Thomas Manteuffel - US grants

This research program consists of a number of projects related to supercomputing. They include the Fast Adaptive Composite Method, Multilevel Load Balancing, Analysis of Preconditionings, Algebraic Multigrid, Multi-level Methods for Optimization, etc. The research results will have applications in many branches of science and engineering.

This is a grant under the Scientific Computing Research Equipment for the Mathematical Sciences program of the Division of Mathematical Sciences of the National Science Foundation. This program supports the purchase of special purpose computing equipment dedicated to the conduct of research in the mathematical sciences. This equipment is required for several research projects and would be difficult to justify for one project alone. Support from the National Science Foundation is coupled with discounts and contributions from manufacturers and with substantial cost-sharing from the institutions submitting the proposal. This program is an example of academic, corporate, and government cooperation in the support of basic research in the mathematical sciences. This equipment will be used to support five projects in the Department of Mathematics at the University of Colorado at Denver: Multilevel Methods for Transport Models, directed by T. Manteuffel; Asynchronous Fast Adaptive Composite Grid Methods for Parallel Computers, directed by S. McCormick; Multilevel Reconstruction Algorithms, directed by W. Briggs; Fast Solvers for the p-Version Finite Element Method, directed by J. Mandel; and Flows in Heterogeneous Porous Media, directed by T. Russell.

This award supports a conference on iterative methods for the solution of linear and nonlinear systems of equations. The conference will be held in Copper Mountain, Colorado at the Village Square Lodge from April 1 to April 5, 1990. The topics of emphasis are nonsymmetric linear systems, parallel implementations, and novel preconditioning techniques. The speakers include Olof Widlund, Michael Overton, Ahmed Sameh, Nick Trefethen, and Paul Saylor. Papers submitted at this conference will be refereed and published in the SIAM Journal of Scientific and Statistical Computing.

A new symmetrization method for the numerical solution of partial differential equations on emerging parallel computers will be studied. The algorithmic development of the method in the context of massive parallelization will be emphasized. The method of symmetrization is attractive for an elliptic partial differential operator, A=S+BT in that it transforms the problem in A to one in an operator that is self-adjoint and that closely resembles S, the self-adjoint part. Thus, by invoking symmetry even for nonself-adjoint problems, it allows the use of efficient and robust versions of multilevel and conjugate gradient algorithms for the solution of the arising linear systems of equations. The use of multilevel and conjugate gradient algorithms in the implementation of the method on a parallel computer will provide the opportunity to study parallelization in a context where a multilevel algorithm itself is used as in inner solver for an algorithm composed of inner and outer iterations. The algorithms will be tested in a massively parallel environment offered by the Connection Machine.

This grant will help fund the Fifth Copper Mountain Conference on Multigrid Methods to be held March 31 through April 5, 1991. The topics of emphasis are hypersonic flow, adaptive methods, large structures, transport processes, particle physics, parallel computation, and large-scale applications software. These topics will be addressed by the invited speakers, in the special sessions and the workshops, and by the panel. Other sessions of contributed papers will also be held. In order to provide the novice with a basic understanding of multigrid and its processes, four one-hour introductory lectures on multigrid techniques will be offered the first morning of the conference.

The area of research for this project is the iterative solution of large sparese linear systems of equations on vector and parallel computers. The goal of this research is to develop iterative algorithms which are adapted to parallel computers and which exploit the properties of parallel architectures. Three particular topics will be studied. First adaptive procedures for polynomial preconditioning of symmetric indefinite systems will be developed and tested. Second, research will be done on an adaptive polynomial method for solving nonsymmetric systems of equations for which the specturm of the matrix lies in an arbitrary region in the complex plane. A third topic of research will be the parallelization of restarted generalized conjugate gradient algorithms such as the GMRES algotithm. The goal of each of these topics of research is to develop algorithms which are better suited to highly parallel architectures.

