Ludmil Zikatanov - US grants

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 research in this proposal is on the study and applications of
efficient algebraic multigrid methods for the solution of linear
algebraic systems arising from the discretization of second order
partial differential equations by the generalized finite element
method. The proposed research will focus on the development and
analysis of adaptive techniques in the construction of hierarchy of
nested spaces and the choice of approximate subspace solvers that lead
to the efficient and robust multigrid methods applicable to wide range
of generalized finite element discretizations.

The rapid increase in the power of today's supercomputers has made it
feasible for the scientific community to use numerical simulations to
model physical phenomena to produce meaningful results. One of the
modern techniques that can deliver quantitative results via such
simulations is the generalized finite element method. This method has
proved to be a very robust discretization tool, applicable in various
branches of engineering and sciences, for example, in simulating and
determining the elastic, electromagnetic and other important physical
properties of heterogeneous materials. Like most other discretization
techniques, most often the majority of computation in such simulations
is devoted to the solution of the resulting linear systems of
equations. Hence, it is very important to develop efficient solvers
for these systems. The results from the proposed research are thus
expected to have a broad and noticeable impact by providing the much
needed iterative multilevel solution techniques for the discrete
linear systems arising from numerical models in many applications.
The proposed research is also expected to have an educational impact
as it will provide a solid base for training of graduate students in
the modern theoretical and practical aspects of numerical methods for
problems in science and engineering.

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

The primary goal of this collaborative proposal is to develop
theoretically based algebraic multigrid (AMG) solvers for Hermitian
(and, where possible, non-Hermitian) positive-definite problems. The
team aims to improve understanding of the performance of the family of
AMG algorithms and, with this improved knowledge, to develop AMG
methods that offer provable, computable, a priori information on the
algorithm's performance. The project team represents a close
collaboration of experts in this area, each of whom has made
contributions in the field. Over the past several years, the team has
begun to work collectively on developing new multilevel solvers and
rigorous theoretical results for the convergence and complexity
analysis thereof. Together, the team will have the capability to take
a step toward answering some of the fundamental research questions
associated with these two essential aspects of the analysis and design
of efficient algorithms.

We expect the work proposed here to: (1) directly impact computational
simulation codes currently employing multi-level solvers, by providing
faster and more reliable computational tools for the numerical
computations at the core of physical simulations; and (2) allow for
simulation of phenomena for which suitable solvers are currently
unavailable. The results from the proposed research will, thus, have
a direct impact on scientific and engineering problems, including
those from energy, through both the simulation of particle physics and
processing of data from oil reservoir models, biophysics, in surgical
simulation, and the environment, in climate prediction and contaminant
remediation models. The algorithms to be investigated here are
already in use in many of these fields, but are often considered to be
"expert-only" tools. The goal of this proposal is to develop more
reliable and robust versions of these tools. The proposed research
will have a strong educational impact as well, as it provides for a
solid base for training of graduate students in the modern theoretical
and practical aspects of numerical methods for modeling of
applications arising in science and engineering.

The most common origin and manifestation of anomalous phenomena in complex fluids are different ``elastic'' effects. These can be ascribed to the elasticity of deformable particles, elastic repulsion between charged liquid crystals, polarization of colloids or multi-component phases, elastic effects due to microstructures formation, or bulk elastic effects due to the presence of polymer molecules in viscoelastic complex fluids. Mathematically, such elastic effects can be represented in terms of internal variables. Examples of such internal variables are: the orientational order parameter in liquid crystals, the distribution density function in the dumb-bell model for polymeric materials, the magnetic field in magnetohydrodynamic fluids, and the volume fraction in mixtures of different materials. The different rheological and hydrodynamic properties of these materials can in turn be attributed to the special coupling between the transport of these internal variables and the induced elastic stresses which typically manifest on all scales, and to a large extent determine the specific properties of the system, such as the stability and regularity of particle configurations and the likelihood of specific pattern formations in the system. The understanding of such complex mechanisms which couple different physical scales is crucial in designing accurate mathematical and numerical models and algorithms in order to simulate such systems. This workshop is based on the premise that gaining significant new results and thereby obtaining deeper insight and understanding of materials described by complex fluids (e.g., polymers, emulsions, liquid crystals, magnetorheological fluids, blood suspensions) requires a combined approach consisting of experiment, modeling, analysis, and simulation. The primary objective of the workshop is thus to gather students, junior faculty, and experts to discuss new integrated modeling techniques that properly address the fundamental unresolved issues common to studies of complex fluids: consistency of models with physics, rigorous analysis of the models, numerical and scientific computing issues, and the experimental verification and validation of the predictive capabilities of the mathematical and numerical models.

Modeling and simulating the rich pool of designer and smart materials described by complex fluids requires marshaling interdisciplinary research forces to develop and analyze mathematical models and computational tools for such problems. Recently, numerous methodologies and frameworks have been developed in an aim to capture the various time and length scales involved in studies of such complex materials. This workshop will gather key contributors to the development of these new multiscale modeling and simulations techniques, with the aim of introducing these ideas to students and young researchers and also promoting interdisciplinary research on complex fluids applications amongst participants in the future. The workshop will highlight topics from interrelated research areas for both deterministic and stochastic computational approaches, concentrating on: an Energetic Variational approach for multiscale modeling and simulation; computationally and experimentally guided model validation and further development of robust adaptive numerical models and solution methods.

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.

Numerical Modeling of Fluids and Structures at the 9th International Conference on Large-Scale Scientific Computations (LSSC) in Sozopol, Bulgaria from June 3-7, 2013.

