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

数学为理解物理和生物现象提供了一个优雅的框架，并为探索这些系统的详细行为提供了工具。在这个研究项目中，我将从数学的角度研究一些具体的物理和生物系统。一个关键的目标是开发健壮和高效的计算技术，以帮助我们进一步了解。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(最近的重点是相关的谱问题)，以及(B)数学在跨学科问题中的应用。这两个领域之间有一种天然的数学联系，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洞察。计算散射涉及研究波传播的算法的发展和分析。我在(A)中提出的研究计划继续调查和创建快速而准确的结构保存算法，以研究目标散射中具有挑战性的问题。我的研究已经发展到包括本征值问题(EVP)。一个新的方向是计算光谱几何的相关领域，它涉及到数值模拟物体形状与其光谱之间的相互作用。最直观的例子是鼓手的基本音调：给定一个鼓，我们能说出它的音调吗？反过来，如果我们想要设计一个具有给定音调的鼓头，我们如何找到它的形状？其中一些问题是经典的，另一些则与社会直接相关。然而，这些计算工具的发展带来了许多有趣的数学和计算挑战，我们将在程序中解决这些挑战。我们将探索的途径之一是利用贝叶斯优化的思想来开发新的谱计算策略。建议的程序涉及几个相关主题的研究，包括对流固相互作用中出现的异常约束EVP的分析和近似。并开发了混合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，我与科学和工业中的研究人员合作，将数学工具应用于跨学科问题。现有的一些问题涉及(A)中的想法的应用。在其他方面，我将继续努力在根本层面上理解肌肉是如何产生力量的：肌肉结构、质量和组织属性如何改变力量输出？数学可以用来阐明这些系统和其他系统的行为。

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.