9403540 Curry The Program in Applied Mathematics at the University of Colorado will purchase computational equipment which will be dedicated to the support of research in the mathematical sciences. The equipment will be used for several research projects, including in particular: 1.) Numerically Induced Chaos, Sympletic Integrators and Nonlinear Wave Equations; 2.) Fast Numerical Algorithms: Wavelets; 3.) Dynamical Systems: Computational Experiments with Maps; 4.) Multilevel Methods for First Order System Least Squares Discretization of Partial Differential Equations; and 5.) Nonlinear Ocean Waves: Simulation, Chaos, and Field Data

The investigators continue work to develop numerical methods for solving problems of flows in porous media and to develop and analyze fast algorithms for partial differential equations, for computational fluid dynamics, and for several linear algebra problems. Specifically, they (a) develop fast parallel numerical methods for the solution of the Boltzmann equation used to model the transport of neutral and charged particles in 1 and 3 spatial dimensions; (b) develop multilevel zonal techniques for large scale computational fluid dynamics applications on parallel computers; (c) study discretization and solution techniques for fluid flows in porous media that involve complex geometry with heterogeneous material properties; (d) develop a new methodology for the solution of systems of partial differential equations by introducing new dependent variables and forming a least-squares functional. In each case, they exploit and extend multi-level computational techniques. These projects are focused on important applications, such as calculating radiation doses for cancer treatment, ion implantation for semiconductor development, global weather modeling, fluid flow over airfoils and through rocket nozzles, oil recovery and contaminant transport in subsurface water supplies. They reflect the growing recognition that effective numerical modeling of physical phenomena involves several stages - formulation of the mathematical model, transforming the model into a discrete problem, and developing algorithms for the numerical solution of the discrete problem on advanced computers - none of which should be attempted without consideration for the others. Each project examines all components of the process, reformulating the model and discretization when necessary to yield discrete problems that can be efficiently solved numerically on parallel computers.

Manteuffel The researcher and his colleagues organize a conference devoted to iterative methods for the solution of linear and nonlinear systems of equations. Particular themes are nonsymmetric linear systems, globally convergent algorithms for nonlinear systems, algorithms of optimal order of work, and applications on multiprocessor computer architectures. The mathematical description of real-world problems leads inevitably to equations to solve. Many times the equations are nonlinear ones, arising because the underlying problem is nonlinear. Often, however, the original problem is linear and the mathematical equations are too. Such problems arise in all areas of science and technology, and are of special interest in biology, materials, environmental studies, and manufacturing. Progress in these areas requires solution of more comprehensive modeling equations, or more accurate solution of existing models. In either case, these needs increase the size of the equations or their nonlinearity. The conference addresses advances in algorithms to deal with the bigger size of the equations, nonlinearities, the amount of time and work needed to get a solution, and the possible use of multiprocessor computers. ***

McCormick 9521625 The principal investigator and his colleagues organize the seventh Copper Mountain conference on multigrid methods. Numerical methods for solving a differential equation usually begin by imposing a grid on the region where the equation holds. From the differential equation, algebraic equations are then developed; their solution represents the qolution of the differential equation. The accuracy of the approximate solution commonly is measured by the fineness of the grid. Multigrid methods are numerical methods for solving partial differential equations that systematically exploit the relationship between approximate solutions on different grids to arrive at a solution whose accuracy is consistent with the finest grid but for considerably less work. The methods are often dramatically more efficient than others. Research in the past dozen years has extended the methods to a broad range of problems of considerable practical import in engineerine, manufacturing, materials, physics, and fluid dynamics.

Manteuffel 9528039 The researcher and his colleagues organize a conference devoted to iterative methods for the solution of linear and nonlinear systems of equations. Particular themes are nonsymmetric linear systems, globally convergent algorithms for nonlinear systems, algorithms of optimal order of work, and applications on multiprocessor computer architectures. The mathematical description of real-world problems leads inevitably to equations to solve. Many times the equations are nonlinear ones, arising because the underlying problem is nonlinear. Often, however, the original problem is linear and the mathematical equations are too. Such problems arise in all areas of science and technology, and are of special interest in biology, materials, environmental studies, and manufacturing. Progress in these areas requires solution of more comprehensive modeling equations, or more accurate solution of existing models. In either case, these needs increase the size of the equations or their nonlinearity. The conference addresses advances in algorithms to deal with the bigger size of the equations, nonlinearities, the amount of time and work needed to get a solution, and the possible use of multiprocessor computers.

