Computational Applied Mathematics and Operations Research

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2D and 2.5D Kirchhoff Inversion Using Upwind Finite Difference Amplitudes
(1996-07) Araya, Kidane; Symes, William W.
Finite difference solution of the transport equation provides an efficient and accurate method for computation of 2.5D geometric acoustics amplitudes. These amplitudes can be used in simulation, migration and inversion formulas. Remodeled data based on high frequency asymptotic inversion using these amplitudes shows excellent agreement with both synthetic and field input data.
A Block Coordinate Descent Method for Multi-Convex Optimization with Applications to Nonnegative Tensor Factorization and Completion
(2012-08) Xu, Yangyang; Yin, Wotao
This paper considers block multi-convex optimization, where the feasible set and objective function are generally non-convex but convex in each block of variables. We review some of its interesting examples and propose a generalized block coordinate descent method. Under certain conditions, we show that any limit point satisfies the Nash equilibrium conditions. Furthermore, we establish its global convergence and estimate its asymptotic convergence rate by assuming a property based on the Kurdyka-Lojasiewicz inequality. The proposed algorithms are adapted for factorizing nonnegative matrices and tensors, as well as completing them from their incomplete observations. The algorithms were tested on synthetic data, hyperspectral data, as well as image sets from the CBCL and ORL databases. Compared to the existing state-of-the-art algorithms, the proposed algorithms demonstrate superior performance in both speed and solution quality. The Matlab code is available for download from the authors' homepages.
A Branch and Cut Algorithm for Nonconvex Quadratically Constrained Quadratic Programming
(1999-01) Audet, Charles; Hansen, Pierre; Jaumard, Brigitte; Savard, Gilles
We present a branch and cut algorithm that yields in finite time, a globally epsilon-optimal solution (with respect to feasibility and optimality) of the nonconvex quadratically constrained quadratic programming problem. The idea is to estimate all quadratic terms by successive linearizations within a branching tree using Reformulation-Linearization Techniques (RLT). To do so, four classes of linearizations (cuts), depending on one to three parameters, are detailed. For each class, we show how to select the best member with respect to a precise criterion. The cuts introduced at any node of the tree are valid in the whole tree, and not only within the subtree rooted at that node. In order to enhance the computational speed, the structure created at any node of the tree is flexible enough to be used at other nodes. Computational results are reported. Some problems of the literature are solved, for the first time with a proof of global optimality.
A Branch Decomposition Algorithm for the p-Median Problem
(INFORMS, 2017) Fast, Caleb C.; Hicks, Illya V.
In this paper, we use a branch decomposition technique to improve approximations to the p-median problem. Starting from a support graph produced either by a combination of heuristics or by linear programming, we use dynamic programming guided by a branch decomposition of that support graph to find the best p-median solution on the support graph. Our results show that when heuristics are used to build the support graph and the support graph has branchwidth at most 7, our algorithm is able to provide a solution of lower cost than any of the heuristic solutions. When linear programming is used to build the support graph and the support graph has branchwidth at most 7, then our algorithm provides better solutions than popular heuristics and is faster than integer programming. Thus, our algorithm is a useful practical tool when support graphs have branchwidth at most 7.
A C++ Class Supporting Adjoint-State Methods
(2009-09) Enriquez, Marco
The adjoint-state method is widely used for computing gradients in simulation- driven optimization problems. The adjoint-state evolution equation requires access to the entire history of the system states. There are instances, however, where the required state for the adjoint-state evolution is not readily accessible; consider large- scale problems, for example, where the entire simulation history is not saved to con- serve memory. This thesis introduces a C++ state-access class, StateHistory, to support a myriad of solutions to this problem. Derived StateHistory classes im- plement a (simulation) time-altering function and data-access functions, which can be used in tandem to access the entire state history. This thesis also presents a derived StateHistory class, GriewankStateHistory, which uses Griewank's opti- mal checkpointing scheme. While only storing a small fraction of simulation states, GriewankStateHistory objects can reconstitute unsaved states for a small computa- tional cost. These ideas were implemented in the context of TSOpt, a time-stepping library for simulation-driven optimization algorithms.
