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Seismic imaging of reservoir flow properties: Time-lapse pressurechanges

Description: Time-lapse fluid pressure and saturation estimates are sensitive to reservoir flow properties such as permeability. In fact, given time-lapse estimates of pressure and saturation changes, one may define a linear partial differential equation for permeability variations within the reservoir. The resulting linear inverse problem can be solved quite efficiently using sparse matrix techniques. An application to a set of crosswell saturation and pressure estimates from a CO{sub 2} flood at the Lost Hills field in California demonstrates the utility of this approach. From the crosswell estimates detailed estimates of reservoir permeability are produced. The resulting permeability estimates agree with a permeability log in an adjacent well and are in accordance with water and CO{sub 2} saturation changes in the interwell region.
Date: April 8, 2003
Creator: Vasco, Don W.
Partner: UNT Libraries Government Documents Department

Time-periodic solutions of the Benjamin-Ono equation

Description: We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which, in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase and the real part of one of the Fourier modes at t = 0. We use our method to study global paths of non-trivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached (or until the solution blows up). By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODE's governing the evolution of solitons using the ansatz suggested by the numerical simulations.
Date: April 1, 2008
Creator: Ambrose , D.M. & Wilkening, Jon
Partner: UNT Libraries Government Documents Department

Performance and scaling of locally-structured grid methods forpartial differential equations

Description: In this paper, we discuss some of the issues in obtaining high performance for block-structured adaptive mesh refinement software for partial differential equations. We show examples in which AMR scales to thousands of processors. We also discuss a number of metrics for performance and scalability that can provide a basis for understanding the advantages and disadvantages of this approach.
Date: July 19, 2007
Creator: Colella, Phillip; Bell, John; Keen, Noel; Ligocki, Terry; Lijewski, Michael & Van Straalen, Brian
Partner: UNT Libraries Government Documents Department

Steepest descent for partial differential equations of mixed type

Description: The method of steepest descent is used to solve partial differential equations of mixed type. In the main hypothesis for this paper, H, L, and S are Hilbert spaces, T: H -> L and B: H -> S are functions with locally Lipshitz Fréchet derivatives where T represents a differential equation and B represents a boundary condition. Define ∅(u)=1/2IIT(u)II^2. Steepest descent is applied to the functional ∅. A new smoothing technique is developed and applied to Tricomi type equations (which are of mixed type).
Date: August 1992
Creator: Kim, Keehwan
Partner: UNT Libraries

International Conference on Multiscale Methods and Partial Differential Equations.

Description: The International Conference on Multiscale Methods and Partial Differential Equations (ICMMPDE for short) was held at IPAM, UCLA on August 26-27, 2005. The conference brought together researchers, students and practitioners with interest in the theoretical, computational and practical aspects of multiscale problems and related partial differential equations. The conference provided a forum to exchange and stimulate new ideas from different disciplines, and to formulate new challenging multiscale problems that will have impact in applications.
Date: December 12, 2006
Creator: Hou, Thomas
Partner: UNT Libraries Government Documents Department

Dimensional reduction as a tool for mesh refinement and trackingsingularities of PDEs

Description: We present a collection of algorithms which utilizedimensional reduction to perform mesh refinement and study possiblysingular solutions of time-dependent partial differential equations. Thealgorithms are inspired by constructions used in statistical mechanics toevaluate the properties of a system near a critical point. The firstalgorithm allows the accurate determination of the time of occurrence ofa possible singularity. The second algorithm is an adaptive meshrefinement scheme which can be used to approach efficiently the possiblesingularity. Finally, the third algorithm uses the second algorithm untilthe available resolution is exhausted (as we approach the possiblesingularity) and then switches to a dimensionally reduced model which,when accurate, can follow faithfully the solution beyond the time ofoccurrence of the purported singularity. An accurate dimensionallyreduced model should dissipate energy at the right rate. We construct twovariants of each algorithm. The first variant assumes that we have actualknowledge of the reduced model. The second variant assumes that we knowthe form of the reduced model, i.e., the terms appearing in the reducedmodel, but not necessarily their coefficients. In this case, we alsoprovide a way of determining the coefficients. We present numericalresults for the Burgers equation with zero and nonzero viscosity toillustrate the use of the algorithms.
Date: June 10, 2007
Creator: Stinis, Panagiotis
Partner: UNT Libraries Government Documents Department

A New Stabilized Nodal Integration Approach

Description: A new stabilized nodal integration scheme is proposed and implemented. In this work, focus is on the natural neighbor meshless interpolation schemes. The approach is a modification of the stabilized conforming nodal integration (SCNI) scheme and is shown to perform well in several benchmark problems.
Date: February 8, 2006
Creator: Puso, M; Zywicz, E & Chen, J S
Partner: UNT Libraries Government Documents Department

