Brute-force Computation Research Articles

A brute force calculation of the general QFT diagram

A brute force calculation of the general QFT diagram

Exact Output Rate of Generalized Peres Algorithm for Generating Random Bits from Loaded Dice

We report a computation of the exact output rate of recently-discovered generalization of Peres algorithm for generating random bits from loaded dice. Instead of resorting to brute-force computation for all possible inputs, which becomes quickly impractical as the input size increases, we compute the total output length on equiprobable sets of inputs by dynamic programming using a recursive formula.

Computing many-body wave functions with guaranteed precision: The first-order Møller-Plesset wave function for the ground state of helium atom

We present an approach to compute accurate correlation energies for atoms and molecules using an adaptive discontinuous spectral-element multiresolution representation for the two-electron wave function. Because of the exponential storage complexity of the spectral-element representation with the number of dimensions, a brute-force computation of two-electron (six-dimensional) wave functions with high precision was not practical. To overcome the key storage bottlenecks we utilized (1) a low-rank tensor approximation (specifically, the singular value decomposition) to compress the wave function, and (2) explicitly correlated R12-type terms in the wave function to regularize the Coulomb electron-electron singularities of the Hamiltonian. All operations necessary to solve the Schrödinger equation were expressed so that the reconstruction of the full-rank form of the wave function is never necessary. Numerical performance of the method was highlighted by computing the first-order Møller-Plesset wave function of a helium atom. The computed second-order Møller-Plesset energy is precise to ~2 microhartrees, which is at the precision limit of the existing general atomic-orbital-based approaches. Our approach does not assume special geometric symmetries, hence application to molecules is straightforward.

Ultrasmall metal-insulator-metal nanoresonators: impact of slow-wave effects on the quality factor

We study the quality factor variation of three-dimensional Metal-Insulator-Metal nanoresonators when their volume is shrunk from the diffraction limit (λ/2n)3 down to a deep subwavelength scale (λ/50)3. In addition to rigorous fully-vectorial calculations, we provide a semi-analytical expression of the quality factor Q obtained with a Fabry-Perot model. The latter quantitatively predicts the absorption and radiation losses of the nanoresonator and provides an in-depth understanding of the mode lifetime that cannot be obtained with brute-force computations. In particular, it highlights the impact of slow-wave effects on the Q-factor as the size of the resonator is decreased. The Fabry-Perot model also evidences that, unexpectedly, wave retardation effects are present in metallic nanoparticles, even for deep subwavelength dimensions in the quasi-static regime.

Halo clustering and gNL-type primordial non-gaussianity

A wide range of multifield inflationary models generate non-Gaussian initial conditionsin which the initial adiabatic fluctuation is of the form (ζG+gNLζG3).We study halo clustering in these models using two different analytic methods: the peak-backgroundsplit framework, and brute force calculation in a barrier crossing model, obtaining agreementbetween the two.We find a simple, theoretically motivated expression for halo bias which agreeswith N-body simulations and can be used to constrain gNL from observations. We discuss practical caveats to constraining gNL using only observable properties of atracer population, and argue that constraints obtained from populations whose observed biasis ≲2.5 are generally not robust to uncertainties in modeling the halo occupationdistribution of the population.

Recognizing Human Actions by Learning and Matching Shape-Motion Prototype Trees

A shape-motion prototype-based approach is introduced for action recognition. The approach represents an action as a sequence of prototypes for efficient and flexible action matching in long video sequences. During training, an action prototype tree is learned in a joint shape and motion space via hierarchical K-means clustering and each training sequence is represented as a labeled prototype sequence; then a look-up table of prototype-to-prototype distances is generated. During testing, based on a joint probability model of the actor location and action prototype, the actor is tracked while a frame-to-prototype correspondence is established by maximizing the joint probability, which is efficiently performed by searching the learned prototype tree; then actions are recognized using dynamic prototype sequence matching. Distance measures used for sequence matching are rapidly obtained by look-up table indexing, which is an order of magnitude faster than brute-force computation of frame-to-frame distances. Our approach enables robust action matching in challenging situations (such as moving cameras, dynamic backgrounds) and allows automatic alignment of action sequences. Experimental results demonstrate that our approach achieves recognition rates of 92.86 percent on a large gesture data set (with dynamic backgrounds), 100 percent on the Weizmann action data set, 95.77 percent on the KTH action data set, 88 percent on the UCF sports data set, and 87.27 percent on the CMU action data set.

