Jinchao Xu - US grants

The Seventh International Conference on Domain Decomposition Methods in Scientific and Engineering Computing will be held at Penn State University from October 27 to 30, 1993. Domain decomposition refers to a class of methods for the simulating large-scale scientific and engineering systems on computers. Using a divide-and conquer strategy to obtain the solution to a massive problem by combining the solutions to numerous smaller ones, domain decomposition offers the possibility of simulating systems that are too large and complex even for today's powerful supercomputers using conventional solution methods. Such problems include long term climate prediction and the study of global change and the design of space vehicles and structures. Among computational scientist, engineers, and mathematicians, domain decomposition techniques are of particularly intense interest now because they are well adapted to emerging high performance computer architectures. This conference will convene academic and industrial researchers in domain decomposition and allied areas in order to disseminate recent advances and stimulate further development in areas ranging from basic theoretical research on the analysis and validation of domain decomposition algorithms to specific industrial applications. The conference will enhance the technology transfer between academia and industry, and provide opportunities for students and young scientists and engineers to become involved in this important new class of computational methods. Graduate students, young researchers, women, minorities and people with disabilities are being especially encouraged to attend the conference and to contribute papers. The proceedings of this conference will be published by the American Mathematical Society and are expected to become an important resource for researcher and users of domain decomposition.

9706949 Jinchao Xu The proposed research is on the development, analysis and applications of advanced numerical methods for solving partial differential equations arising in sciences and engineering. The focus of research is on efficient multigrid and domain decomposition methods and other relevant algorithms that are suited for parallel and high performance computers. One major object is to try to make multigrid and domain decomposition methods be more practical and more easily used. In particular, efficient and practical multigrid methods will be developed for unstructured grids and also for grids that are currently available from existing (commercial) finite element software. Special techniques under development include the agglomeration method and auxiliary space (grid) methods. Theoretical analysis will also be carried out to justify the efficiency of various methods and also to motivate the development of more sophisticated methods. A major portion of the proposed research will be devoted to the development of efficient multigrid methods for convection-dominated convection-diffusion problems, Navier-Stokes equations and hyperbolic equations and also to the theoretical justifications of the efficiency of some special multigrid methods which use some carefully designed domain decomposition methods as smoothers. Some of the proposed research will be carried out in collaboration with computational scientists and engineers to develop practical and efficient methods for solving some real life problems. Because of the advent of high performance computers, it becomes more and more feasible to use computers to simulate real life problems. In fact it has become a more and more common practice to use computer simulations to replace the traditional and oftentimes very expensive laboratory experiments. Most of the real life problems can be modeled and described by the so-called partial differential equations. These equations can become more and more complicated if we want to have more and more accurate and more realistic modeling of the real life situations. Hence it is a constant challenge to develop efficient numerical methods for solving these equations. The proposed research is precisely for the study of a class of most advanced numerical algorithms for effectively solving partial differential equations on parallel and high performance computers. Our research has been directly tired to various practical applications such as environment protection and the design of medical material and devise. In fact, for example, one numerical package that we have helped develop (based on research related to this proposal) have been adopted by U.S. EPA for environmental assessment and protection.

This project will investigate efficient multigrid (MG) and domain decomposition (DD) algorithms for elliptic and non-elliptic problems on arbitrary unstructured meshes, which are suitable for distributed and shared memory parallel computing architectures. Three aspects of DD and MG will be emphasized in this work: (1) Performance of various DD and MG algorithms on parallel computers with particular attention paid to communication and cache memory latency. (2) Design of optimized DD and MG "gray box" libraries for elliptic and advection dominated problems, which exploit optional information supplied by the user. (3) Theoretical analysis and practical design of DD and MG algorithm appropriate for advection dominated problems. Particular emphasis will be placed on DD and MG algorithms for solving the discretization matrices arising from a variety of large scale scientific computing problems, such as computational fluid dynamics (CFD) for advection dominated problems, image processing, photolithography, and the modeling of microchip performance and fabrication. The non-elliptic behavior of these practical problems renders the known multigrid and decomposition theory inadequate and serves to motivate a balanced effort consisting of algorithmic development, theoretical analysis, and practical application.

