2006 — 2010 |
Zhao, Shan |
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information |
Fast Simulation of Wave Scattering and Propagation in Inhomogeneous Media With Complex Geometries @ Michigan State University
The goal of the proposed project is to develop innovative numerical approaches to produce fourth order accurate simulation of electromagnetic waves in inhomogeneous media with complex geometries, by using only simple Cartesian grids. The rapid growth of computer capability in the past few decades not withstanding, our ability to model three-dimensional wave propagation and scattering involving geometrically complicated dielectric interfaces is severely limited. Mathematically, the wave solutions are usually non-smooth or even discontinuous across the material interfaces, so that our effort in designing efficient algorithms is easily foiled, unless the complex interfaces are properly treated. The complex interfaces and geometries are commonly tackled by using body-fitted grids in the literature. Even though considerable progress has been made in grid generation, the formation of a good quality body-fitted grid system in geometrically complex domain remains a difficult and time-consuming task. Alternatively, in this project, the investigator will explore how to accommodate dielectric interfaces with complex geometries by using Cartesian grids including the staggered Yee grids. The resulting Cartesian grid methods, which in some sense fit the numerical differentiation operators to the complicated geometries, are less well studied in the literature, in contrast to the body-fitted grid methods. The development of high order Cartesian grid methods with complex interfaces being accurately treated, is of imminent practical importance to efficient wave simulations, but remains unsolved. In this project, innovative fourth order Cartesian grid approaches will be constructed based on the matched interface and boundary (MIB) method newly developed by the investigator and his collaborators for solving partial differential equations (PDEs) involving material interfaces or inhomogeneous media. To address a widespread variety of electromagnetic applications, a complete set of fourth order MIB methods will be developed for different electromagnetic formulations including the Helmholtz equation, the wave equation, and Maxwell's equations, and for different scenarios including the transverse magnetic mode, the transverse electric mode, and fully three-dimensional mode.
Computational electromagnetics (CEM), an interdisciplinary field where one witnesses mutual contributions from mathematicians and engineers is of paramount importance for a wide range of applications, including analysis and synthesis of antenna, calculation of radar cross section (RCS), simulation of ground or surface penetrating radar, to name only a few. The proposed numerical approaches aim to address challenging CEM applications involving large-scale and irregularly shaped structures, for which currently existing methods encounter great difficulties. By delivering more accurate and efficient wave simulations, the proposed methods will lead to breakthroughs in resolving long-standing problems in the real CEM applications. Moreover, the proposed methods will have considerable impact on other challenging interface problems in scientific computing, such as the immersed interface and moving interface problems in fluid dynamics, electrostatic interface problems for structural prediction of large biomolecules in computational biology.
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0.913 |
2006 — 2010 |
Zhao, Shan Bates, Peter Wei, Guowei [⬀] |
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information |
Mathematical Modeling of Biomolecular Surfaces @ Michigan State University
Molecular surfaces are paramount to the stability and solubility of macromolecules, such as proteins, DNAs and RNAs, because their features determine how other molecules interact with them to produce the desired function. Important applications of molecular surfaces include protein folding, protein-protein interfaces, protein-membrane interaction, DNA binding and bending, macromolecular docking, enzyme catalysis, solvation energy and molecular dynamics. Despite extensive effort in the past three decades, the generation and analysis of molecular surfaces remain a bottleneck in the structure prediction of biomolecules. To overcome this difficulty, investigators introduce a partial differential equation (PDE) based approach of molecular surface modeling. A set of curvature-controlled PDEs is designed for molecular surface generation. Two new concepts, molecular multiresolution surfaces and minimal molecular surfaces, which naturally arise in the proposed PDE modeling, are introduced to the field of molecular biology. The proposed molecular multiresolution surfaces provide a unified description of the van der Waals surface, solvent accessible surface, and solvent excluded surface. The proposed minimal molecular surface is also consistent with surface free energy minimization. Numerical techniques, including PDE solution algorithms and isosurface extraction schemes will be investigated. Numerical tests will be carried out to validate the proposed ideas.
