ACM Trans Math Softw Research Articles

The effect of temperature-dependent thermal conductivity on the geothermal structure of the Sydney Basin

The thermal structure of continental crust is a critical factor for geothermal exploration, hydrocarbon maturation and crustal strength, and yet our understanding of it is limited by our incomplete knowledge of its geological structure and thermal properties such as hydrogeologic and thermo-mechanical feedbacks that come into play. One of the most critical parameters in modelling upper crustal temperature is thermal conductivity, which itself exhibits strong temperature dependence. In this study, we integrate new laboratory measurements of the thermal conductivity of Sydney Basin rocks under varying temperatures, with finite-element geothermal models of the Sydney Basin using deal.II (Bangerth et al. in ACM Trans Math Softw (TOMS) 33:24, 2007). Basin geometry and structure are adapted from Danis et al. (Aust J Earth Sci 58:517–42, 2011), which quantified the extent of Triassic sediment, Permian coal measures, Carboniferous volcanics and thickness of the crystalline crust. We find that temperature-dependent thermal conductivity results in lower lateral variations in temperature compared to constant thermal conductivity models. However, the average temperatures at depth are significantly higher when temperature-dependent thermal conductivity effects are included. A number of regions within the Sydney Basin demonstrate temperatures above 150 °C at depths of less than 2000 m in these models, for instance NW of Singleton, exhibits a strong thermal anomaly, demonstrating the potential for geothermal prospectivity of the region from experimentally constrained thermal parameters.

On the nonmonotonicity degree of nonmonotone line searches

The nonmonotone globalization technique is useful in difficult nonlinear problems, because of the fact that it may help escaping from steep sided valleys and may improve both the possibility of finding the global optimum and the rate of convergence. This paper discusses the nonmonotonicity degree of nonmonotone line searches for the unconstrained optimization. Specifically, we analyze some popular nonmonotone line search methods and explore, from a computational point of view, the relations between the efficiency of a nonmonotone line search and its nonmonotonicity degree. We attempt to answer this question how to control the degree of the nonmonotonicity of line search rules in order to reach a more efficient algorithm. Hence in an attempt to control the nonmonotonicity degree, two adaptive nonmonotone rules based on the morphology of the objective function are proposed. The global convergence and the convergence rate of the proposed methods are analysed under mild assumptions. Numerical experiments are made on a set of unconstrained optimization test problems of the CUTEr (Gould et al. in ACM Trans Math Softw 29:373–394, 2003) collection. The performance data are first analysed through the performance profile of Dolan and More (Math Program 91:201–213, 2002). In the second kind of analyse, the performance data are analysed in terms of increasing dimension of the test problems.

On the improvement of a scalable sparse direct solver for unsymmetrical linear equations

This paper focuses on the application level improvements in a sparse direct solver specifically used for large-scale unsymmetrical linear equations resulting from unstructured mesh discretization of coupled elliptic/hyperbolic PDEs. Existing sparse direct solvers are designed for distributed server systems taking advantage of both distributed memory and processing units. We conducted extensive numerical experiments with three state-of-the-art direct linear solvers that can work on distributed-memory parallel architectures; namely, MUMPS (MUMPS solver website, http://graal.ens-lyon.fr/MUMPS), WSMP (Technical Report TR RC-21886, IBM, Watson Research Center, Yorktown Heights, 2000), and SUPERLU_DIST (ACM Trans Math Softw 29(2):110---140, 2003). The performance of these solvers was analyzed in detail, using advanced analysis tools such as Tuning and Analysis Utilities (TAU) and Performance Application Programming Interface (PAPI). The performance is evaluated with respect to robustness, speed, scalability, and efficiency in CPU and memory usage. We have determined application level issues that we believe they can improve the performance of a distributed-shared memory hybrid variant of this solver, which is proposed as an alternative solver [SuperLU_MCDT (Many-Core Distributed)] in this paper. The new solver utilizing the MPI/OpenMP hybrid programming is specifically tuned to handle large unsymmetrical systems arising in reservoir simulations so that higher performance and better scalability can be achieved for a large distributed computing system with many nodes of multicore processors. Two main tasks are accomplished during this study: (i) comparisons of public domain solver algorithms; existing state-of-the-art direct sparse linear system solvers are investigated and their performance and weaknesses based on test cases are analyzed, (ii) improvement of direct sparse solver algorithm (SuperLU_MCDT) for many-core distributed systems is achieved. We provided results of numerical tests that were run on up to 16,384 cores, and used many sets of test matrices for reservoir simulations with unstructured meshes. The numerical results showed that SuperLU_MCDT can outperform SuperLU_DIST 3.3 in terms of both speed and robustness.