9706866 McCormick The main thrust of the proposed research is the development of efficient numerical techniques for simulation of a wide variety of physical processes. Important applications include aerodynamics, meteorology, elasticity, electromagnetics, porous media, and particle transport. The general goal is to develop accurate discretization methods and fast algebraic solvers for the partial differential equations that govern these and other applications. The research topics include: multilevel first-order system least squares, which involves reformulation of partial differential equations as well-posed minimization principles to allow for robust and efficient solution methods; porous media problems, which will be treated by Eulerian-Lagrangian localized adjoint methods that have been successful for such multiparty and reactive flows; transport phenomena, which will be simulated using efficient and robust least-squares methods for the three-dimensional Boltzmann transport equations; and iterative methods, which is aimed at developing effective algebraic solvers for the equations that arise in many applications. The focus of this project is research in the field of computational mathematics. The purpose is to improve our understanding of the mathematics behind computer simulation of complex physical phenomena. Such simulations are key to the study and control of many important processes, including groundwater flow, global change, energy production, and material science. One of the challenges in such simulations is the development of improved mathematical methods for solving the equations that arise in these models. The basic aim of this research is dramatic improvement in our ability to model increasingly more complicated and sophisticated processes with much greater accuracy and efficiency. This will pave the way for simulations that can provide scientists, engineers, and policy-makers with much more powerful tools to understand and improve our industry, science, and environm ent.

Manteuffel 9727525 The researcher and his colleagues organize a conference devoted to iterative methods for the solution of linear and nonlinear systems of equations. Particular themes are nonsymmetric linear systems, globally convergent algorithms for nonlinear systems, algorithms of optimal order of work, saddle point problems, and applications on multiprocessor computer architectures. The mathematical description of real-world problems leads inevitably to equations to solve. Many times the equations are nonlinear ones, arising because the underlying problem is nonlinear. Often, however, the original problem is linear and the mathematical equations are too. Such problems arise in all areas of science and technology, and are of special interest in biology, materials, environmental studies, and manufacturing. Progress in these areas requires solution of more comprehensive modeling equations --- such models are usually more nonlinear than simpler models ---, or more accurate solution of existing models --- increasing the size of the numerical problem to be solved. In certain problems, notably in fluid flow and in elasticity, computational methods for the equations describing the phenomena yield numerical systems having a saddle point; the solution is on a surface that looks locally like a saddle. Computational methods for such problems offer special difficulties. The conference addresses advances in algorithms to deal with larger numerical systems, nonlinearities, the amount of time and work needed to get a solution, saddle points, and the use of multiprocessor computers.

The Department of Applied Mathematics of the University of Colorado Boulder will purchase SUN Microsystems hardware and associated software which will be used to support research in the mathematical sciences. The equipment will be used in 5 research projects, including computational problems related to: Wavelets and Fast Algorithms, Iterating Lie Point Symmetries, problems in geophysics, development of first order systems least squares for PDEs, and the dynamics of multidimensional mappings.

McCormick
9816592

The investigator and his colleagues organize a series of annual Copper Mountain Conferences. The subject of these meetings alternates between multigrid methods in odd-numbered years and iterative methods in even-numbered years. The Copper Mountain Conferences provide a forum for the exchange of ideas in these two closely related fields. The program for a conference in this series consists of tutorials, invited lectures, and contributed papers, as well as time for scientific interaction among the participants. Topics of emphasis for the conferences include advanced architectures, algebraic-type methods, and nonsymmetric linear systems. This grant provides support for students to participate in the conferences.