ABSTRACT

(NSF proposal: DMS-0074299, PI: Jinchao Xu)

The proposed project is on the study of advanced solution methods for partial differential equations that arise from scientific and engineering applications. The theme of research is on the development, application and analysis of multigrid methods. The multigrid method is among the most powerful techniques for solving large scale linear and nonlinear systems arising from the discretization of partial differential equations. But the method has not been used in practice as often as they should be (because it is often not easy to code and to use), nor as efficiently as they could be (because it is often not easy to get the method work correctly). Our research is, on one hand, to develop special type of multigrid methods that are relatively easier to use for general users for some standard applications and is, on the other hand, to develop multigrid methods that are carefully tailored for some special class of practically interesting problems. One major component of the proposed research is a systematical investigation on various fundamental theoretical issues related to multigrid methods in general and also theoretical questions related to the algorithms to be developed for several specific problems of practical interests.

The multigrid methods we propose to develop and study are expected to be applicable to a large class of practical problems including numerical simulations for electrochemical power devices (batteries) and advanced materials such as lattice block materials and liquid crystalline materials. These multigrid methods are expected to make a major impact in these and related applications and in particular to make it possible to simulate these problems in three dimensions in such a way that other traditional approaches may not be feasible. For example, the proposed study of advanced numerical methods for simulating electrochemical batteries has been and will be making a significant contribution to advanced battery technologies and manufactures that are vitally important in our everyday life, from the watch and the camera flash to electromobilies, modern space vehicles and wireless communications in information technology. Because of the practical backgrounds of these problems, the proposers and their research associates and graduate students are expected to actively interact and collaborate with physicists, engineers, computational scientists and practitioners from industries.

DMS Award Abstract
Award #: 0209497
PI: Xu, Jinchao
Institution: Pennsylvania State University
Program: Computational Mathematics
Program Manager: Catherine Mavriplis

Title: Multiscale Methods for Partial Differential Equations

The focus of this work is on the development and applications of a two-scale discretization technique, namely the finite element method based on partition of unity. One main application is on the design of efficient discretization for nonmatching (either overlapping or nonoverlapping) grids. The main idea of nonmatching grids is to divide a physical domain into a set of overlapping or nonoverlapping subregions which can accommodate smooth, simple, easily generated grids. In this approach, a grid generation for complex geometries can be made simple, refinement grids can be added or removed without changing other grids, different equations/numerical methods may be used on different grids, efficient structured grid solvers may be used. Furthermore, overlapping grids are well suited for parallelization and vectorization. The proposed generalized finite element method based on partition of unity provides a general and powerful discretization framework for this type of grids. Another major task is the development of a multigrid iterative method for solving the resulting algebraic systems for these new discretization schemes. As divide and conquer techniques, the proposed multiscale algorithms are suitable for parallel and high-performance computers.

A class of new multiscale techniques are proposed to study for efficient numerical solution of partial differential equations. Multiscale methods in general are proven to be among the most powerful mathematical tools for the investigation of a broad range of models that are described by partial differential equations. Their pivotal role in the design of fast, reliable, and robust numerical methods for the solution of various problems places them among the most important research areas in the applied mathematics in the recent years. Since these methods are in some sense problem-independent, they are expected to have many important applications in science and engineering such as composite materials and subsurface flows in environmental applications.


Date: May 28, 2002

NSF proposal DMS-0215392
PIs: Xu, Belmonte, Du, Li and Zikatanov

ABSTRACT

The Department of Mathematics at the Pennsylvania State
University will purchase a 64-node parallel PC cluster
to be dedicated to the support of research and teaching
in the mathematical sciences. In particular, the PC
cluster will be used to support the research projects
of faculty members in the areas of the numerical
solution of partial differential equations in fluid
dynamics and material sciences, and computational
finance and in the studies of general numerical
techniques such as parallel multigrid algorithms and
quasi-Monte Carlo methods. In particular, the projects
include studies of important issues concerning modeling
and simulations of non-Newtonian flows, liquid crystals,
quantized vortices, water waves and fuel cells. Much of
the research efforts rely critically on the
establishment of the proposed PC cluster.

The proposal involves an integrated collaboration
between the numerical work to be performed in the PC
cluster and the experimental work to be performed in
the W. G. Pritchard Laboratories of the Department of
Mathematics. The new equipment will make it possible to
numerically simulate the various complex physical
phenomena observed in the fluid lab and will greatly
enhance collaborations among researchers in computational
and applied mathematics at Penn State. The PC cluster
will also be the basis for the creation of a new
computational laboratory, which together with the
Pritchard fluid lab, will provide a unique environment
for multidisciplinary research as well as for (both
undergraduate and graduate) student training.

