A linear-time eigenvalue solver for finite-element-based analysis of large-scale wave propagation problems in on-chip interconnect structures

The problem of wave propagation along a transmission system composed of an overhead wire parallel to the (plane) surface of the earth, in spite of its great technical importance, does not appear to have been satisfactorily solved.1 While a complete solution of the actual problem is impossible, on account of the inequalities in the earth's surface and its lack of conductive homogeneity, the solution of the problem, where the actual earth is replaced by a plane homogeneous semi-infinite solid, is of considerable theoretical and practical interest. The solution of this problem is given in the present paper, together with formulas for calculating inductive disturbances in neighboring transmission systems.

Power flow in distribution networks with earth return

The next generation VLSI circuits will be designed with millions of densely packed interconnect segments on a single chip. Inductive effects between these segments begin to dominate signal delay as the clock frequency is increased. Modern parasitic extraction tools to estimate the onchip inductive effects with high accuracy have had limited impact due to large computational and storage requirements. This work describes a parallel software package for inductance extraction called ParIS, which is capable of analyzing interconnect configurations involving several conductors within reasonable time. The main component of the software is a novel preconditioned iterative method that is used to solve a dense complex linear system of equations. The linear system represents the inductive coupling between filaments that are used to discretize the conductors. A variant of the fast multipole method is used to compute dense matrix-vector products with the coefficient matrix. ParIS uses a two-tier parallel formulation that allows mixed mode parallelization using both MPIand OpenMP. An MPI process is associated with each conductor. The computation within a conductor is parallelized using OpenMP. The parallel efficiency and scalability of the software is demonstrated through experiments on the IBM p690 and Intel and AMD Linux clusters. These experiments highlight the portability and efficiency of the software on multiprocessors with shared, distributed, and distributed-shared memory architectures.

A meshless Chebyshev collocation method for eigenvalue problems of the Helmholtz equation

Binary classification refers to supervised techniques that split a set of points in two classes, with respect to a training set of points whose membership is known for each class. Binary classification plays a central role in the solution of many scientific, financial, engineering, medical and biological problems. Many methods with good classification accuracy are currently available. This work shows how a binary classification problem can be expressed in terms of a generalized eigenvalue problem. A new regularization technique is proposed, which gives results that are comparable to other techniques in use, in terms of classification accuracy. The advantage of this method relies in its lower computational complexity with respect to the existing techniques based on generalized eigenvalue problems. Finally, the method is compared with other methods using benchmark data sets.

A proposed approach is specialized for large scale vehicle routing problems(VRPs) and based on area segmentation and gradual area integration mechanisms so as to avoid combinatorial explosion. The purpose of the proposed approach is to deconstruct large scale problem into small size sub-problems and gradually restore these to original state. Firstly, an original large scale problem is divided into some small sub-areas and optimal solutions in each sub-area are derived. When a best incumbent solution remains unchanged for a certain period, subareas are gradually integrated and new optimal solutions in a new integrated sub-area are newly searched through use of the obtained solutions in previous sub area. This gradual integration and optimization are iterated until every sub-area are integrated into the one (the original problem), and the optimal solution of original problem can be obtained at this time. The proposed approach aims to deconstruct large scale problem into small size sub-problems and perform more efficient search. Through some typical test problems, it was demonstrated that our approach could derive better results more effectively than conventional approach.

This paper investigates the problem of joint base station selection and resource allocation in an orthogonal frequency division multiple access (OFDMA) heterogeneous cellular network. The original throughput maximization problem is NP-hard and we propose solving it by using game theoretic stochastic learning approaches. To this end, we first transform the original problem into a tractable form, which has a weighted utility function. Then we prove that an exact potential game applies and it exists the best Nash equilibria which is a near optimal solution of the original problem when an efficient solution method of the weights is employed. To obtain the optimal solution, we redesign the utility function by leveraging a state space to formulate the original problem into an ordinal state based potential game, which is proved that it exists a recurrent state equilibrium point that maximizes system throughput. Furthermore, we propose two different variants of Max-logit learning algorithm based on these two games respectively: one is a simultaneous learning algorithm with less information exchange, which achieves the best Nash equilibria point of the exact potential game and the other is an efficient learning algorithm for the ordinal state based potential game, which can converge to the global optimization solution. Finally, numerical results are given to validate that theoretical findings.

Feedback stabilization of a second-order system: A nonmodal approach

In this study, we transform an optimization problem into a game problem by multiple decision makers. This transformation is called gamification of the optimization problem. Then, we discuss correspondence of the dynamics to search the rational solution of the game problem with the multi-point type optimization method to search the optimal solution of the original problem. Specifically, we propose a condition to compose objective functions of multiple decision makers who interfere with each other from the objective function of the original problem so that equivalence of the Nash equilibrium solution of the game problem to the optimal solution of the original problem is guaranteed. Then, correspondence of the gradient dynamics to solve the game problem with the multi-point type optimization method is discussed. We also show some numerical examples in order to explain the gamification of the optimization problem.

Carbon nanotubes (CNTs) and carbon nanotube field effect transistors (CNFETs) have demonstrated extraordinary properties and are widely accepted as the building blocks of next generation VLSI circuits. However, no nanoelectronic architecture has been proposed which is solely based on carbon nanotubes and carbon nanotube field effect transistors. The reasons include lack of a self-assembly technology which could form complex carbon nanotube structures, or, absence of a reconfigurable carbon nanotube device which could provide functionality, reliability, and performance via reconfigurability. In this paper, I propose a novel double gate carbon nanotube field effect transistor (RDG-CNFET), which is reconfigurable to be open, short, FET, or via. Layers of orthogonal carbon nanotubes with electrically bistable molecules sandwiched at each crossing form a dense array of RDG-CNFETs and programmable interconnects, and constitute a nanoelectronic architecture of manufacturability (via regularity), reliability (via reconfigurability), and performance (via device density). Simulation based on CNFET and molecular device compact models demonstrates superior logic density, reliability, performance and power consumption of the proposed RDGCNFET based nanoelectronic circuits compared with the existing, e.g., molecular diode/MOSFET based nanoelectronic circuits.

