每个感兴趣的数量的分辨率之间迭代；（ii）研究了能够描述耦合系统各分量之间非线性相互作用的近似的新概念；（iii）推导出椭圆型问题、鞍点系统和时间相关问题的最优收敛衰减率；（iv）在活细胞运动的背景下挑战新算法，其中数值方法必须面对所涉及的众多现象的复杂性及其与细胞几何结构的相互作用。现代算法能够优化和平衡计算工作，以捕获小细节，而不会过度解析感兴趣的数量。然而，当需要逼近几个相互作用的过程时，由于两个主要障碍，所建立的理论无法适用：(i)算法需要在不完全了解所有相互作用量的情况下做出决策；（ii）逼近系统各组成部分的能力以高度非线性的方式纠缠在一起。我们建议对一些问题进行系统的研究，特别强调与活细胞运动有关的物理模型。细胞运动建模的难点在于克服多尺度、多维、多分量现象所固有的巨大计算开销。因此，在这种情况下，高效和灵活的算法至关重要。对细胞运动的理解影响了生物物理学的几个领域，如胚胎发育、组织再生、免疫反应和多细胞生物的伤口愈合。此外，拟议的研究实际上会使能源（石油开采和二氧化碳封存）、环境（地下水污染）和材料科学（斗篷和过滤器设计）等战略部门受益。

The design of adaptive algorithms with provable optimal error decay rates on elliptic problems are well understood, encouraging results are available for parabolic equations while few results are derived in hyperbolic regimes. Although coupled problems are ubiquitous in science and engineering, their adaptive treatment is in its infancy. In fact, ad hoc adaptivity without rigorous justification is very popular but its efficiency suffers from solid mathematical grounding. Yet, the increasingly amount of resources involved in coupled systems makes adaptive algorithms even more essential. The aim of the proposed research is to design, analyze and implement adaptive algorithms tailored to coupled problems. The following objectives are put forward: (i) develop a systematic framework for the design of explicit adaptive algorithms iterating between the resolution of each quantity of interest; (ii) study a new concept of approximation able to describe the nonlinear interactions between each component of the coupled systems; (iii) derive optimal convergence decay rates in the context of elliptic problems, saddle point systems, and time dependent problems; (iv) challenge the new algorithms in the context of living cell motility where numerical methods must confront the complexity of the numerous phenomena involved and their interactions with the cell geometry. Modern algorithms are able to optimize and balance the computational effort to capture small details without over-resolving the quantity of interest. However, when several processes interacting with each other need to be approximated, the established theory fails to apply due to two major obstructions: (i) the algorithm is required to make decisions without complete knowledge of all interacting quantities; (ii) the abilities to approximate each component of the system are tangled together in a highly nonlinear fashion. We propose to initiate a systematic study of couple problems with special emphasis to physical models related to living cell motility. The difficulty of modeling cell locomotion is to overcome the inherent great computational expense when considering multi-scale, multi-dimensional and multi-component phenomena. Efficient and flexible algorithms are thus critical in this context. The understanding of cell locomotion has impact on several areas of bio-physics such as in embryonic development, tissue regeneration, immune response and wound healing in multi-cellular organisms. In addition, the proposed study will actually benefit strategic departments such as energy (oil recovery and carbon dioxide sequestration), environment (groundwater contamination) and material science (cloaking and filter design).