The mathematical description of real-world problems leads inevitably to equations to solve. Many times the equations are nonlinear ones, arising because the underlying problem is nonlinear. Often, however, the original problem is linear and the mathematical equations are too. Such problems arise in all areas of science and technology, and are of special interest in biology, materials, environmental studies, and manufacturing. Progress in these areas requires solution of more comprehensive modeling equations --- such models are usually more nonlinear than simpler models ---, or more accurate solution of existing models --- increasing the size of the numerical problem to be solved. Many problems lead to equations that are not symmetric; computational methods for such problems offer special difficulties. Numerical methods for solving a differential equation usually begin by imposing a grid on the region where the equation holds. From the differential equation, algebraic equations are then developed; their solution represents the solution of the differential equation. The accuracy of the approximate solution commonly is measured by the fineness of the grid. Multigrid methods are numerical methods for solving partial differential equations that systematically exploit the relationship between approximate solutions on different grids to arrive at a solution whose accuracy is consistent with the finest grid but for considerably less work. The methods are often dramatically more efficient than others. The conferences address advances in iterative methods and in multigrid methods to deal with larger numerical systems, nonsymmetric and nonlinear systems, and the use of multiprocessor computers. The methods are of great practical import in engineering, manufacturing, materials, physics, and fluid dynamics. This project supports student participants at the Copper Mountain conferences on multigrid methods and on iterative methods. The students present a talk on their research in the regular sessions of the conference. The primary objective here is to encourage student participation in these rapidly evolving areas, and to provide an excellent opportunity for these students to demonstrate their new results, to learn more about the field from its experts, and to become a more integral part of the discipline. Supporting student participation is critical for developing the next generation of scientists and engineers.

DESCRIPTION: The applicant proposes to develop and test a new general computer simulation algorithm that can be used for modeling the behavior of physiological systems with components of different material properties (multiphysical problems). Examples are blood flow, aqueous flow, joint mechanics. Solid-fluid interactions pose challenges too great for traditional approaches to computer simulation. The applicant proposes to apply a new algorithm, called first-order system least-squares (FOSLS). FOSLS transforms a mathematical problem into a larger but simpler form, which allow existing high-powered numerical methods to solve the problem much faster than the original form permits. The applicant describes how his method could clarify questions of possible interaction between aqueous humor flow and a deformable iris.

Abstract

This is a renewal proposal to continue development of first--order system least squares (FOSLS) for numerical solution of partial differential equations (PDEs). It combines theoretical analysis, algorithm design, and software development, driven by several real applications, including aerodynamics, meteorology, elasticity, electromagnetics, particle transport, and porous flow. The goal is to develop accurate discretizations and fast solvers for the governing PDEs. The focus will be on the continued development of the FOSLS methodology, with special attention on developing methods that allow non-smooth problem character and solutions, and on further implementation of the methodology in the software package FOSPACK. Applications will include coupled systems, especially those arising in biological simulation, and porous media flow. Successful progress of this project would enable numerical simulations beyond current capabilities in many important applications of national interest.

The central aim of this project is research in the field of computational mathematics. The purpose is to improve our understanding of the mathematics behind numerical computer simulation of complex physical phenomena. Such simulations are key to the study and control of many important processes, including groundwater flow, global change, energy production, biological modeling, and material science. One of the challenges in such simulations is the development of improved computational methods for solving the mathematical equations that arise in these models. The basic aim of this research is dramatic improvement in our ability to model increasingly more complicated and sophisticated processes with much greater accuracy and efficiency. This should pave the way for simulations that can provide scientists, engineers, and policy-makers with much more powerful tools to understand and improve our industry, science, and environment.