This project will lead to reliable methods in the structure prediction of biomolecules. Structural information is particularly important for rational drug design efforts specifically targeting proteins and/or membranes. It can be expected that the availability of more protein/membrane structures will lead to accelerated drug development efforts. The mathematical analysis proposed in this project will also impact image processing, pattern recognition, and computer vision. The results of the proposed project can be used for detailed control of PDE surfaces according to industrial and scientific needs and objectives. In particular, nonlinear diffusivities in the proposed equations can be used to control the shape of a PDE surface locally according to its curvature. The proposed research has a strong educational component. The project will support the training of a Ph.D. student researcher in mathematical biology. PDE-based molecular surface modeling is a suitable topic for graduate theses and undergraduate projects. Project results will be incorporated in graduate courses in biological modeling and simulation.
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0.901 |
2010 — 2014 |
Zhao, Shan |
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information |
Modeling, Algorithms and Computation of Electromagnetic Wave Interacting With Dispersive Interface @ University of Alabama Tuscaloosa
The goal of the proposed project is to develop novel mathematical and simulation tools for studying electromagnetic wave interacting with arbitrarily curved dispersive interface. Great challenges exist in developing efficient and reliable numerical methods for such interactions. Physically, jumps in wave solution and its derivatives across the dispersive interface are time dependent. Numerically, the existing algorithms suffer a serious accuracy reduction due to their incapability to handle such time variant jumps. Computationally, this interface error will be significantly amplified when coupling with the staircasing approximation in treating curved interface. Due to these challenges, an extremely expensive mesh resolution of about 100 grid points per wavelength was commonly practiced in the metamaterial simulations. In this project, the investigator will rigorously analyze the time dependence and cross coupling of electromagnetic field components at the dispersive interface. Novel formulations will be derived for commonly used dispersive material and metamaterial models to convert time dependent jump conditions into time independent ones and to minimize the cross coupling. Building on these mathematical modeling, a second order accurate interface algorithm will be developed to deal with arbitrarily curved dispersive interface, by using only a simple Cartesian grid. This higher order of accuracy will promise a higher numerical resolution, so that the computational burden of the existing simulations can be significantly relieved.
Dispersive media are ubiquitous in nature, such as in biological tissues, rocks, soils, and plasma. The numerical simulation of dispersive media is crucial to a wide range of electromagnetic and optical applications, such as microwave imaging for early detection of breast cancer, double negative metamaterial based subwavelength imaging system, and cloaking devices. The proposed mathematical modeling, algorithm development, and numerical computations will address key scientific challenges in an interdisciplinary filed lying at the interface of computational mathematics, physics, and electric engineering. The planned research activities will bring new advances to computational mathematics and lead to reliable simulation tools for the characterization, analysis, and design of various practical engineering devices and systems. These tools in turn may offer a better means for analyzing or calibrating some basic physical laws, such as the one governing the resolution limit of the sub-diffraction imaging system. In addition, this project will provide an interdisciplinary research training environment which could inspire and promote more students to purse careers in science and engineering.
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0.913 |
2013 — 2017 |
Zhao, Shan Geng, Weihua (co-PI) [⬀] |
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information |
Matched Alternating Direction Implicit (Adi) Schemes For Solving the Nonlinear Poisson-Boltzmann Equation With Complex Dielectric Interfaces @ University of Alabama Tuscaloosa
The goal of the proposed project is to develop second order interface methods embedded in the alternating direction implicit (ADI) framework for solving the 3D nonlinear Poisson-Boltzmann (PB) equation with complex dielectric interfaces. Efficiency and accuracy are known to be the two major difficulties for solving the nonlinear PB equation numerically. The efficiency concern stems from the needs for solving the PB equation in demanding applications, such as one-time solution to systems with large spatial degrees of freedom, and/or million-time solutions in dynamical simulations. The accuracy concern is due to various challenging features of the PB model, including piecewisely-defined dielectric constants, a strong nonlinearity, singular point charges, and complex dielectric interfaces. Without addressing these features, fine meshes have to be used for a reliable simulation, which in turn impairs efficiency. In this project, a new pseudo-transient continuation formulation will be constructed based on a suitable regularization setting so that the singular charges are represented analytically. The nonlinear term of the PB equation will be integrated exactly with time splitting techniques. To deal with piecewise dielectric constants, a tensor product decomposition of 3D interface conditions will be carried out to derive essentially 1D jump conditions so that the dielectric interface can be accommodated along each Cartesian direction in an alternating manner. Fast algebraic solvers will be developed for solving matrices of each Cartesian direction. Consequently, the proposed matched ADI approaches not only maintain both the simplicity of Cartesian grids and the efficiency of the Thomas algorithm, but also achieve spatially second order accuracy in resolving complex dielectric interfaces.