The purpose of this project is to develop advanced computational
techniques in order to perform large-scale, state of the art
simulations of two-phase transport problems arising in proton exchange
membrane (PEM) fuel cells. Because of the complexity of the
underlying mathematical models for fuel cells, current solution
techniques are far from being satisfactory, and therefore more
efficient numerical techniques are urgently needed. While there is
still a long way before we can solve all the coupled systems
efficiently, this proposal will be devoted to solution techniques for
an important subsystem posted on the gas diffusion layers and the gas
channel. This subsystem of equations possesses a number of critical
numerical difficulties caused by anisotropy, large discontinuity,
degeneracy and nonlinearity. The goal of the proposed project is to
address these difficulties simultaneously by developing proper
discretization techniques and robust iterative methods for solving the
discretized systems. The discretization techniques to be developed
will be mainly based on adaptive finite element/volume methods and the
iterative methods will be based on multigrid techniques. The accuracy
of the discretization scheme and the efficiency of the iterative
methods for solving the discretized system will be studied.


The importance of the fuel cell technology can hardly be
overemphasized as PEM fuel cell engines can potentially replace
internal combustion engines in the future. Since a PEM fuel cell
simultaneously involves electrochemical reactions, current
distribution, two-phase flow multi-component transport and heat
transfer, comprehensive mathematical modeling and computational
simulation are required in order to: (1) understand the many
interacting, complex electrochemical and transport phenomena that
cannot be measured experimentally; (2) identify limiting steps and
components; (3) simulate dynamic responses under vehicle driving
conditions; and (4) provide a computer-aided tool for design of future
fuel cell engines with much higher power density (kW/liter) and lower
cost. The integration of the different expertise of the PI and co-PI
is expected to lead to significant progress and likely breakthroughs
in the field of fuel cell simulations. Newly developed numerical
techniques will be immediately employed in the existing library of
numerical codes that have been developed for years by the Penn State
Electrochemical Engine Center (ECEC), lead by the co-PI. It is hoped
that the new numerical techniques to be developed will lead to at
least an order of magnitude improvement over the existing methods.
Application and impact to national security/enviroment and to industries
are naturally expected for this research because of the close tie of ECEC
with national labs and automobile manufactures. Moreover, this work
will provide a unique interdisciplinary research opportunity for
graduate as well as undergraduate education.

A group of researchers in the Department of Mathematics at Penn State
will purchase a symmetrical multiprocessor (SMP) cluster, and a
visualization server. The new equipment will be used to support
various ongoing mathematical research projects including number
theory, numerical solution of partial differential equations,
mathematical modeling and numerical simulations for hydrogen fuel
cells, modeling in fluid dynamics and material sciences, mathematical
modeling and numerical simulations in biological sciences, analysis
and implementation of new cutting-edge numerical techniques and
parallel algorithms. Most of the models and applications from these
projects are described by systems of partial or ordinary differential
equations and their numerical solutions rely on intensive computations
that require high memory capacity and high memory bandwidth. The
shared memory computer to be purchased will serve well for these
requirements and will significantly enhance the proposers' research
capabilities.


The research projects that benefit from the new equipment range from
the very basic mathematical research such as number theory to very
practical applications such as mathematical modeling and numerical
simulations of hydrogen fuel cells that have vital national interests
(for security and environment). The proposers communicate and
collaborate on a various research subjects within the department as
well as across the university. Such collaborations provide a
stimulating environment for the training of graduate students,
postdoctoral fellows, as well as junior faculty members. The proposed
equipment will also aid existing, and provide solid computational base
for new joint projects with researchers from other sciences and
engineering

