Peijun Li - US grants

The grandeur of mountain topography has for millennia captured the attention of poets, artists, and scientists. How plate tectonic processes of mountain building and mountain erosion by surface processes interact to produce topography over millions of years is now at the forefront of Earth science research. A fundamental question that arises when studying the evolution of mountains is: what did the past topography of mountain ranges look like? This question has proven very difficult to answer. Recent developments in both computer modeling of mountain building and erosional processes, and developments in geochemistry have made progress in reconstructing paleotopography. Advances in new geochemical techniques and mathematics (inverse problem theory) now allow a means of testing computer model predictions with geochemical (thermochronometer) data from rocks exposed at the Earth's surface today. These data record the cooling history of rocks as they are exhumed to the surface by erosion and faulting. This interdisciplinary project is addressing questions and hypotheses that are fundamental to quantifying the evolution of mountain topography including: (1) How can geologically meaningful interpretations of tectonic and geomorphic processes influencing mountain topography be improved from an integration of thermochronometer data, computer modeling, and mathematics? (2) How sensitive are thermochronometer data to different mountain building and erosional processes and how can sampling strategies be optimized to improve interpretations? and (3) What is the magnitude and rate of topographic change that can be resolved from mathematical inversion of thermochronometer data?

To address these questions, this project investigates the forward and inverse problems of mountain topographic evolution with a comprehensive model. Coupled 3D thermal, hydrologic, and kinematic computer models are under development in addition to a surface process model accounting for glacial, fluvial, and hillslope erosional processes. The coupled model is used to explore the sensitivity of thermochronometer data to different processes and mathematically invert a dense network of new and existing thermochronometer samples from the southern Coast Mountains, B.C., for the regional paleotopography. Field work is in progress for the collection of additional data. Several novel mathematical techniques are also under development. In particular, a low pass filter technique and a regularized iterative method are being used to solve the notoriously ill-posed backward parabolic equation and large scale, nonlinear inverse heat transport equation. These problems are by nature interdisciplinary and in the forefront of predicting and interpreting thermochronometer data and mountain topography.

The research objective of this proposal is to examine mathematical issues and develop computational methods for solving important classes of direct and inverse scattering problems that arise in near-field optics modeling. The approach is to treat first the direct scattering problem and then the inverse scattering problem, and increase the complexity of the modeled system as far as possible. The proposed research concerns the following topics: (1) based on a global model for scanning tunneling microscopy, develop an adaptive coupling of the finite element and boundary integral method with error control by an a-posteriori error estimate to solve the direct problem; (2) extend the adaptive coupling of the finite element and boundary integral method to the vector theory of electromagnetic scattering; (3) develop an adaptive treecode algorithm to accelerate the boundary integral evaluations and thus to provide more efficient direct solvers; (4) develop a novel continuation method to solve the inverse problem at a fixed wavenumber. The proposed research will result in a suite of nice modeling and computational techniques, suitable for qualitative and quantitative study of various experimental configurations in
near-field optical systems. Particularly, these techniques will contribute towards better understandings of the complex physical and mathematical problems in near-field optics, and provide valuable information for industry to design and fabricate new optical devices.

Near-field optics has developed dramatically in recent years as an effective approach to breaking the diffraction limit and obtaining images with subwavelength resolution, which leads to vast applications in modern science and technology, including biology, chemistry, materials science, and information storage. Guided by the increasingly accurate and realistic numerical simulations, the significant advances of the near-field optical microscopies have led to integration and miniaturization of optical devices, and many original and reproducible measurements in the vicinity of complex lithographically designed nanostructures. Reciprocally, the practical applications and scientific developments have driven the need for rigorous mathematical models and analysis to describe the scattering of complicated structures, and to accurately compute electromagnetic vector fields and thus to predict the performance of a given structure in near-field and nano optics, as well as to carry out optimal design of new structures. The research lies at the interface of mathematics, physics, engineering, and materials science. It has significant potential for advancing the frontiers of applied and computational mathematics, and for evolving new mathematics and science.

