Given a power network consisting of nodes (generators/loads) and edges (lines), there exist a set of constraints that must be satisfied in order for the system to be operational. When one or more power lines are cut, the bus phases and load/generator power values may need to be altered in order to restore the system to operation. The load shedding problem is to find the smallest adjustment to the loads that achieves this restoration. In this work, we show how to solve this nonlinear optimization problem by solving a sequence of linear programs. On random graphs with 1500 edges, we find that our method is at least 40 times faster than competing nonlinear optimization methods. We show that our method is capable of solving load shedding problems for a real graph with 19840 edges, and that the method scales with an O(n2) running time where n is the number of edges. For real subgraphs with hundreds of edges, we are able to rapidly solve all possible load shedding problems in which at most two lines are deleted. This work takes a first step towards a scalable load shedding algorithm capable of handling large networks that will arise in the future.

We describe an efficient implementation of an algorithm for computing selected elements of a general sparse symmetric matrix A that can be decomposed as A = LDLT , where L is lower triangular and D is diagonal. Our implementation, which is called SelInv , is built on top of an efficient supernodal left-looking LDLT factorization of A . We discuss how computational efficiency can be gained by making use of a relative index array to handle indirect addressing. We report the performance of SelInv on a collection of sparse matrices of various sizes and nonzero structures. We also demonstrate how SelInv can be used in electronic structure calculations.

We describe the design and implementation of KSSOLV, a MATLAB toolbox for solving a class of nonlinear eigenvalue problems known as the Kohn-Sham equations . These types of problems arise in electronic structure calculations, which are nowadays essential for studying the microscopic quantum mechanical properties of molecules, solids, and other nanoscale materials. KSSOLV is well suited for developing new algorithms for solving the Kohn-Sham equations and is designed to enable researchers in computational and applied mathematics to investigate the convergence properties of the existing algorithms. The toolbox makes use of the object-oriented programming features available in MATLAB so that the process of setting up a physical system is straightforward and the amount of coding effort required to prototype, test, and compare new algorithms is significantly reduced. All of these features should also make this package attractive to other computational scientists and students who wish to study small- to medium-size systems.

We present a new linearly scaling three-dimensional fragment (LS3DF) method for large scale ab initio electronic structure calculations. LS3DF is based on a divide-and-conquer approach, which incorporates a novel patching scheme that effectively cancels out the artificial boundary effects due to the subdivision of the system. As a consequence, the LS3DF program yields essentially the same results as direct density functional theory (DFT) calculations. The fragments of the LS3DF algorithm can be calculated separately with different groups of processors. This leads to almost perfect parallelization on over one hundred thousand processors. After code optimization, we were able to achieve 60.3 Tflop/s, which is 23.4% of the theoretical peak speed on 30,720 Cray XT4 processor cores. In a separate run on a BlueGene/P system, we achieved 107.5 Tflop/s on 131,072 cores, or 24.2% of peak. Our 13,824-atom ZnTeO alloy calculation runs 400 times faster than a direct DFT calculation, even presuming that the direct DFT calculation can scale well up to 17,280 processor cores. These results demonstrate the applicability of the LS3DF method to material simulations, the advantage of using linearly scaling algorithms over conventional O(N3) methods, and the potential for petascale computation using the LS3DF method.

Object-oriented programming is a relatively new tool in the development of optimization software. The code extensibility and the rapid algorithm prototyping capability enabled by this programming paradigm promise to enhance the reliability, utility, and ease of use of optimization software. While the use of object-oriented programming is growing, there are still few examples of general purpose codes written in this manner, and a common approach is far from obvious. This paper describes OPT++, a C++ class library for nonlinear optimization. The design is predicated on the concept of distinguishing between an algorithm-independent class hierarchy for nonlinear optimization problems and a class hierarchy for nonlinear optimization methods that is based on common algorithmic traits. The interface is designed for ease of use while being general enough so that new optimization algorithms can be added easily to the existing framework. A number of nonlinear optimization algorithms have been implemented in OPT++ and are accessible through this interface. Furthermore, example applications demonstrate the simplicity of the interface as well as the advantages of a common interface in comparing multiple algorithms.