Local convex hulls for a special class of integer multicommodity flow problems.

Based on previous work in rolling stock scheduling problems (Alfieri et al. in Transp Sci 40:378–391, 2006; Cacchiani et al. in Math Progr B 124:207–231, 2010; Lin and Kwan in Electron Notes Discret Math 41:165–172, 2013; Schrijver in CWI Q 6:205–217, 1993; Ziarati et al. in Manag Sci 45:1156–1168, 1999), we generalize a local convex hull method for a class of integer multicommodity flow problems, and discuss its feasibility range in high dimensional cases. Suppose a local convex hull can be divided into an up hull, a main hull and a down hull if certain conditions are met, it is shown theoretically that the main hull can only have at most two nonzero facets. The numbers of points in the up and down hull are explored mainly on an empirical basis. The above properties of local convex hulls have led to a slightly modified QuickHull algorithm (the “2-facet QuickHull”) based on the original version proposed by Barber et al. (ACM Trans Math Softw 22:469–483, 1996). As for the feasibility in applying this method to rolling stock scheduling, our empirical experiments show that for the problem instances of ScotRail and Southern Railway, two major train operating companies in the UK, even in the most difficult real-world or artificial conditions (e.g. supposing a train can be served by any of 11 compatible types of self-powered unit), the standard QuickHull (Barber et al. in ACM Trans Math Softw 22:469–483, 1996) can easily compute the relevant convex hulls. For some even more difficult artificial instances that may fall outside the scope of rolling stock scheduling (e.g. a node in a graph can be covered by more than 11 kinds of compatible commodities), there is evidence showing that the “2-facet QuickHull” can be more advantageous over the standard QuickHull for our tested instances. When the number of commodity types is even higher (e.g. >19), or the number of points in a high dimensional space (e.g. 15 dimensions) is not small (e.g. >2000), the local convex hulls cannot be computed either by the standard or the 2-facet QuickHull methods within practical time.

Semidefinite programming relaxation methods for global optimization problems with sparse polynomials and unbounded semialgebraic feasible sets

We propose a hierarchy of semidefinite programming (SDP) relaxations for polynomial optimization with sparse patterns over unbounded feasible sets. The convergence of the proposed SDP hierarchy is established for a class of polynomial optimization problems. This is done by employing known sums-of-squares sparsity techniques of Kojima and Muramatsu Comput Optim Appl 42(1):31---41, (2009) and Lasserre SIAM J Optim 17:822---843, (2006) together with a representation theorem for polynomials over unbounded sets obtained recently in Jeyakumar et al. J Optim Theory Appl 163(3):707---718, (2014). We demonstrate that the proposed sparse SDP hierarchy can solve some classes of large scale polynomial optimization problems with unbounded feasible sets using the polynomial optimization solver SparsePOP developed by Waki et al. ACM Trans Math Softw 35:15 (2008).

Modelling and numerical investigations of the mechanical behavior of polyurethane under the influence of moisture

In the present paper, a viscoelastic elastomer is investigated with respect to the influence of moisture on its mechanical behavior. Uniaxial relaxation tests are executed at room temperature with different levels of moisture content. The model consists of a finite viscoelastic and incompressible approach wherein the mechanical parameters are coupled to the moisture content. For numerical investigations, all necessary balance and constitutive equations are implemented in the open-source C++ finite element code deal.II (Bangerth et al. in ACM Trans Math Softw 33(4):24/1–24/27 2007). With the help of this implementation and by comparing experimental results for dry, intermediate and fully saturated specimens and the results gained from simulations, the precise material parameters of the used polymer can be identified.