Chapter 2 - The Method of Multipliers for Equality Constrained Problems

Subject.The canonical task of transport logistics in the distributed system "suppliers - consumers" is considered. Goal. Development of an accurate algorithm for solving this problem according to the probabilistic criterion in the assumption of the random nature of transportation costs has been done. Tasks. 1. Development of an exact method for solving the problem of finding a plan that minimizes the total cost of transportation in conditions when their costs are given by their distribution densities. 2. Development of a method for solving the problem when the distribution density of the cost of transportation is not known. Methods. A computational scheme for solving the problem is proposed, which is implemented by an iterative procedure for sequential improvement of the transportation plan. The convergence of this procedure is proved. In order to accelerate the convergence of the computational procedure to the exact solution, an alternative method is proposed based on the solution of a nontrivial problem of fractional nonlinear programming. The method reduces the original complex problem to solving a sequence of simpler problems. The original problem is supplemented by considering a situation that is important for practice when, in the conditions of a small sample of initial data, there is no possibility of obtaining adequate analytical descriptions for the distribution densities of the random costs of transportation. To solve the problem in this case, a minimax method is proposed for finding the best transportation plan in the most unfavorable situation, when the distribution densities of the random cost of transportation are the worst. To find such densities, the modern mathematical apparatus of continuous linear programming was used. Results. A mathematical model and a method for solving the problem of transport logistics in conditions of uncertainty of the initial data are proposed. The desired plan is achieved using the solution of the fractional nonlinear programming problem. Conclusions: The problem of forming a transportation plan is considered, provided that their costs are random values. Also, a method for solving the problem of optimization of transportation for a situation where the density of distribution of random cost cannot be correctly determined is considered.

In this article, we give conditions on parameters k, l that the generalized eigenvalue problem x″″ + kx″ + lx = λh(t)x, 0 < t < 1, x(0) = x(1) = x′(0) = x′(1) = 0 possesses an infinite number of simple positive eigenvalues { λ k } k = 1 ∞ and to each eigenvalue there corresponds an essential unique eigenfunction ψ k which has exactly k - 1 simple zeros in (0,1) and is positive near 0. It follows that we consider the fourth-order two-point boundary value problem x″″ + kx″ + lx = f(t,x), 0 < t < 1, x(0) = x(1) = x′(0) = x′(1) = 0, where f(t, x) ∈ C([0,1] × ℝ, ℝ) satisfies f(t, x)x > 0 for all x ≠ 0, t ∈ [0,1] and lim |x|→0 f(t,x)/x = a(t), lim |x|→+∞ f(t,x)/x = b(t) or lim x→-∞ f(t,x)/x = 0 and lim x→+∞ f(t,x)/x = c(t) for some a(t), b(t), c(t) ∈ C([0,1], (0,+∞)) and t ∈ [0,1]. Furthermore, we obtain the existence and multiplicity results of nodal solutions for the above problem. The proofs of our main results are based upon disconjugate operator theory and the global bifurcation techniques.MSC (2000): 34B15.

Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N3) with the size of the investigated problem, N (e.g. the electron count in electronic structure theory), and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on a few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the Eigenvalue soLvers for Petascale Applications (ELPA) library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as generalized eigenvalue problems, relying on the well documented matrix layout of the Scalable Linear Algebra PACKage (ScaLAPACK) library but replacing all actual parallel solution steps with subroutines of its own. For these steps, ELPA significantly outperforms the corresponding ScaLAPACK routines and proprietary libraries that implement the ScaLAPACK interface (e.g. Intel’s MKL). The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding backtransformation of the eigenvectors. ELPA offers both a one-step tridiagonalization (successive Householder transformations) and a two-step transformation that is more efficient especially towards larger matrices and larger numbers of CPU cores. ELPA is based on the MPI standard, with an early hybrid MPI-OpenMPI implementation available as well. Scalability beyond 10 000 CPU cores for problem sizes arising in the field of electronic structure theory is demonstrated for current high-performance computer architectures such as Cray or Intel/Infiniband. For a matrix of dimension 260 000, scalability up to 295 000 CPU cores has been shown on BlueGene/P.

Dielectric waveguide formulations usually result in generalized eigenvalue problems which are non-Hermitian in nature. We solve this class of problems by a two stage process. In the first stage, a standard lumped eigenvalue problem is solved for a few eigenpairs of interest (usually the guided modes) by Arnoldi's method accelerated by Chebyshev polynomials. The solution is then refined for the actual unlumped generalised eigenvalue problem by using inflated inverse iteration. After the eigenvalues and eigenvectors have been calculated at the first point (preferably the highest frequency), the eigenpairs for the rest of the frequencies of interest are calculated for progressively smaller frequencies by inflated inverse iteration alone, using the eigenvectors of the last step as the initial guess. To demonstrate the efficacy of this method, we use a variational formulation for anisotropic, dielectric waveguides based only on the E/sub s/ components or only on the H/sub s/ components of the electromagnetic fields that was presented by Chew and Nasir (1989). In this formulation it was shown that due of the imposition of the divergence condition the spurious waveguide modes were eliminated.