The investigators and his colleagues are organizing a special session on Numerical Modeling of Fluids and Structures at the 9th International Conference on Large-Scale Scientific Computations (LSSC) in Sozopol, Bulgaria from June 3-7, 2013. This minisymposium focuses on developing, investigating, and applying fundamental mathematical theories and advanced modeling and simulation techniques to various multiphysics and multiscale problems, especially fluid-structure interactions. Key scientific questions addressed in this session have a wide range of applications, including magnetorheological fluids, aerodynamics, biomedical applications, micro-electro-mechanical systems, and ground water modeling.

This special session supports participation of students, postdocs, and junior researchers from the United States, who are working in areas related to the modeling and simulation of fluids, structures, and their interactions. The workshop environment promotes contacts between researchers from the United States and other countries, including theorists and experimentalists from applied mathematics, computational science and engineering. Mutually beneficial discussions between junior and senior researchers and mathematicians and engineers are expected. The special session is mainly organized by junior researchers and, along with other young participants, makes their research highly visible to the rest of the scientific community, providing them the access to various application fields for which they can contribute many advancements.

This project focuses on the integration of recent theoretical and algorithmic advances in numerical models, which describe the behavior of elastic materials on multiple scales, exhibiting stochastic behavior. The project will include algorithmic design, convergence and complexity analysis, as well as issues that arise in the performance of the upscaling and multilevel algorithms in realistic simulations. The proposed research: (1) aids the development of new and robust methods for upscaling that provide reliable calculations and predictions in structural mechanics; (2) supports the migration of such methods into real-life scientific and engineering simulations; and (3) engages the broader scientific community through research and educational activities, highlighting the integrated approach in numerical modeling of elastic materials from adaptive discretizations to robust solvers and back.

The proposed research aims to improve understanding of the interplay between the techniques from differential geometry and topology, which lead to discretizations compatible with the geometric and topological structures inherited from the physical/mathematical model. Based on this, the PIs plan to develop agglomeration methods that offer provable optimal algorithm performance. The novel efficient and accurate upscaling techniques for elasticity problems have potential applications in material sciences and geosciences. In addition, accurate coarse discretizations yield efficient multilevel solvers for the linear systems coming from corresponding discretizations of linear elasticity. Such solvers enable simulations with finer spatial resolution and/or reduce the necessary computational resources for such simulations. Finally, the design of upscaling techniques has many similarities with the design of discretizations in general. The success of the project will facilitate accurate discretization schemes and robust solvers for linear elasticity equations based on element agglomeration.

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.

The project focuses on the development of new mathematical tools which improve our understanding and application of advanced computational methods. The research explores novel as well as established practical algorithms which are then validated and verified in a variety of hydrogeological scenarios. One major goal of computational mathematics research, in general, is to improve our understanding of numerical algorithms and to make them more efficient and accurate. However, the tasks of improving methods and then applying them are often completed by distinct groups of researchers and have long transfer times between development and application. Often, the theoretical findings and the heuristic algorithms developed by practitioners follow different trajectories. Indeed, many valuable numerical techniques have been proposed and used by practitioners without much theoretical justification or for a narrow set of problems. Concurrent improvements based on new mathematical insights are often unrelated to practical problems. This research addresses the disparity between the two disciplinary trajectories by reconnecting advanced and abstract mathematical theories with practice. The project has the potential to impact a wide range of applications, including for example the simulation of variably saturated flow.

This project is concerned with the development and analysis of adaptive, conservative and monotone discretizations that are extendable to any order for the solution of nonlinear partial differential equations, such as Richards' equation used to simulate variably saturated flow and Biot's model in poroelasticity. Typically, such discretization methods result in large-scale, ill-conditioned linear systems. The efficient solution of such systems, generally non-symmetric and indefinite, is crucial for the performance of overall numerical simulation as it consumes the larger part of the computing resources. Part of this project includes the development of a class of efficient and adaptive multilevel solvers capable of generating flexible hierarchies of spaces. Such hierarchies are useful also in filtering and representing sparse data sets, robust with respect to the structure and the type of data: smooth, oscillatory or combinations of the two. In the targeted applications in hydrogeology, the research will be on techniques which efficiently approximate elevation, groundwater head, precipitation, and other relevant hydrogeological spatial data.

The overall goal of this research project is to develop, analyze, and implement stable numerical methods for the simulation of flow in fractured porous media, based on mixed-dimensional modeling. Fractured porous media flow is multi-physics and multi-scale, and the development of robust and effective numerical tools for such systems represents a class of important challenges in computational mathematics. For example, in many practical applications fractures or other features, such as capillaries in the brain, are lower-dimensional and interact in a complex way through the three-dimensional domains encapsulating them. Important applications of fractured porous media include hydraulic fracturing, waste deposition, and models from biomechanics. The project aims to provide new computational paradigms, promote the usage of mixed-dimensional modeling in these and related fields, and alleviate current limitations in computer simulations. The techniques will be implemented in an open-source software package and will be made available to the scientific community. <br/><br/>In this project, three important aspects of mixed-dimensional modeling and simulation will be investigated. The first research objective is to design advanced stable discretizations, which couple stabilized schemes for linear elasticity with structure-preserving discretizations for Darcy flow in a mixed-dimensional setting. Stability and mass conservation will be achieved using a minimal number of degrees of freedom. The second objective is adaptive approximations in the mixed-dimensional framework. Space-time adaptivity will be developed and implemented to improve accuracy with solid theoretical foundations. The third objective is to study robust linear solvers for the resulting discrete linear systems. Based on physical and mathematical properties of the mixed-dimensional models and their numerical discretizations, new block preconditioners and monolithic multigrid methods will be developed. Robustness with respect to physical and discretization parameters will be justified both theoretically and numerically. The main application of this project is linear poromechanics in fractured porous media, but possible generalizations to nonlinear formulations will also be investigated, including several applications in physics and engineering.<br/><br/>This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.