CAREER: Explicit Adaptive Methods for Coupled Problems

椭圆型问题中具有可证明的最优误差衰减率的自适应算法的设计是众所周知的，对于抛物型方程有令人鼓舞的结果，而对于双曲型方程则很少有结果。 虽然耦合问题在科学和工程中普遍存在，但其自适应处理仍处于起步阶段。事实上，没有严格证明的自适应是非常流行的，但它的效率受到坚实的数学基础。然而，越来越多的资源参与耦合系统使得自适应算法变得更加重要。所提出的研究的目的是设计，分析和实施自适应算法的耦合问题。提出了以下目标：（i）发展一个系统的框架，用于设计显式自适应算法，在每个感兴趣的量的分辨率之间迭代：（ii）研究一个新的近似概念，能够描述耦合系统的每个组件之间的非线性相互作用;（iii）在椭圆问题、鞍点系统和时间相关问题的背景下导出最优收敛衰减率;（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形状的相互作用。 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，当需要近似相互作用的几个过程时，由于两个主要障碍，所建立的理论无法应用：（i）算法需要在没有完全了解所有相互作用量的情况下做出决策;（ii）近似系统每个组件的能力以高度非线性的方式纠缠在一起。我们建议启动一个系统的研究夫妇的问题，特别强调与活细胞运动的物理模型。 细胞运动建模的难点在于克服考虑多尺度、多维和多组分现象时固有的巨大计算开销。因此，高效和灵活的算法在这方面至关重要。对细胞运动的理解对生物物理学的几个领域产生了影响，例如多细胞生物体中的胚胎发育，组织再生，免疫反应和伤口愈合。此外，拟议中的研究实际上将使能源（石油回收和二氧化碳封存）、环境（地下水污染）和材料科学（隐身和过滤器设计）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).