Proposal Title: Collaborative Research: Multigrid QCD at the Petascale
Institution: Trustees of Boston University
Abstract Date: 10/10/07
0749300
Brower
0749202
Brannick
0749317
McCormick
Numerical solutions to Quantum Chromodynamcs on a lattice are critical to high
precision experimental tests of the standard model and an ab-initio understanding of
nuclear matter. The core of these calculations involves inverting a Dirac matrix which
becomes increasingly ill conditioned as the lattice is refined. Consequently while
Terascale computing hardware has exposed this new physics, it is incapable of fully
accommodating it. On the other hand, if lattice QCD algorithms are reformulated to
exploit and reveal the physics at this finer microsale, Petascale hardware does have the
potential for opening up a new era of physics discovery. This award brings together a
close collaboration of leading experts in applied mathematics and theoretical physics to
meet this challenge by the application of new multi-level algorithms for QCD
simulations. The central mission of the proposed Multigrid Quantum
Chromodynamics at the Petascale project (MGQCD) is: to develop new and significantly
more robust multigrid methods for enabling more complex and higher fidelity physics for
lattice QCD calculations; to support their migration into Petascale simulations; and to
engage the broader scientific community through collaborative research and
educational activities that highlight the multigrid methodology.
NATIONAL SCIENCE FOUNDATION
Proposal Abstract

In this proposal, various algorithms and theories will be developed to partially realize the following four-stage strategy for developing user-friendly solvers and solver-friendly discretizations: (1) develop user-friendly optimal solvers and relevant theories for a small number of basic solver-friendly systems, namely the discrete Poisson's equation and its variants; (2) extend the list of solver-friendly partial differential equations (such as the discrete Stokes and Maxwell equations) by reducing them to the solution of a handful of basic solver-friendly systems (for which optimal and user-friendly solvers can be applied); (3) develop solver-friendly discretization techniques for more complicated PDEs (systems) such that the discretized systems will join the list of solver-friendly systems (such as the Eulerian-Lagrangian method for the Navier-Stokes equation, the Johnson-Segalman equations, and the magnetohydrodynamics equations); and (4) solve the discretized system from a general discretization by using a solver-friendly discretization (if it is not a satisfactory discretization to obtain the numerical solution by itself) as an auxiliary discretization that can be used as a preconditioner or a means for obtaining a good initial guess for a linear or nonlinear iteration. These techniques will be developed with the purpose of making them effective for solving complicated problems such as non-Newtonian models and fuel cell model equations. Parallel implementations will be one major consideration in the design of these algorithms. Theoretical issues---such as the most fundamental open problem concerning the optimal convergence of algebraic multigrid methods---will be carefully investigated.


Many problems in scientific and engineering computing can be reduced to the numerical solution of certain partial differential equations. Over the last few decades, researchers have expended significant effort on developing efficient iterative methods for solving discretized partial differential equations. Though these efforts have yielded many mathematically optimal solvers such as the multigrid method, the unfortunate reality is that multigrid methods have not been much used in practical applications. This marked gap between theory and practice is mainly due to the fragility of traditional multigrid methodology and the complexity of its implementation. This proposal aims to develop theories and techniques that will narrow this gap, specifically by developing mathematically optimal solvers that are robust and easy to use in practice. The proposed study will focus on an integrated application of user-friendly solvers and solver-friendly discretizations for various basic partial differential equations that arise in many applications; therefore, the results of this proposal are expected to be directly applicable in many areas of computational and applied mathematics. The solver and discretization techniques we produce, including mathematical algorithms, analyses, and software, will provide powerful tools for exploring multiscale models in physics, chemistry, and engineering. Through the accompanying Matrix-Solver Community Project (http://www.multigrid.org/solvers/), the results of this proposal will lead to timely and broad impacts. The proposed project will have a strong educational impact as well, as it focuses on training graduate students in theoretical and practical aspects of modern computational science and interdisciplinary applications.

The goal of this project is to develop and study a special class of multilevel methods that combine techniques from the Geometric Multigrid (GMG) and Algebraic Multigrid (AMG) methodologies, which we refer to as the "single-grid multilevel method" (SGML). The focus is discretized partial differential equations, for which detailed information on the underlying geometric grid is generally available to the user. The research team is designing solvers that use information from the finest grid (hence termed the single-grid method) to select a simple and fixed coarsening that allows for explicit control of the overall grid and operator complexities of the multilevel solver. The central new idea that we are investigating concerns the design and analysis of algorithms for adaptive construction of the MG relaxation scheme when used as a smoother. In contrast to existing AMG methods, in which the smoother is fixed and coarsening is the key component in the setup phase, SGML will construct the smoother in the setup phase to complement its simple geometry-based coarsening process. It should be noted that the algebraic construction of the smoother can also benefit from using properties of the geometric grid, for example, to obtain a suitable partitioning of the unknowns in parallel. The SGML approach (together with the many of the promising algebraic techniques for constructing the MG interpolations developed over the last decade) is also under consideration. The PI and co-PIs, though, are focusing on the SGML method because of its ability to explicitly control complexity, which in turn allows for (nearly) optimal load balancing and predictable communication patterns, such that the method is well suited for parallel computing.