Final Report on Subcontract B591217: Multigrid Methods for Systems of PDEs

Description: Progress is summarized in the following areas of study: (1) Compatible relaxation; (2) Improving aggregation-based MG solver performance - variable cycle; (3) First Order System Least Squares (FOSLS) for LQCD; (4) Auxiliary space preconditioners; (5) Bootstrap algebraic multigrid; and (6) Practical applications of AMG and fast auxiliary space preconditioners.
Date: October 25, 2011
Creator: Xu, J; Brannick, J J & Zikatanov, L
Partner: UNT Libraries Government Documents Department

Operator-Based Preconditioning of Stiff Hyperbolic Systems

Description: We introduce an operator-based scheme for preconditioning stiff components encoun- tered in implicit methods for hyperbolic systems of partial differential equations posed on regular grids. The method is based on a directional splitting of the implicit operator, followed by a char- acteristic decomposition of the resulting directional parts. This approach allows for solution to any number of characteristic components, from the entire system to only the fastest, stiffness-inducing waves. We apply the preconditioning method to stiff hyperbolic systems arising in magnetohydro- dynamics and gas dynamics. We then present numerical results showing that this preconditioning scheme works well on problems where the underlying stiffness results from the interaction of fast transient waves with slowly-evolving dynamics, scales well to large problem sizes and numbers of processors, and allows for additional customization based on the specific problems under study.
Date: February 9, 2009
Creator: Daniel R. Reynolds, Ravi Samtaney, and Carol S. Woodward
Partner: UNT Libraries Government Documents Department

The analysis of a sparse grid stochastic collocation method for partial differential equations with high-dimensional random input data.

Description: This work describes the convergence analysis of a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems and, as such, the derived strong error estimates for the fully discrete solution are used to compare the computational efficiency of the proposed method with the Monte Carlo method. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo.
Date: December 1, 2007
Creator: Webster, Clayton; Tempone, Raul (Florida State University, Tallahassee, FL) & Nobile, Fabio (Politecnico di Milano, Italy)
Partner: UNT Libraries Government Documents Department

Overture: an objectoriented framework for solving partial differential equations on overlapping grids

Description: The Overture framework is an object-oriented environment for solving partial differential equations in two and three space dimensions. It is a collection of C++ libraries that enables the use of finite difference and finite volume methods at a level that hides the details of the associated data structures. Overture can be used to solve problems in complicated, moving geometries using the method of overlapping grids. It merges geometry, grid generation, difference operators, boundary conditions, data-base access and graphics into an easy to use high level interface.
Date: September 22, 1998
Creator: Brown, D L; Henshaw, W D & Quinlan , D J
Partner: UNT Libraries Government Documents Department

Overture: An Object-Oriented Framework for Overlapping Grid Applications

Description: The Overture framework is an object-oriented environment for solving partial differential equations on over-lapping grids. We describe some of the tools in Overture that can be used to generate grids and solve partial differential equations (PDEs). Overture contains a collection of C++ classes that can be used to write PDE solvers either at a high level or at a lower level for efficiency. There are also a number of tools provided with Overture that can be used with no programming effort. These tools include capabilities to: repair computer-aided-design (CAD) geometries and build global surface triangulations; generate surface and volume grids with hyperbolic grid generation; generate composite overlapping grids; generate hybrid (unstructured) grids; and solve particular PDEs such as the incompressible and compressible Navier-Stokes equations.
Date: April 4, 2002
Creator: Henshaw, W. D.
Partner: UNT Libraries Government Documents Department

Numerical Methods for Stochastic Partial Differential Equations

Description: This is the final report of a Laboratory Directed Research and Development (LDRD) project at the Los Alamos National laboratory (LANL). The objectives of this proposal were (1) the development of methods for understanding and control of spacetime discretization errors in nonlinear stochastic partial differential equations, and (2) the development of new and improved practical numerical methods for the solutions of these equations. The authors have succeeded in establishing two methods for error control: the functional Fokker-Planck equation for calculating the time discretization error and the transfer integral method for calculating the spatial discretization error. In addition they have developed a new second-order stochastic algorithm for multiplicative noise applicable to the case of colored noises, and which requires only a single random sequence generation per time step. All of these results have been verified via high-resolution numerical simulations and have been successfully applied to physical test cases. They have also made substantial progress on a longstanding problem in the dynamics of unstable fluid interfaces in porous media. This work has lead to highly accurate quasi-analytic solutions of idealized versions of this problem. These may be of use in benchmarking numerical solutions of the full stochastic PDEs that govern real-world problems.
Date: July 8, 1999
Creator: Sharp, D.H.; Habib, S. & Mineev, M.B.
Partner: UNT Libraries Government Documents Department