This proposal is for development of an effective hp-adaptive First-Order
System Least-Squares (FOSLS) method for nonlinear partial differential
equations (PDEs) that may exhibit singularities and other non-smoothness
properties. The general FOSLS methodology is already a powerful tool for
the solution of PDE-based problems in science and engineering. One of its
main advantages is that it can be used to transform a given set of
equations into a loosely coupled system of scalar equations that can be
treated easily by multilevel finite elements. The reformulation is done
by recasting the equations as a least-squares principle associated
with a carefully derived first-order system. Another advantage of FOSLS is
that the associated functional itself provides a natural sharp local error
estimator, which can be employed for effective adaptive refinement. In
previous work, the FOSLS methodology was developed and analyzed for
problems that include linear elasticity, linear transport, and
incompressible fluid dynamics. Under standard H2-type smoothness
assumptions on the original PDE, FOSLS has been proved theoretically
and numerically to exhibit optimal finite element and multigrid
convergence properties. However, two principal difficulties currently
prevent FOSLS from being applied effectively to more complicated problems:
nonsmooth solutions and nonlinearity. The project aim is to develop an
adaptive weighted-functional coupled with adaptive nested iteration
schemes to treat these difficulties. The goal is to obtain a scheme that
can solve complex nonlinear PDE systems with a total cost equivalent to a
small number of relaxation steps on the finest grid.

Successful completion of this project will enhance our ability to model
complex physical processes on high-end computing platforms. In particular,
this project addresses computational models of coupled fluid/structure
interactions that occur, for example, in biomechanical systems such as
blood flow in compliant vessels. Other areas that could potentially
benefit from this project are aerodynamics, astrophysics, geophysics and
meteorology. As modern computer architectures become more complex,
harnessing 10s of thousands of processors working together on a single,
complex problem, it becomes even more important to develop optimal
numerical algorithms. Successful completion of this project will widen the
class of problems for which optimal algorithms exist.

A fundamental research goal in earth-surface science (geomorphology, hydrology, sedimentology) is to develop mathematical (usually numerical) models that describe the formation of river basins over geologic time and their response to natural and anthropogenic forcing over human time scales. Within the last two decades, the potential for building, testing, and applying such models has advanced significantly thanks to several parallel developments, including (1) cosmogenic nuclide analysis for measuring erosion rates, (2) development of improved rate laws for processes such as soil production, and (3) rapid advances in digital mapping technology, such as LiDAR and the Shuttle Radar Topography Mission (SRTM), that have led to the widespread availability of high-resolution terrain data. However, computational efficiency is now a key limiting factor. Because of nonlinearities in the governing equations, current models cannot, for example, handle an area larger than a few hectares at the resolution of a typical LiDAR image (about
1 square kilometer). To address this obstacle, a 4-year collaborative effort between geoscientists and applied mathematiciansis is proposed, aimed at developing efficient and parallelized numerical algorithms for solving the equations that govern the evolution of a topographic surface.

River basins and their networks occupy the vast majority of the earth's land surface. Most of us live in a river basin and depend on its ability to gather water over a large area and focus it into a narrow channel.
However, the same mechanism gathers sediment and contaminants and can produce natural disasters such as floods and landslides. Efficient and accurate landscape evolution and surface-process models are needed to address numerous environmental problems and many other problems of societal relevance. They also provide outstanding educational and research opportunities that bridge the gap between applied mathematics and geoscience since they embody such a rich variety of challenging mathematical problems. These problems represent exciting intellectual frontiers in both geoscience and mathematics and it is hoped that the proposed collaborative work will attract the attention of mathematicians and lead to further advances.

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.

Particle Image Velocimetry (PIV) is a method for obtaining a fluid velocity field based on the translation of particles between images with a known time span between them. A potential limitation of PIV is that only two-dimensional velocity field along a single plane can be obtained from a two-dimensional image. This limitation is normally overcome by designing the experimental flow system so that the third velocity component is either zero or unimportant. In many applications, however, it is not possible to simplify the fully three-dimensional velocity field (e.g., in the left ventricle of the heart), and the two-dimensional limitations associated with PIV analysis are a significant problem that must be overcome. Would it be possible, however, to combine the two-dimensional PIV data together with a fully three-dimensional numerical approximation to the Navier-Stokes equations and obtain a sufficiently accurate three-dimensional velocity field in a domain such as the left ventricle of the heart? The use of least-squares finite element methods (LSFEMs) is proposed here to approximately solve the Navier-Stokes equations, and, significantly, to weakly constrain the solution along the PIV plane to match the experimental data. The PIV data would basically act as an internal boundary, and the numerical solution would weakly match the data with a variable weighting that determines the strength of the coupling between the data and numerical solution. LSFEMs are uniquely well suited for solving an over-constrained problem like this in a computationally efficient manner.