Study of the asymptotic Cramér–Rao Bound for the COLD uniform linear array

In this paper, we study the Cocentered Orthogonal Loop and Dipole pairs Uniform Linear Array (COLD-ULA) which is sensitive to the source polarization in the context of the localization of time-varying narrow-band far-field sources. We derive and analyze nonmatrix expressions of the deterministic Cramér–Rao Bound (CRB(COLD)) for the direction and the polarization parameters under the assumption that all the sources are lying in the azimuthal plane. We denote this bound by ACRB(COLD), where the “A” stands for Asymptotic, meaning that the presented results are derived under the assumption that the number of sensors is sufficiently large. While, to our knowledge, closed-form (nonmatrix) expressions of the CRB(COLD) for multiple time-varying polarized sources signal do not exist in the literature, we show that the ACRB(COLD) takes a closed-form (nonmatrix) expression in this context and is a good approximation of the CRB(COLD) even if the number of sensor is moderate (about ten), if the source signals are not spatially too close. Our approach has two important advantages: (i) the computational complexity of the proposed closed-form of the bound is very low, compared to the brute force computation of a matrix-based deterministic CRB in case of time-varying model parameters and (ii) useful informations can be deduced from the closed-form expression on the behavior of the bound. In particular, we prove that the ACRB(COLD) for the direction parameter is not affected by the knowledge or the lack of it concerning the polarization parameters. Another conclusion is that with a COLD-ULA, more model parameters can be estimated than for the uniformly polarized ULA without degrading the estimation accuracy of the localization parameter. Finally, we also study the ACRB(COLD) for a priori known complex amplitudes.

Anomalous thermal transport in disordered harmonic chains and carbon nanotubes

We report the coherent potential approximation method of treating quantum thermal transport properties of nanoscale systems with mass disorder. Instead of massive efforts required in brute-force calculations, configuration averaging of disordered systems can be efficiently handled in a self-consistent manner by setting up the phonon version of nonequilibrium vertex correction theory. The accuracy of the method is verified by comparing with the exact results and Monte Carlo experiments in one-dimensional atomic chains. Results obtained for disordered harmonic chains and carbon nanotubes provide evidence of anomalous thermal transport in such systems. We also observe crossover in the transport where phonon scattering by disorder becomes important. Our results show that disorder plays a role in thermal conductance reduction.

Coarse-Grained (Multiscale) Simulations in Studies of Biophysical and Chemical Systems

Recent years have witnessed an explosion in computational power, leading to attempts to model ever more complex systems. Nevertheless, there remain cases for which the use of brute-force computer simulations is clearly not the solution. In such cases, great benefit can be obtained from the use of physically sound simplifications. The introduction of such coarse graining can be traced back to the early usage of a simplified model in studies of proteins. Since then, the field has progressed tremendously. In this review, we cover both key developments in the field and potential future directions. Additionally, particular emphasis is given to two general approaches, namely the renormalization and reference potential approaches, which allow one to move back and forth between the coarse-grained (CG) and full models, as these approaches provide the foundation for CG modeling of complex systems.

A nonperturbative approximation for the moderate Reynolds number Navier–Stokes equations

The nonlinearity of the Navier-Stokes equations makes predicting the flow of fluid around rapidly moving small bodies highly resistant to all approaches save careful experiments or brute force computation. Here, we show how a linearization of the Navier-Stokes equations captures the drag-determining features of the flow and allows simplified or analytical computation of the drag on bodies up to Reynolds number of order 100. We illustrate the utility of this linearization in 2 practical problems that normally can only be tackled with sophisticated numerical methods: understanding flow separation in the flow around a bluff body and finding drag-minimizing shapes.