Control and Analysis of a Single-Link Flexible Beam with Experimental Verification

Description: The objective of this report is to ascertain the general conditions for the avoidance and reduction of residual vibration in a flexible manipulator. Conventional manipulators usually have a 1.5 to 2-m reach, and their associated dynamic models typically are composed of lumped parameter elements; the major compliance emanates from the, drive trains because of torsional loading effects. The energy storage of the drive system is predominantly potential energy because of the low inertia in the drive tram; thus simple spring models have been adequate. A long-reach manipulator with a large aspect ratio (length to diameter) is a fundamentally different problem. Energy storage for this type of manipulator is distributive by nature because of the potential energy resulting from bending and the kinetic energy due to deflection rates. Instead of ordinary differential equations, partial differential equations are required to describe this system, making the analysis more difficult. The general flexibility problem associated with a distributive dynamic system, with specific emphasis on flexible manipulator, will be addressed in this report. Furthermore, three control schemes will be discussed and demonstrated on, a single flexible manipulator to determine their general merits.
Date: January 1, 1992
Creator: Jansen, J.F.
Partner: UNT Libraries Government Documents Department

Using automatic differentiation for second-order matrix-free methods in PDE-constrained optimization.

Description: Classical methods of constrained optimization are often based on the assumptions that projection onto the constraint manifold is routine but accessing second-derivative information is not. Both assumptions need revision for the application of optimization to systems constrained by partial differential equations, in the contemporary limit of millions of state variables and in the parallel setting. Large-scale PDE solvers are complex pieces of software that exploit detailed knowledge of architecture and application and cannot easily be modified to fit the interface requirements of a blackbox optimizer. Furthermore, in view of the expense of PDE analyses, optimization methods not using second derivatives may require too many iterations to be practical. For general problems, automatic differentiation is likely to be the most convenient means of exploiting second derivatives. We delineate a role for automatic differentiation in matrix-free optimization formulations involving Newton's method, in which little more storage is required than that for the analysis code alone.
Date: November 20, 2000
Creator: Hovland, P. D.; Keyes, D. E.; McInnes, L. C. & Samyono, W.
Partner: UNT Libraries Government Documents Department

Users' Guide to ADIC 1.1.

Description: This guide describes the use of the Automatic Differentiation in C (ADIC) system. ADIC is a suite of tools and libraries that automates the process of generating derivatives for scientific programs. In the context of solving PDEs, optimizations, sensitivity analysis, and inverse problems, researchers often require the derivatives {partial_derivative}f/{partial_derivative}x of a function f expressed as a program with respect to some input parameter(s) x. Automatic differentiation (AD) techniques augment the program with derivative computation by applying the chain rule of calculus to elementary operations in an automated fashion. ADIC uses sophisticated compiler techniques to augment the input C programs with derivative computation capability in an automatic fashion. It also provides a finer control of derivative code generation process via control scripts and pragmas. Another significant capability of ADIC is its component architecture, AIF, that allows ADIC's capability to be extended via plug-in modules.
Date: August 31, 2004
Creator: Hovland, P. D. & Norris, B.
Partner: UNT Libraries Government Documents Department

Reducing the memory requirement in reverse mode automatic differentiation by solving TBR flow equations.

Description: The fast computation of gradients in reverse mode Automatic Differentiation (AD) requires the generation of adjoint versions of every statement in the original code. Due to the resulting reversal of the control flow certain intermediate values have to be made available in reverse order to compute the local partial derivatives. This can be achieved by storing these values or by recomputing them when they become required. In any case one is interested in minimizing the size of this set. Following an extensive introduction of the ''To-Be-Recorded'' (TBR) problem the authors present flow equations for propagating the TBR status of variables in the context of reverse mode AD of structured programs.
Date: January 11, 2002
Creator: Naumann, U.
Partner: UNT Libraries Government Documents Department

Spectral Representations of Uncertainty: Algorithms and Applications

Description: The objectives of this project were: (1) Develop a general algorithmic framework for stochastic ordinary and partial differential equations. (2) Set polynomial chaos method and its generalization on firm theoretical ground. (3) Quantify uncertainty in large-scale simulations involving CFD, MHD and microflows. The overall goal of this project was to provide DOE with an algorithmic capability that is more accurate and three to five orders of magnitude more efficient than the Monte Carlo simulation.
Date: April 24, 2005
Creator: Karniadakis, George Em
Partner: UNT Libraries Government Documents Department

Variational particle scheme for the porous medium equation and for the system of isentropic Euler equations