The main goal of the proposed work is to develop new models, computational methodologies and related mathematical theory for remote sensing with applications in chemical and biological threat detections.
In those applications, data is usually gathered by optical sensors and then processed to reconstruct or analyze the properties of sources, such as chemical or biological plume. The tasks are difficult due to a number of challenges. For examples, the problems are ill-posed, the source functions are often random in nature and the data is noisy and incomplete. The PIs and collaborators proposed to investigate three different, but closely related, aspects of some newly emergent remote sensing techniques. In data acquisition, they work on inverse random source problems for the Helmholtz equation. Such problems exist in a wide range of applications in optical science, remote sensing and medical imaging. They aim to develop novel and efficient strategies to reconstruct the distributions of random source functions from incomplete boundary data measurements and perform uncertainty assessments. In data processing, they develop wavelet based multiscale methods in conjunction with the PDE based non-local mean methods for image denoising and information extraction of 3-D Lidar images. The methods integrate several high level mathematical tools, such as geometrical partial differential equations (PDE's), multiscale wavelet transforms and calculus of variation, together with some special properties of the Lidar imagery to achieve better results with fast computations. In data analysis stage, the PIs and collaborators study a novel nonlinear de-mixing method based on Hilbert transform and empirical mode decomposition (EMD) for signal analysis. EMD are designed to handle nonlinear and non-stationary signals, which cannot be easily processed by the traditional wavelet or Fourier based methods. By using EMD, they can extract useful but hidden information through techniques such as instantaneous frequency analysis.

Remote sensing techniques have gained unprecedent attentions due to new challenges in many disciplines including homeland security, military, geosciences, medical science and engineering. Specially, they have become one of the primary tools for data collections in unreachable, unfriendly or hazardous environments. For instance, a most recent advance in chemical or biological threat detection technology uses laser beams and optical sensors to collect signals from targets, such as aerosol plumes. Then the gathered data is processed to identify harmful agents. A key step to succeed is to determine the material properties of the sources, such as whether there exist certain chemical or biological agents, from the collected data sets. This requires solving the so called inverse problems. In practice, they are challenging due to a number of issues. The collected data is often incomplete, random and noisy, the aerosol plume is too thick to ``see'' signals from the center parts , the signatures of the harmful agents and normal aerosol particles are mixed and hard to be separated. In this proposal, the PIs focus their studies in three aspects of the most recent advances in remote sensing techniques with applications in chemical and biological threat detections. They aim to develop novel, robust and efficient computational methods and related mathematical theory to solve the inverse random source problems from incomplete data sets, to remove noise from the signals, and to separate the signatures of different aerosol particles so that harmful agents can be easily identified from the signals. In addition, another major objective is to integrate the research activities with education and training of undergraduate, graduate students and postdocs through seminars and courses.

In scattering theory, due to the complexity of material properties and uncertainty in physical models and parameters, precise modeling and accurate computing present challenging and significant mathematical and computational questions. The PI proposes to develop mathematical models, examine mathematical issues, and design computational methods for new and important classes of direct and inverse problems that arise from the acoustic and electromagnetic wave propagation in complex and random environments. The mathematical modeling techniques and computational methods developed in this project address several key scientific challenges in applied and computational mathematics, which include: (1) multi-scale modeling and computation of the wave propagation in a heterogeneous medium; (2) computational stochastic direct and inverse scattering problems; (3) numerical solution of Maxwell's equations and well-posedness of associated models; (4) global uniqueness, local stability, and numerical solution of the ill-posed inverse scattering problems. The educational plan is to foster greater awareness of the broad and important applications of mathematics so as to attract more students in pursuing a major, a minor, or a graduate degree in mathematics. The proposed education activities include: (1) undergraduate and graduate courses and curriculum development; (2) mentoring of undergraduate, graduate, and postdoc research; (3) organizing summer schools, seminars, and workshops.

The dramatic growth of computational capability and the development of fast algorithms have transformed the methodology for scientific investigation and industrial applications in the field of scattering theory. Reciprocally, the practical applications and scientific developments have driven the need for more sophisticated mathematical models and numerical algorithms to describe the scattering of complicated structures, and to accurately compute acoustic and electromagnetic fields and thus to predict the performance of a given structure, as well as to carry out optimal design of new structures. The proposed computational models and tools are highly promising for qualitative and quantitative study of the complex physical and mathematical problems in optics and electromagnetics, and provide an inexpensive and easily controllable virtual prototype of the structures in the design and fabrication of optical and electromagnetic devices. The research is multidisciplinary by nature and lies at the interface of mathematics, physics, engineering, and materials sciences. In addition, it has significant potential to advance the frontiers of applied and computational mathematics, and even to have impact on other branches of science.