Echocardiologists have developed methods for introducing microbubbles into circulating blood that can be resolved using ultrasound. The location of the microbubbles combined with the high temporal resolution of ultrasound allows the local blood velocity to be determined using Particle Image Velocimetry, but the high temporal resolution requirement also limits the ultrasound scans (and, hence, the velocity field data) to two dimensions. One goal of using the FDA approved microbubbles is to use the blood velocity data to calculate physiologically important information such as pressure gradients and energy loss for the blood flow, but these calculations require a full three-dimensional velocity field. The goal of the proposed research is to develop mathematical techniques that combine computational fluid dynamics with experimental velocity data, such as that from microbubbles, to obtain a full, three-dimensional velocity field, thus providing greater insight into the dynamics of the flow.

Abstract

#1249858 / Manteuffel, Thomas
#1249950 / Jeffrey Heys


The role of computer simulations in scientific discovery continues to grow in many fields, including biology, finance, chemistry, and medicine. In many cases, these computer simulations are based on the solution of partial differential equations, which are mathematical equations that can only be approximately solved on large computers. Scientific discovery is also continuing through the use of more advanced experimental techniques that are enabling us to obtain more data than ever before and obtain new data that was not available previously. A critical need for scientists now is an approach for combining the computer simulations with the abundant data that is now available. To help address this need, we are proposing the development of advanced least-square finite element methods, which have a number of advantages for solving this problem of combining computer simulations with experimental data. First, the approach is flexible enough that it can assimilate data from any location in the simulation. If you are simulating blood flow through a vessel, the experimental data can be located anywhere, including near the wall or near the center of the vessel. Second, the approach can account for the accuracy of the experimental data. If the blood flow data is more accurate near the center of the vessel than near the wall, the simulation will more closely match the accurate data near the center, and it will not match the data near the wall as closely because the data likely contains significant error. A final advantage for the proposed approach is that it is computationally efficient. It has been designed from the beginning to work well with scalable, multilevel mathematical techniques and work well on modern, multiprocessor computer architectures. This is not an approach that will be overwhelmed by large, complex problems, but it will be efficient on today's computers and tomorrow's computers.

If a mechanic wishes to assess the condition of a cars engine, they will open up the hood and inspect the critical parts of the engine. The assessment of the health of the heart is a much more challenging problem because we cannot easily and safely open up the hood. An alternative approach is for a cardiologist to inject FDA approved microbubbles, these are bubbles that are smaller than red blood cells, into the blood, and then these microbubbles can be safely visualized using an external ultrasound machine. The movement of these microbubbles gives an indication of the blood flow in the heart, but more information is needed to properly assess the health of the heart. Specifically, cardiologists are interested in pressure changes in the heart and the overall efficiency of the heart. To obtain this additional information, we can simulate the flow of blood in the heart on a computer, and, ideally, combine the data from the moving microbubbles with the computer simulation so that we can obtain information specific to each individuals own heart. Problems that require us to combine a computer simulation with experimental data are becoming increasing common in fields from medicine to microbiology to meteorology. The work described in this proposal will provide a powerful new tool for combining experimental data with computer simulations. The approach will allow scientist to account for the accuracy of the data so that the more accurate data will be matched closely by the simulation and less accurate data will not match the simulation as well. The approach will be efficient on the next generation of computers because it supports advanced mathematic techniques. Overall, the positive impact of this approach should extent to many different scientific fields. The project will involve graduate and undergraduate students at both Montana State University and the University of Colorado-Boulder, and these students will interact extensively between universities. The project will also include the development of new engineering and mathematics course content and support scientific conferences.