Description: We introduce variational particle schemes for the porous medium equation and the system of isentropic Euler equations in one space dimension. The methods are motivated by the interpretation of each of these partial differential equations as a 'steepest descent' on a suitable abstract manifold. We show that our methods capture very well the nonlinear features of the flows.
Date: December 10, 2008
Creator: Westdickenberg, Michael & Wilkening, Jon
Partner: UNT Libraries Government Documents Department

Some free boundary problems in potential flow regime usinga based level set method

Description: Recent advances in the field of fluid mechanics with moving fronts are linked to the use of Level Set Methods, a versatile mathematical technique to follow free boundaries which undergo topological changes. A challenging class of problems in this context are those related to the solution of a partial differential equation posed on a moving domain, in which the boundary condition for the PDE solver has to be obtained from a partial differential equation defined on the front. This is the case of potential flow models with moving boundaries. Moreover the fluid front will possibly be carrying some material substance which will diffuse in the front and be advected by the front velocity, as for example the use of surfactants to lower surface tension. We present a Level Set based methodology to embed this partial differential equations defined on the front in a complete Eulerian framework, fully avoiding the tracking of fluid particles and its known limitations. To show the advantages of this approach in the field of Fluid Mechanics we present in this work one particular application: the numerical approximation of a potential flow model to simulate the evolution and breaking of a solitary wave propagating over a slopping bottom and compare the level set based algorithm with previous front tracking models.
Date: December 9, 2008
Creator: Garzon, M.; Bobillo-Ares, N. & Sethian, J.A.
Partner: UNT Libraries Government Documents Department

ROSE: Compiler Support for Object-Oriented Frameworks

Description: ROSE is a preprocessor generation tool for the support of compile time performance optimizations in Overture. The Overture framework is an object-oriented environment for solving partial differential equations in two and three space dimensions. It is a collection of C++ libraries that enables the use of finite difference and finite volume methods at a level that hides the details of the associated data structures. Overture can be used to solve problems in complicated, moving geometries using the method of overlapping grids. It has support for grid generation, difference operators, boundary conditions, database access and graphics. In this paper we briefly present Overture, and discuss our approach toward performance within Overture and the A++P++ array class abstractions upon which Overture depends, this work represents some of the newest work in Overture. The results we present show that the abstractions represented within Overture and the A++P++ array class library can be used to obtain application codes with performance equivalent to that of optimized C and Fortran 77. ROSE, the preprocessor generation tool, is general in its application to any object-oriented framework or application and is not specific to Overture.
Date: November 17, 1999
Creator: Qainlant, D.
Partner: UNT Libraries Government Documents Department

The Distinct Element Method - Application to Structures in Jointed Rock

Description: The Distinct Element Method (DEM) is a meshfree method with applications to rock mechanics, mining sciences, simulations of nuclear repositories, and the stability of underground structures. Continuum mesh-based methods have been applied successfully to many problems in geophysics. Even if the geology includes fractures and faults, when sufficiently large length scales are considered a continuum approximation may be sufficient. However, a large class of problems exist where individual rock joints must be taken into account. This includes problems where the structures of interest have sizes comparable with the block size. In addition, it is possible that while the structure may experience loads which do no measurable damage to individual blocks, some joints may fail. This may launch smaller blocks as dangerous projectiles or even cause total failure of a tunnel. Traditional grid-based continuum approaches are wholly unsuited to this class of problem. It is possible to introduce discontinuities or slide lines into existing grid-based methods, however, such limited approaches can break down when new contacts form between blocks. The distinct element method (DEM) is an alternative, meshfree approach. The DEM can directly approximate the block structure of the jointed rock using arbitrary polyhedra. Using this approach, preexisting joints are readily incorporated into the DEM model. In addition, the method detects all new contacts between blocks resulting from relative block motion. We will describe the background of the DEM and review previous application of the DEM to geophysical problems. Finally we present preliminary results from a investigation into the stability of underground structures subjected to dynamic loading.
Date: November 30, 2001
Creator: Morris, J.P.; Glen, L.; Blair, S. & Heuze, F.
Partner: UNT Libraries Government Documents Department

Overture: Object-Oriented Tools for Application with Complex Geometry

Description: The Overture framework is an object-oriented environment for solving partial differential equations in two and three space dimensions. It is a collection of C++ libraries that enables the use of finite difference and finite volume methods at a level that hides the details of the associated data structures. Overture can be used to solve problems in complicated, moving geometries using the method of overlapping grids. It has support for grid generation, difference operators, boundary conditions, data-base access and graphics. Short sample code segments are presented to show the power of this approach.
Date: May 31, 1999
Creator: Brown, D.; Henshaw, B. & Quinlan, D.
Partner: UNT Libraries Government Documents Department