Systematic Coarse Graining: “Four Lessons and A Caveat” from Nonequilibrium Statistical Mechanics

AbstractWith the guidance offered by nonequilibrium statistical thermodynamics, simulation techniques are elevated from brute-force computer experiments to systematic tools for extracting complete, redundancy-free, and consistent coarse-grained information for dynamic systems. We sketch the role and potential of Monte Carlo, molecular dynamics, and Brownian dynamics simulations in the thermodynamic approach to coarse graining. A melt of entangled linear polyethylene molecules serves us as an illustrative example.

PDE and Materials

This meeting brought together a carefully selected group of mathematicians, physicists, material scientists and engineers to discuss and review the impact of modern mathematical methods on the understanding of material behaviour and the design of new materials. A key issue is that the material behaviour is determined by physical processes at many different spatial and temporal scales which cannot be simulataneously resolved by brute force computation. There are a number of heuristic approaches (such as the quasicontinuum method or accelerated dynamics) to address the multiscale problem, but a mathematical theory which allows one to understand when and it what sense these approaches provide a good approximation to the full problem is still in its infancy. Central themes of the meeting were (spatial) microstructures, the relation between atomistic and continuum models, the treatment of multiple temporal scales, and the influence of randomness and the use of methods from statistical mechanics. The recent theory of microstructures in crystalline materials which undergo solid-solid phase transitions has lead to a very inspiring bridge between materials science and the mathematical analysis of regularity and of oscillations in nonlinear partial differential equations, in particular the theory of compensated compactness by F. Murat and L. Tartar. At the meeting L. Székelyhidi presented a recent breakthrough in this area, a characterization of all compact sets in the space of 2 \times 2 matrices which support a nontrivial microstructure. Other important contributions include new regularity results by C. Melcher and the use of new rigidity estimates and topological constructions to derive theories of lower-dimensional elastic objects (G. Dolzmann). Regularizations, e.g., due to interfacial energies or strain-gradient terms in elasticity or exchange energy terms in magnetic materials, prevent the formation of the mathematically idealized infinitely fine microstructures and break the dilation invariance of the problem by introducing a length scale. Interestingly this single additional scale can lead to the formation of complex patterns on a multitude of scales (‘domain branching’) and S. Conti presented interesting new results for type-I superconductors, based on subtle interpolation inequalities. In a different direction J.M. Robbins showed how the control of topological singularities through the shape of the container might lead to new switching mechanics for liquid crystals. E. Salje presented new experimental data and theoretical analyses on the interaction between mechanical microstructure and chemical behaviour and E. Quandt showed how the evolution of domain structures in magnetostrictive devices influences the performance of engineering devices. The connection between atomistic and continuum models is mathematically at its very beginning, not the least because the powerful theory of pde and the calculus of variations at the continuum level has so far no equally powerful counterpart for discrete systems. C. Ortner presented first results from his PhD thesis for a one-dimensional numerical analysis of the quasicontinuum method, a numerical scheme which tries to switch adaptively between continuum and atomistic models. G. Friesecke showed how a rigorous analysis of screening effects leads to a linear scaling of the energy with the number of atoms (‘stability of matter’). F. Theil presented a new approach to derive the Boltzmann equation from a discrete model of interacting atoms (without recollisions). This was closely related to an inspiring lecture by O. Penrose on the relation between microscopic reversibility and macroscopic irreversibility, which sparked a very lively discussion. More generally new rigorous results in statistical mechanics (the oldest multiscale method) played a key role at the meeting, notably in the talks by E. Presutti and R. Kotecký. S. Luckhaus presented a new model Hamiltonian that can serve as a starting point for a statistical mechanics of solids, including a description of defects, such as dislocations or grain boundaries. The influence of randonmess and heterogeneity on the behaviour at macroscopical scales was analyzed in the talks by N. Dirr, G. Menon, B. Niethammer and A. Planes. The strong separation of time scales is one of the major bottlenecks in the use of molecular dynamics simulations. Indeed often the processes which are most interesting from the point of view of applications in chemistry, physics and biology occur at times scales which are well beyond any MD simulation with realistic potentials. The organized a special session to contrast some of the current approaches in the mathematics, physics and computational communities (C. Schütte, E. Vanden-Eijnden, A. Voter). This was complemented by A. Mielke's rigorous analysis of two-scale Hamiltonian systems and M. Ortiz new approach to microstructure evolution by a global minimization principle in space-time. In addition to the lectures the very special environment of Oberwolfach provided an excellent opportunity for informal discussions, allowing for an open exchange of ideas between people which usually attend separate groups of conferences.