Modified nonlinear conjugate gradient method with sufficient descent condition for unconstrained optimization

In this paper, an efficient modified nonlinear conjugate gradient method for solving unconstrained optimization problems is proposed. An attractive property of the modified method is that the generated direction in each step is always descending without any line search. The global convergence result of the modified method is established under the general Wolfe line search condition. Numerical results show that the modified method is efficient and stationary by comparing with the wellknown Polak-Ribiere-Polyak method, CG-DESCENT method and DSP-CG method using the unconstrained optimization problems from More and Garbow (ACM Trans Math Softw 7, 17-41, 1981), so it can be widely used in scientific computation. Mathematics Subject Classification (2010) 90C26 · 65H10

How to Maximize the Likelihood Function for a DSGE Model

This paper extends two optimization routines to deal with objective functions for DSGE models. The optimization routines are (1) a version of Simulated Annealing developed by Corana A, Marchesi M, Ridella (ACM Trans Math Softw 13(3):262---280, 1987), and (2) the evolutionary algorithm CMA-ES developed by Hansen, Muller, Koumoutsakos (Evol Comput 11(1), 2003). Following these extensions, we examine the ability of the two routines to maximize the likelihood function for a sequence of test economies. Our results show that the CMA-ES routine clearly outperforms Simulated Annealing in its ability to find the global optimum and in efficiency. With ten unknown structural parameters in the likelihood function, the CMA-ES routine finds the global optimum in 95% of our test economies compared to 89% for Simulated Annealing. When the number of unknown structural parameters in the likelihood function increases to 20 and 35, then the CMA-ES routine finds the global optimum in 85 and 71% of our test economies, respectively. The corresponding numbers for Simulated Annealing are 70 and 0%.

HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method

HOM4PS-2.0 is a software package in FORTRAN 90 which implements the polyhedral homotopy continuation method for solving polynomial systems. It updates its original version HOM4PS in three key aspects: (1) new method for finding mixed cells, (2) combining the polyhedral and linear homotopies in one step, (3) new way of dealing with curve jumping. Numerical results show that this revision leads to a spectacular speed-up, ranging up to 1950s, over its original version on all benchmark systems, especially for large ones. It surpasses the existing packages in finding isolated zeros, such as PHCpack (Verschelde in ACM Trans Math Softw 25:251–276, 1999), PHoM (Gunji et al. in Computing 73:57–77, 2004), and Bertini (Bates et al. in Software for numerical algebraic geometry. Available at http://www.nd.edu/~sommese/bertini), in speed by big margins.

Advanced service trading for scientific computing over the grid

One of the great benefits of computational grids is to give access to a wide range of scientific software and computers with different architectures. It is then possible to use a huge variety of tools for solving the same problem and even to combine these tools in order to obtain the best solution. Grid service trading (searching for the best combination of software and execution platform according to the user requirements) is thus a crucial issue. Trading relies on the description of available services and computers, on the current state of the grid, and on the user requirements. Given the large amount of services that may be deployed over the grid, this description cannot be reduced to a simple service name. A sophisticated service specification approach similar to algebraic data type is presented in this paper. Services are described in terms of their algebraic and semantic properties. This is nothing else than proceeding to a description of algorithms and objects properties for a given application domain. We then illustrate how this specification can be used to determine the service or the combination of services that best answer a user request. As a major benefit, users are not required to explicitly call grid-services, but instead manipulate high-level domain-specific expressions. Our approach is fully generic and can be used in almost all application domains. We illustrate this approach and its possible limitations within the framework of dense linear algebra. More precisely, we focus on Level 3 BLAS (ACM Trans Math Softw 16:1---17, 1990; ibid 16:18---28, 1990) and LAPACK (Society for Industrial and Applied Mathematics, Philadelphia, 1999). Some examples in nonlinear optimization are also given to demonstrate how generic our approach is and report on experiments where both domains interact to show the multi-domain possibilities.