Overall, the iterative solvers under development are designed to be implemented in open source parallel codes and made available to the scientific computing community. This will provide a computational framework for future algorithm research and development in related areas as well as powerful tools for simulation. In summary, the proposed methodology constructs solvers using all the information available to increase the efficiency of numerical modeling and simulation of physical phenomena on parallel multi-core computing architectures. Educational activities include the training of graduate students.

A primary goal of this project is to create framework for developing more robust and user-friendly solvers for linear systems of equations, which are ubiquitous in science, engineering, and industrial applications. The investigators will carry out integrated analysis and development of geometric multigrid and algebraic multigrid methodologies. Geometric multigrid (GMG) methods form a class of multilevel solvers designed to solve linear systems of equations arising from certain classes of discretized partial differential equations (PDEs). Algebraic multigrid (AMG) methods are also multilevel solvers, though these techniques avoid any dependence on information regarding an underlying grid geometry or PDE. As a result, GMG methods are effective tools for solving a more restricted class of linear systems with a strong theoretical backing for their performance, whereas AMG solvers apply to more general linear systems, though do not share the same mathematical rigor in justifying their performance. This research project will employ functional analysis as a natural framework for studying GMG and AMG methods in a unified setting. The resulting theory will establish strong guiding principles for analyzing and developing robust, efficient, and scalable multilevel solvers. The iterative solvers under development will be implemented in open source parallel codes, made available to a broader scientific computing community, providing powerful tools for simulation and a foundation for future algorithm research and development. Moreover, this project provides Ph.D. students with opportunities to participate in a variety of education and research activities, in which they will receive advanced training, participate in conferences, and collaborate with researchers from industry and Department of Energy laboratories.

This research project investigates a unifying framework for analyzing GMG and AMG methods in a functional analysis setting. This framework provides a strong foundation for understanding the operators, relationships between spaces, and other core ingredients involved in these multilevel solvers, such as the construction of coarse spaces and their respective bases, and a convergence analysis that applies to a broad variety of existing AMG methods. Furthermore, for problems originating from discretized PDEs, the integration of geometric information for constructing auxiliary grid-based preconditioners and highly effective smoothers on each level can be seamlessly integrated, allowing for more flexible and aggressive algebraic coarsening. More benefits of incorporating geometric information are realized in nearly optimal load balancing and predictable communication patterns for parallel implementations. For more general linear systems, the project studies the use of (relaxed) compressed sensing techniques to preserve sparsity for coarse space operators, which can be supplemented with information from an underlying geometric grid or adjacency graphs of the matrix to gain control over the computational complexity. It is expected that these techniques will be useful in solving problems with non-quasiuniform underlying grids or problems with matrices that are not symmetric and positive-definite (SPD). To verify the efficacy of the developed methodologies, the resulting solvers will be applied to fluid-structure interaction problems and nearly singular SPD problems.

Scientific computing is an important and general interdisciplinary research field and plays a crucial role in many application areas in science and engineering. In recent years, there has been tremendous growth in both the need for large-scale scientific computing and the amount of computing power, thanks to new high performance computing (HPC) clusters with millions of cores. Domain decomposition methods represent one of the most powerful and versatile techniques for the efficient parallel solution of such large-scale scientific problems. The goal of the current project is to provide financial support for US-based early career researchers, i.e., graduate students and post-docs, to attend the 26th International Conference on Domain Decomposition Methods, to be held in Hong Kong from December 2-6, 2019. The conference is part of a series that is considered one of the most successful in Applied and Computational Mathematics. The NSF-funded US participants will benefit from learning about cutting-edge developments in domain decomposition methods and latest worldwide trends in high performance computing. They will also have the opportunity to develop new collaborations with other researchers from Asia and around the world. The proposal will support up to 10 participants to travel to this workshop and some of these 10 are expected to be graduate students at US universities.