The electrostatic interactions are vital not only for the study of biological and chemical systems and processes at the molecular level, but also for the design of semiconductor devices at the nanoscale. The PB model, in which the electrostatic interactions are computed implicitly via a mean force approach, can surprisingly well describe the electrostatics of a charged system. This model finds broad applications in science and engineering, such as modeling the charged polymers and surfactants in interface and colloid science, studying transistors on very large scale integration (VLSI) semiconductor devices in nanotechnology, and analyzing structure, function, and dynamics of solvated biomolecules including proteins and DNAs in molecular biology. The proposed mathematical modeling, algorithm development, and numerical computations will address key scientific challenges in interdisciplinary fields involving computational mathematics, chemistry, biology, and electrical engineering. The planned research activities will bring new advances to computational mathematics and lead to reliable simulation tools for the electrostatic analysis of various physical, chemical, and biological systems/devices. In addition, this project will provide interdisciplinary research and training opportunities for students pursing careers in science and engineering.
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0.913 |
2018 — 2020 |
Zhao, Shan Geng, Weihua (co-PI) [⬀] |
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information |
Cbms Conference: Mathematical Molecular Bioscience and Biophysics @ University of Alabama Tuscaloosa
This award supports the 2019 NSF-CBMS conference "Mathematical Molecular Bioscience and Biophysics," hosted by University of Alabama, Tuscaloosa, during May 13-17, 2019. The conference will feature Professor Guowei Wei of Michigan State University as the Principal Lecturer. Mathematical molecular bioscience and biophysics has been emerging as a promising interdisciplinary research area at the interface of mathematics and biology, driven by the trends of contemporary life sciences that transform biosciences from macroscopic to microscopic or molecular, and from qualitative and phenomenological to quantitative and predictive. The conference will promote biological studies for solving cutting edge problems at molecular level, so that mathematics can play a more important role in addressing fundamental challenges in molecular biosciences and biophysics. This conference consists of ten principal lectures, together with supplemental presentations by other experts and round table discussions. The conference aims to attract many junior mathematicians, including undergraduate and graduate students, postdoctoral fellows, and young faculty, to enter this new interdisciplinary field. In addition, the meeting will benefit the hosting university by enhancing its research program and raising its visibility to peer institutes in the southeastern region.
Mathematical Molecular Bioscience and Biophysics (MMBB) concerns the development of mathematical theories, models, methods, schemes, and algorithms for elucidating molecular mechanisms and for solving open problems at the forefront of molecular biosciences and biophysics. The lecture series will provide a thorough overview of the MMBB literature to mathematical and biological societies. Numerous areas of mathematics, including differential equations, functional analysis, harmonic analysis, Lie groups, Lie algebras, geometry, graph theory, topology, combinatorics, multiscale modeling, inverse problems, optimization, machine learning, stochastic analysis, uncertainty quantification, fuzzy logic, statistical inference, and nonparametric regression, have found important applications in MMBB and many successful applications will be illustrated in this conference. Biological open problems at the forefront of the MMBB, such as those associated with drug design and discovery, which will stimulate new research directions in modeling and computation of biomolecular structure, function, dynamics and transport, will be identified. For more information, please refer to the conference webpage: http://cbms.ua.edu
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.
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0.913 |
2018 — 2021 |
Zhao, Shan |
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information |
Collaborative Research: a Regularized Poisson Boltzmann Model For Fast Computation of the Ensemble Average Polar Solvation Energy @ University of Alabama Tuscaloosa
This collaborative research project will develop mathematical models and simulation tools for studying the interactions between large biological molecules, for example proteins, and surrounding water molecules modeling an aqueous environment. The energy computation for characterizing such interactions is complex because the structures of biomolecules are not completely fixed or rigid, and the surrounding water molecules are also in constant motion. Thus, to deliver quantities that are comparable with experimentally-measurable energies, one must account for these conformational changes in the corresponding mathematical description. A new theoretical model will be formulated in this project by combining appropriate biophysical considerations with mathematical advances, allowing simulations to mimic the effect of conformational changes in both macromolecule and water atoms. The proposed mathematical development will benefit researchers in molecular biosciences and biophysics. Moreover, the proposed models and algorithms will be implemented in DelPhi, which is distributed free of charge to academic users, to ensure extensive usage by practitioners from mathematics, chemistry, physics, and biology. In addition, this project will provide interdisciplinary research and training opportunities for undergraduate and graduate students in biological modeling, computation and mathematical analysis.