A Characteristic-Mixed Method for Contaminant Transport and Miscible Displacement
(1992-02) Arbogast, Todd; Chilakapati, Ashokkumar; Wheeler, Mary F.
Recently, Arbogast and Wheeler have formulated and analyzed a modified method of characteristics-mixed method for approximating solutions to convection-diffusion equations. This scheme is theoretically mass conservative over each grid cell; it is approximately so in implementations. We consider application of this procedure to contaminant transport and to miscible displacement with unfavorable mobility ratio. Results in one, two, and three space dimensions are discussed.
A Characteristics-Mixed Finite Element Method for Advection Dominated Transport Problems
(1992-11) Arbogast, Todd; Wheeler, Mary F.
We define a new finite element method, called the characteristics-mixed method, for approximating the solution to an advection dominated transport problem. The method is based on a space-time variational form of the advection-diffusion equation. Our test functions are piecewise constant in space, and in time they approximately follow the characteristics of the advective (i.e., hyperbolic) part of the equation. Thus the scheme uses a characteristic approximation to handle advection in time. This is combined with a low order mixed finite element spatial approximation of the equation. Boundary conditions are incorporated in a natural and mass conservative fashion. The scheme is completely locally conservative; in fact, on the discrete level, fluid is transported along the approximate characteristics. A post-processing step is included in the scheme in which the approximation to the scaler unknown is improved by utilizing the approximate vector flux. This has the effect of improving the rate of convergence of the method. We show that it is optimally convergent to order one in time and at least suboptimally convergent to order 3/2 in space.
A Chemical Compositional Reservoir Simulator on Distributed Memory Parallel Computers: Comparative Parallel-UTCHEM Simulation Performance Study (Part I)
(1994-11) Rame, M.; Delshad, M.
This paper presents the application of distributed memory parallel computers to field scale reservoir simulations using a parallel version of UTCHEM, The University of Texas Chemical Flooding Simulator. The model is a general purpose highly vectorized chemical compositional simulator that can simulate a wide range of displacement processes at both field and laboratory scales. The original simulator was modified to run on both distributed memory parallel machines (Intel iPSC/860 and Delta, Connection Machine 5, Kendall Square 1 and 2, and CRAY T3D) and a cluster of workstations. A domain decomposition approach has been taken towards parallelization of the code. A portion of the discrete reservoir model is assigned to each processor by a set-up routine that attempts a data layout as even as possible from the load-balance standpoint. Each of these subdomains is extended so that data can be shared between adjacent processors for stencil computation. The added routines that make parallel execution possible are written in a modular fashion that makes the porting to new parallel platforms straight forward. Results of the distributed memory computing performance of Parallel simulator are presented for field scale applications such as tracer flood and polymer flood. A comparison of the wall-clock times for same problems on a vector supercomputer is also presented.
A Closer Look at Drawbacks of Minimizing Weighted Sums of Objectives for Pareto Set Generation in Multicriteria Optimization Problems
(1996-12) Das, Indraneel; Dennis, J.E. Jr.
A standard technique for generating the Pareto set in multicriteria optimization problems is to minimize (convex) weighted sums of the different objectives for various different settings of the weights. However, it is well-known that this method succeeds in getting points from all parts of the Pareto set only when the Pareto curve is convex. This article provides a geometrical argument as to why this is the case. Secondly, it is a frequent observation that even for convex Pareto curves, an evenly distributed set of weights fails to produce an even distribution of points from all parts of the Pareto set. This article aims to identify the mechanism behind this occurrence. Roughly, the weight is related to the slope of the Pareto curve in the objective space in a way such that an even spread of Pareto points actually corresponds to often very uneven distributions of weights. Several examples are provided showing assumed shapes of Pareto curves and the distribution of weights corresponding to an even spread of points on those Pareto curves.