Copper Mountain Conference on Iterative Methods, Copper Mountain, Colorado, April 6 -- April 11, 2014. The purpose of this proposal is to support the participation of 20 students and 10 women, post docs, minorities, and disabled in the 2014 Copper Mountain Conference on Iterative Methods. Begun in 1983 and alternating between Multigrid Methods (odd-numbered years) and Iterative Methods (even-numbered years), with substantial and growing overlap in both programs, this series is an important forum for exchange of ideas in these two closely related fields. The meeting will continue to promote the established tradition of a very high level of student participation and to increase efforts to promote participation by members of groups underrepresented in the mathematical sciences. The proposal is to support travel and local expenses for these groups, to foster an egalitarian structure with no invited speakers and all talks of equal length, to continue to organize tutorials and themed evening workshops, and to encourage broad representation of participants from academia, national labs, and industry.

Iterative methods and multilevel algorithms are of critical importance for the development of efficient simulations in all domains of science and technology. The conference series is concerned with all aspects of these methods, including traditional techniques of convergence analysis, implementation and development of mathematical software, and use of such ideas in new settings, including advanced computer architectures and new applications. Building on previous success, the plan is to take a proactive role in recruiting students and female and minority participants. This support will have significant impact on science and technology because, for many simulations performed in these important fields, iterative solvers dominate overall calculation time and constrain capabilities. As simulations continue to grow in complexity, the need for more effective solvers becomes increasingly acute. This conference will facilitate the development of efficient iterative solvers and their dissemination to users in industry and national labs. Moreover, the continued high level of participation of students and other young scientists will create and nurture a community of researchers and practitioners who are welcomed to the field.

Conference web page:
http://grandmaster.colorado.edu/~copper/2014/

Copper Mountain Conference On Multigrid Methods, Copper Mountain, Colorado, March 22-27, 2015.This grant is to support participation in the 2015 Copper Mountain Conference on Multigrid Methods. The funding from this award is specifically for students, women, and scientists who are members of underrepresented groups. The Copper Mountain Conferences have graduate students forming a very large fraction of the attendees (typically 30%-40%), many of them full participants who co-author papers and give presentations. Women and minorities are growing fractions of the attendees, but further increases are desired, so this award also targets their participation. Support for the students, women, and minority scientists from this award is in the form of reduced registration fees and local and travel expenses. A hallmark of this conference is a "Student Paper Competition," which draws entries from a significant fraction of the student attendees and results in extraordinarily high-quality papers on scientific discovery from the students, working in tandem with and under the guidance of their faculty advisors. These results, like the conferences as a whole, span wide-ranging and important theoretical and applications areas, such as techniques of convergence analysis, implementation and development of mathematical software, and use of such ideas in novel settings, including advanced computer architectures and new applications such as uncertainty quantification. The aim here is to continue the increase in participation by members of these underrepresented groups so that the multigrid discipline and science in general are improved by the professional development of these talented researchers.

The Copper Mountain Conferences form arguably the premier conferences in two closely related mathematical fields: iterative and multigrid methods. These two fields provide computational support for numerical simulation of a very wide host of human endeavors, including environmental and energy research, medical and biological applications, and many other areas critical to the U.S. and international science and engineering community. This award supports the participation of students, women, and scientists who are members of underrepresented groups at the 2015 Conference. The Conferences traditionally work to ensure the future vitality of the fields of iterative and multigrid methods by facilitating development and nurturing of a community of capable graduate students and entry-level scientists. Through their egalitarian structure, with no invited speakers and all talks of equal length, the Conferences provide mechanisms for young people to meet each other and all participants in a relaxed yet scientifically rigorous setting; these mechanisms include topical tutorials, themed evening workshops, and access to the broad representation of participants from academia, national laboratories, and industry. The Conferences have a tradition of a very high level of student participation (typically 30-40% of attendees), and will cultivate this through supporting students' local and travel expenses. Emphasis is also placed on engendering diversity through support of women and minority scientists. These conferences bring together the world's leading practitioners in these critical fields and result in high-level publications, effective applications codes, and the establishment of long-term collaborative research partnerships.