Experimentally-observable solvation energies are ensemble averaged. However, direct Poisson-Boltzmann (PB) calculations of such energies require the generation of a representative ensemble of structures in terms of thousands of snapshots, which is computationally very expensive. Tremendous savings in computational time can be achieved if one can calculate the ensemble average solvation energy by employing a single structure by mimicking the effect of conformation changes of macromolecules via heterogeneous dielectric distributions. In this project, a novel super-Gaussian PB model will be formulated embodying three key innovations: (1) incorporation of environment-dependent atomic characteristics of macromolecules within the continuum electrostatic partial differential equation (PDE); (2) development of a novel, regularized formulation to treat singular charges, with new elliptic PDEs developed through rigorous mathematical analysis: partial charges and water molecules (inside cavities, bonded to the protein, and in bulk solvent) will no longer be modeled as homogeneous spatial regions as in the current Gaussian model, but will reflect the flexibility of the entire solute-solvent system; and (3) elimination of the need to determine a sharp molecular surface, for macromolecules in both vacuum and water environments. Model benchmarking and biological applications tests will be carried out for validation, and the resulting tool will be incorporated into the widely-used DelPhi program package for computing ensemble-average solvation energies. This project is supported jointly by the Division of Mathematical Sciences Mathematical Biology Program, the Division of Chemistry Chemical Theory, Models and Computational Methods Program, and the Established Program to Stimulate Competitive Research (EPSCoR).
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.
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0.913 |
2021 — 2024 |
Zhao, Shan |
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information |
Collaborative Research: Implicit Solvent Modeling and Fast Algorithm Development For Simulating Solutes With Atomic Polarizable Multipoles @ University of Alabama Tuscaloosa
This collaborative research project aims to improve the implicit solvent modeling for studying electrostatic interaction between solutes, such as proteins, DNA, and RNA, and their surrounding solvent environment. The research will improve on current approaches and will formulate a new polarizable multipole implicit solvent model with improved and enhanced modeling accuracy. Furthermore, efficient and accurate numerical algorithms will be developed to meet computational challenges of the new model. This research will provide biophysicists a new tool for analyzing electrostatic interactions of solvated biomolecules in the form of models and algorithms implemented in a freely available software package. In addition, this project will offer interdisciplinary research and training opportunities for undergraduate and graduate students in biological modeling, computation, and mathematical analysis.
The project will address limitations in the existing implicit solvent models for studying electrostatic interaction between solutes. These include the facts that the solute charge sources are often modeled as point charges located at atomic centers, and this rough approximation to the quantum mechanical charge density is known to be a major source of the modeling errors. Moreover, polarization, an important physical phenomenon account for the redistribution of the electron density in the presence of an external electric field is missing in this point charge model. The project will develop a novel nonlinear Poisson-Boltzmann (PB) model associated with an atomic polarizable multipole (PM) force field to describe the self-consistent polarization process and study electrostatic interactions among permanent multipoles, induced dipoles, and reaction-field potential. The coupling of PM source with the nonlinear PB (NPB) equation, as opposed to the linearized PB, is challenging in many aspects involving modeling and numerical difficulties such as charge singularities, geometric complexity, interface jumps, nonlinearity, polarization, as well as high computational cost. To overcome such difficulties, a set of efficient, accurate, and seamlessly coupled numerical methods will be developed to resolve numerical challenges associated with the PM-NPB model. In particular, multipole charge singularities are regularized using Green’s function based decomposition; self-consistent polarization, which involves repeatedly solving an NPB equation across the molecular interface, is efficiently realized by a linearized iterative algorithm coupled with a fast 3D Augmented Matched Interface and Boundary (AMIB) method. Finally, model benchmarking and biological applications will be carried out to ensure that the research results provide a robust tool for simulating electrostatic interactions.
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.
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0.913 |