A Combinatorial Abstraction of One Shortest Path Problem and Its Relationship to Greedoids
(1988-05) Boyd, E. Andrew
A natural generalization of the shortest path problem to arbitrary set systems is presented that captures a number of interesting problems, including the usual graph-theoretic shortest path problem and the problem of finding a minimum weight set on a matroid. Necessary and sufficient conditions for the solution of this problem by the greedy algorithm are then investigated. In particular, it is noted that it is necessary but not sufficient for the underlying combinatorial structure to be a greedoid, and the three extremely diverse collections of sufficient conditions taken from the greedoid literature are presented.
A Combined Shape-Newton and Topology Optimization Technique in Real-Time Image Segmentation
(2004-07) Hintermüller, M.
In this paper, for solving a class of shape optimization problems, a new algorithmic concept combining shape and topological sensitivities is presented. The geometry of interest is represented by means of geometrical implicit functions. While the topology optimization phase is based on a gradient descent scheme, the phase based on shape sensitivity uses Newton-type descent flows. The algorithm is applied to edge detector based image segmentation problems, but it can be extended to solve more general shape optimization problems as well.
A Comparison of Three Total Variation Based Texture Extraction Models
(2007-01) Yin, Wotao; Goldfarb, Donald; Osher, Stanley
This paper qualitatively compares three recently proposed models for signal/image texture extraction based on total variation minimization:the Meyer, Vese-Osher, and TV-L1 models. We formulate discrete versions of these models as second-order cone programs (SOCPs) which can be solved efficiently by interior-point methods. Our experiments with these models on 1D oscillating signals and 2D images reveal their differences: the Meyer model tends to extract oscillation patterns in the input, the TV-L1 model performs a strict multiscale decomposition, and the Vese-Osher model has properties falling in between the other two models.
A Compressive Sensing and Unmixing Scheme for Hyperspectral Data Processing
(2011-01) Li, Chengbo; Sun, Ting; Kelly, Kevin; Zhang, Yin
Hyperspectral data processing typically demands enormous computational resources in terms of storage, computation and I/O throughputs, especially when real-time processing is desired. In this paper, we investigate a low-complexity scheme for hyperspectral data compression and reconstruction. In this scheme, compressed hyperspectral data are acquired directly by a device similar to the single-pixel camera based on the principle of compressive sensing. To decode the compressed data, we propose a numerical procedure to directly compute the unmixed abundance fractions of given endmembers, completely bypassing high-complexity tasks involving the hyperspectral data cube itself. The reconstruction model is to minimize the total variational of the abundance fractions subject to a pre-processed fidelity equation with a significantly reduced size, and other side constraints. An augmented Lagrangian type algorithm is developed to solve this model. We conduct extensive numerical experiments to demonstrate the feasibility and efficiency of the proposed approach, using both synthetic data and hardware-measured data. Experimental and computational evidence obtained from this study indicates that the proposed scheme has a high potential in real-world applications.
A Computational Note on Markov Decision Processes Without Discounting
(1987-07) Pfeiffer, Paul E.; Dennis, J.E. Jr.
The Markov decision process is treated in a variety of forms or cases: finite or infinite horizon, with or without discounting. The finite horizon cases and the case of infinite horizon with discounting have received considerable attention. In the infinite horizon case, with discounting, the problem either receives a linear programming treatment or is treated by the elegant and effective policy-iteration procedure by Ronald Howard. In the undiscounted case, however, a special form of this procedure is required, which detracts from the directness and elegance of the method. The difficulty comes in the step generally called the value-determination procedure. The equations used in this step are linearly dependent, so that the solution of the system of linear equations requires some adjustment. We propose a new computational procedure which avoids this difficulty and works directly with the average next-period gains and powers of the transition probability matrix. The fundamental computational tools are matrix multiplication and addition.
A Computational Study of a Gradient-Based Log-Barrier Algorithm for a Class of Large-Scale SDPs
(2001-06) Burer, Samuel; Monteiro, Renato D.C.; Zhang, Yin
The authors of this paper recently introduced a transformation that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods to handle efficiently. Based on the transformation, they proposed a globally convergent, first-order (i.e., gradient-based) log-barrier algorithm for solving a class of linear SDPs. In this paper, we discuss an efficient implementation of the proposed algorithm and report computational results on semidefinite relaxations of three types of combinatorial optimization problems. Our results demonstrate that the proposed algorithm is indeed capable of solving large-scale SDPs and is particularly effective for problems with a large number of constraints.
