Numerical analysis and high-accuracy algorithms for scattering, spectra, and interdisciplinary applications.

适用于散射、光谱和跨学科应用的数值分析和高精度算法。

• 批准号: RGPIN-2019-06886
• 负责人: Nigam, Nilima
• 金额: $ 3.06万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758554
• 关键词: Numerical; analysis; accuracy; algorithms; scattering

Mathematics provides an elegant framework to understand physical and biological phenomena, and the tools with which to probe the detailed behaviour of these systems. In this research program, I will be investigating some specific physical and biological systems from a mathematical viewpoint. A key goal is to develop robust and efficient computational techniques to help further our understanding. My research interests are concentrated in two major areas: (A) numerical analysis for scattering problems  (with a recent focus on associated spectral problems), and (B) applications of mathematics to interdisciplinary problems. There is a natural mathematical connection between these two areas, with novel problems in science and engineering requiring the development of new algorithms. Conversely, careful analysis and simulation can yield deep insight into basic science questions. Computational scattering is concerned with the development and analysis of algorithms for studying wave propagation. My proposed research program in (A) continues the investigation and creation of fast and accurate structure--preserving algorithms to study challenging problems in  scattering. My research has evolved to include eigenvalue problems (EVP). A new direction is the related area of computational spectral geometry, which concerns numerically simulating the interplay between the shape of an object and its spectrum. The most intuitive example is that of the fundamental tones of drumheads: given a drum, can we tell what its tones will be? And conversely, if we wanted to design a drumhead with a given set of tones, how would we find its shape? Some of these questions are classical, and others are of immediate societal relevance.   The development of these computational tools presents many interesting mathematical and computational challenges, which we address in the program.  Amongst the avenues we will explore is the possibility of using ideas from Bayesian optimization to develop new strategies for spectral calculations. The proposed program involves investigation in several related topics, including the analysis and approximation of an unusual constrained EVP arising in fluid--structure interaction. and  the development of high--accuracy computational strategies for mixed Dirichlet--Neumann and Steklov eigenvalue computations. We will provide approximation strategies for  these EVP, and investigate their use for computational spectral geometry.  In area (B),  I bring mathematical tools to bear on interdisciplinary problems, in collaborations with researchers in science and industry. Some existing questions concern applications of ideas from (A). In others,  I will continue efforts to understand at a fundamental level how muscles produce force: how does muscle architecture,  mass and tissue properties change force output? Mathematics can be used to shed light on the behaviour of these and other systems.

数学提供了一个优雅的框架来理解物理和生物现象，以及探索这些系统的详细行为的工具。在这个研究项目中，我将从数学的角度研究一些特定的物理和生物系统。一个关键的目标是开发强大而有效的计算技术，以帮助我们进一步理解。我的研究兴趣集中在两个主要领域：（A）散射问题的数值分析（最近专注于相关的光谱问题），以及（B）数学在跨学科问题中的应用。这两个领域之间有着天然的数学联系，科学和工程中的新问题需要开发新的算法。相反，仔细的分析和模拟可以对基础科学问题产生深刻的见解。计算散射涉及研究波传播的算法的开发和分析。我在（A）中提出的研究计划继续研究和创建快速准确的结构保持算法，以研究散射中具有挑战性的问题。我的研究已经发展到包括特征值问题（EVP）。一个新的方向是计算光谱几何学的相关领域，它涉及数值模拟物体形状及其光谱之间的相互作用。最直观的例子是鼓面的基本音调：给定一个鼓，我们能说出它的音调吗？反过来说，如果我们想设计一个具有给定音调的鼓面，我们如何找到它的形状？其中一些问题是经典的，另一些则具有直接的社会相关性。 这些计算工具的发展提出了许多有趣的数学和计算挑战，我们在程序中解决。我们将探索的途径之一是使用贝叶斯优化的思想来开发光谱计算的新策略的可能性。拟议的程序涉及几个相关主题的调查，包括对流体与结构相互作用中出现的异常约束EVP的分析和逼近。以及发展混合Dirichlet-Neumann和Steklov特征值计算的高精度计算策略。我们将为这些EVP提供近似策略，并研究它们在计算谱几何中的应用。在区域（B）中，我与科学和工业研究人员合作，将数学工具应用于跨学科问题。现有的一些问题涉及（A）中思想的应用。在其他方面，我将继续努力从根本上了解肌肉如何产生力量：肌肉结构、质量和组织特性如何改变力量输出？数学可以用来阐明这些和其他系统的行为。