Domain decomposition methods (DDM) can be regarded as a divide-and-conquer strategy for solving mathematical problems posed in a physical domain, reducing a large problem into many smaller, but easier-to-solve, problems. They are particularly suited for making efficient use of distributed memory architectures, which are able to solve many such smaller problems in parallel. As we approach the dawn of exascale computing, such scalable techniques are vital tools for solving complex problems in physics and engineering that would otherwise be intractable. Since 1987, the International Conferences of Domain Decomposition Methods have been held in 15 countries throughout Asia, Europe and North America. In this conference, now in its 26th edition, researchers in domain decomposition and related fields will present and discuss state-of-the-art methodologies and developments and propose future directions in high performance computing. Early career researchers will be able to participate in one of three ways: as speakers at a minisymposium organized by fellow participants, by giving a contributed talk, or by presenting a poster. The NSF-funded US participants, consisting of graduate students, postdocs and other early career researchers who lack their own funding, will benefit from learning new developments in domain decomposition methods and latest trends in high performance computing. MOre details are available at the conference website: https://www.math.cuhk.edu.hk/conference/dd26/?Conference-Home

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.

This award supports participation in the 2020 Workshop on Mathematical Machine Learning and Application held at Penn State University on April 26-29, 2020. The workshop aims to bring together active scientists in the emerging field of data science to discuss recent advances in the study of algorithm development, theoretical analysis, and applications of machine learning. One focus of the workshop is on theoretical understanding of why and how deep learning works from mathematical viewpoints. This grant provides supports of participation of US-based invited speakers and US-based junior participants (graduate students, postdocs and early career researchers who lack their own funding). The main session of the workshop will take place during the period from April 27 to 29, with about 20 invited talks and a poster session. A short course featuring introductory lectures on the mathematics of deep learning will be held prior to the workshop on Sunday, April 26, with junior participants as the main target audience.

In this workshop, researchers in mathematical machine learning and related fields from the United States and other countries around the world will discuss state-of-the-art methodologies and developments and propose future directions in mathematical data science and its applications. Examples of topics to be discussed in the workshop include: machine learning in physical modeling and computational engineering, non-convex optimization in machine learning, approximation theory of deep neural networks, interactions between deep learning and partial differential equations, architecture design and interpretation of convolutional neural networks, deep learning in computer vision and natural language processing, and deep learning with grammars, automata, and rules. More details of this workshop are available at https://ccma.math.psu.edu/2020workshop/.

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.

This research connects two different fields, machine learning from data science and numerical partial differential equations from scientific and engineering computing, through the comparative study of the finite element method and finite neuron method. Finite element methods have undergone decades of study by mathematicians, scientists and engineers in many fields and there is a rich mathematical theory concerning them. They are widely used in scientific computing and modelling to generate accurate simulations of a wide variety of physical processes, most notably the deformation of materials and fluid mechanics. By contrast, deep neural networks are relatively new and have only been widely used in the last decade. In this short time, they have demonstrated remarkable empirical performance on a wide variety of machine learning tasks, most notably in computer vision and natural language processing. Despite this great empirical success, there is still a very limited mathematical understanding of why and how deep neural networks work so well. We hope to leverage the success of deep learning to improve numerical methods for partial differential equations and to leverage the theoretical understanding of the finite element method to better understand deep learning. The interdisciplinary nature of the research will also provide a good training experience for junior researchers. This project will support 1 graduate student each year of the three year project.

Piecewise polynomials represent one of the most important functional classes in approximation theory. In classical approximation theory and numerical methods for partial differential equations, these functional classes are often represented by linear functional spaces associated with a priori given grids, for example, by splines and finite element spaces. In deep learning, function classes are typically represented by a composition of a sequence of linear functions and coordinate-wise non-linearities. One important non-linearity is the rectified linear unit (ReLU) function and its powers (ReLUk). The resulting functional class, ReLUk-DNN, does not form a linear vector space but is rather parameterized non-linearly by a high-dimensional set of parameters. This function class can be used to solve partial differential equations and we call the resulting numerical algorithms the finite neuron method (FNM). Proposed research topics include: error estimates for the finite neuron method, universal construction of conforming finite elements for arbitrarily high order partial differential equations, an investigation into how and why the finite neuron method gives a much better asymptotic error estimate than the corresponding finite element method, and the development and analysis of efficient algorithms for using the finite neuron method.

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.