Scattering problems are concerned with the effect that an inhomogeneous medium has on an incident field. Driven by significant applications in diverse scientific areas such as radar and sonar, geophysical exploration, nondestructive testing, medical imaging, near-field optical microscopy, and nano-optics,
the scattering problems have been extensively studied by many researchers, especially for acoustic and electromagnetic waves. However, many theoretical analysis and numerical computation are left undone for elastic waves due to the complexity of the underlying model equations. The research is multidisciplinary by nature and lies at the interface of mathematics, physics, engineering, and materials sciences. It will contribute towards better understandings of the complex physical and mathematical problems in scattering theory of elasticity. It has significant potential for advancing the frontiers of applied and computational mathematics, and for evolving new mathematics and science. The results of the proposed research activities will be disseminated through publications, seminars, minisymposia, conferences, and workshops. The PI will introduce an advanced graduate course and a graduate seminar series. These will aid in the recruitment and retention of talented students with diverse backgrounds throughout the academic pipeline. The software codes and new course materials developed in the project will be disseminated on a public website and will be available for download by the scientific community. The research and educational components will be integrated together to help to train a new generation of researchers and foster greater awareness and interests in applied and computational mathematics with particular applications to scattering theory among graduate students and postdocs.

This project outlines a three-year research plan for developing effective mathematical models, examining fundamental mathematical issues, and designing efficient computational methods for new and important classes of direct and inverse scattering problems in elastic waves. The proposed research builds on the PI?s prior research accomplishments in the area of scattering theory for acoustic and electromagnetic waves. It concerns the following three topics: (1) time-domain obstacle scattering problem; (2) time-harmonic medium scattering problem; (3) inverse random source scattering problem. The mathematical modeling and analysis techniques and computational methods developed in this project will address several key scientific challenges and open problems in direct and inverse scattering theory for elastic waves, which include modeling and computation of the elastic wave propagation in an inhomogeneous medium, numerical solution of the elastic wave equations and well-posedness of the associated model, uniqueness and stability of stochastic inverse source scattering problem. The proposed computational models and tools are highly promising for quantitative study of the complex physical and mathematical problems in elasticity. They have great potentials to provide inexpensive and easily controllable virtual prototypes of the structures in the design and fabrication of novel elastic devices.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Scattering problems, which are concerned with the effect that an inhomogeneous medium has on an incident field, are fundamental in many scientific areas, including geophysical inspection, medical imaging, stealth technology, and nondestructive testing. As one of the key topics in modern mathematical physics, scattering problems have been widely investigated, and a large number of mathematical and numerical results are available, especially for acoustic, elastic, and electromagnetic waves. Recently, scattering problems for biharmonic waves have attracted much attention in the engineering and mathematical communities due to significant applications in thin plate elasticity, such as the design of platonic diffraction gratings and ultra-broadband elastic cloaking. The goal of this project is to address scientific challenges posed by scattering problems of the biharmonic plate wave equation. The nature of the proposed research is multidisciplinary, and the results of the proposed work will be actively shared with other researchers in mathematics, physics, engineering, and materials science. The educational plan is centered around providing interdisciplinary student training as well as developing an integrated curriculum from the undergraduate level to the graduate level. <br/><br/>Compared with the second-order acoustic, elastic, and electromagnetic wave equations, many direct and inverse scattering problems for the fourth-order biharmonic wave equation are not well understood. This project will further the modeling, theory, and algorithmic development of the direct and inverse scattering problems of the biharmonic plate wave equation, addressing scattering in periodic structures, scattering by multiple cavities, and inverse scattering for random sources. Specifically, the principal investigator will develop effective mathematical models and examine mathematical issues for the biharmonic plate wave equation in periodic structures, design an efficient computational approach for biharmonic wave propagation in multiple cavities, and establish mathematical theory on the uniqueness and stability of the inverse problems for the stochastic biharmonic wave equation. Results of this project are intended to contribute to our understanding of complex physical and mathematical problems in the scattering theory of thin plate elasticity. The research has the potential for evolving new science and providing the industry with guidance to design and fabricate new elastic devices in thin plates.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.