A Computational Study of Vehicle Routing Applications
(1999-05) Rich, Jennifer L.
This thesis examines three specific routing applications. In the first model, the scheduling of home health care providers from their homes, to a set of patients, and then back to their respective homes, is performed both heuristically and optimally for very small instances. The problem is complicated by the presence of multiple depots, time windows, and the scheduling of lunch breaks. It is shown that the problem can be formulated as a mixed integer programming problem and, in very small instances, solved to optimality with a branch-and-cut procedure. To obtain solutions for larger instances, though, a heuristic is shown to have more success. The second application considers the vehicle routing problem with time windows, or VRPTW. The vehicle routing problem involves finding a set of routes starting and ending at a single depot that together visit a set of customers. In the VRPTW there is an additional constraint requiring that each customer must be visited within a given time window. The best known solution procedures for solving the VRPTW use a set partitioning model with column generation. Within this framework, we present a new approach for generating valid inequalities, specifically k-path cuts, to improve the linear programming relaxation. Computational results are given for the standard library of test instances. In particular, the results include solutions for ten previously unsolved instances. The final application concerns the less-than-truckload, or LTL, trucking industry. An LTL carrier primarily handles shipments that are significantly smaller than the size of a tractor-trailer. Savings are achieved by consolidating shipments into loads at regional terminals and transporting these loads from terminal to terminal. The strategic load plan determines how to route the flow of consolidated loads from origin terminals to destination terminals cost effectively and allowing for certain service standards. To find good solutions to this problem, we apply a dual-ascent procedure to a related uncapacitated network design problem to obtain wcomputational results for three different companies.
A Consortium Proposal: "The Rice Inversion Project"
(1992-11) Symes, William W.
This document details a proposal for an industrially sponsored consortium for research in seismic inversion at Rice University. This consortium project will be directed by Professor William W. Symes in the Department of Computational and Applied Mathematics (formerly called The Department of Mathematical Sciences), George Brown School of Engineering. This project will develop novel approaches pioneered by Professor Symes to velocity and reflectivity estimation from waveform data, and will offer its sponsors both pilot software for state-of-the-art vector and parallel computing platforms, and a database of experience in waveform inversion. Participants will include graduate student assistants, postdoctoral research associates, and (whenever possible) short-and long-term visitors from the sponsoring organizations.
A Convergence Theory for a Class of Quasi-Newton Methods for Constrained Optimization
(1983-05) Fontecilla, Rodrigo; Steihaug, Trond; Tapia, Richard A.
In this paper we develop a general convergence theory for a class of quasi-Newton methods for equality constrained optimization. The theory is set in the framework of the diagonalized multiplier method defined by Tapia and is an extension of the theory developed by Glad. We believe that this framework is flexible and amenable to convergence analysis and generalizations. A key ingredient of a method in this class is a multiplier update. Our theory is tested by showing that a straightforward application gives the best known convergence results for several known multiplier updates. Also a characterization of q-superlinear convergence is presented. It is shown that in the special case when the diagonalized multiplier method is equivalent to the successive quadratic programming approach, our general characterization result gives the Boggs, Tolle and Wang characterization.
A Convergence Theory for the Structured BFGS Secant Method with an Application to Nonlinear Least Squares
(1987-05) Dennis, J.E. Jr.; Martinez, H.J.; Tapia, R.A.
In 1981, Dennis and Walker developed a convergence theory for structured secant methods which included the PSB and the DFP secant methods, but not the straightforward structured version of the BFGS secant method. Here we fill this gap in the theory by establishing a convergence theory for the structured BFGS secant method. A direct application of our new theory gives the first proof of local and q-superlinear convergence of the important structured BFGS secant method for the nonlinear least-squares problem which is used by Dennis, Gay and Welsh in the current version of the popular and successful NL2SOL code.
A Cubically Convergent Method for Locating a Nearby Vertex in Linear Programming
(1989-12) Tapia, R.A.; Zhang, Yin

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