Fast Estimator of Primordial Non‐Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background

Measurements of primordial non-Gaussianity (fNL) open a new window onto the physics of inflation. We describe a fast cubic (bispectrum) estimator of fNL, using a combined analysis of temperature and polarization observations. The speed of our estimator allows us to use a sufficient number of Monte Carlo simulations to characterize its statistical properties in the presence of real-world issues such as instrumental effects, partial sky coverage, and foreground contamination. We find that our estimator is optimal, where optimality is defined by saturation of the Cramer-Rao bound, if noise is homogeneous. Our estimator is also computationally efficient, scaling as O(N3/2), compared to the O(N5/2) scaling of the brute-force bispectrum calculation for sky maps with N pixels. For Planck this translates into a speedup by factors of millions, reducing the required computing time from thousands of years to just hours and thus making fNL estimation feasible for future surveys.

A New Look at the Local Solvability Condition of Inhomogeneous Ordinary Differential Equations

Let P = P (z, d/dz) be a linear ordinary differential operator with holomorphic coefficients defined on a neighbourhood X of the origin O in C. In this paper, we consider the local solvability condition to the inhomogeneous equations P (z, d/dz)u(z) = f(z) in the space OX,O of formal power series at O. It is known that straightforward method based on inditial polynomial computation provides only an unsatisfactory answer to this problem, because it does not reveal structure of the equation itself and as a result, for some cases, brute force computations are needed in this direct approach. We propose in this paper an alternative approach with an intention to establish a new effective method to determine the local solvability conditions. For this purpose, we adopt a duality method used by H. Komatsu ([7, 8]) and by H. Komatsu and T. Kawai ([9]) in the study of index theorems and hyperfunction solutions of an ordinary differential equation. We develop a complex variable version of the duality method to show that the local solvability condition can be written in terms of residues. Then upon using the concept of local cohomology and the theory of D-modules of one variable, we derive, in a constructive manner, a regular singular holonomic system of ordinary differential equations supported at the origin that completely describes the local solvability condition.

Application of Artificial Neural Networks to Broadband Antenna Design Based on a Parametric Frequency Model

An artificial neural network (ANN) is proposed to predict the input impedance of a broadband antenna as a function of its geometric parameters. The input resistance of the antenna is first parameterized by a Gaussian model, and the ANN is constructed to approximate the nonlinear relationship between the antenna geometry and the model parameters. Introducing the model simplifies the ANN and decreases the training time. The reactance of the antenna is then constructed by the Hilbert transform from the resistance found by the neuromodel. A hybrid gradient descent and particle swarm optimization method is used to train the neural network. As an example, an ANN is constructed for a loop antenna with three tuning arms. The antenna structure is then optimized for broadband operation via a genetic algorithm that uses input impedance estimates provided by the trained ANN in place of brute-force electromagnetic computations. It is found that the required number of electromagnetic computations in training the ANN is ten times lower than that needed during the antenna optimization process, resulting in significant time savings

PERFECT STOCHASTIC SUMMATION IN HIGH ORDER FEYNMAN GRAPH EXPANSIONS

We describe a new application of an existing perfect sampling technique of Corcoran and Tweedie to estimate the self energy of an interacting Fermion model via Monte Carlo summation. Simulations suggest that the algorithm in this context converges extremely rapidly and results compare favorably to true values obtained by brute force computations for low dimensional toy problems. A variant of the perfect sampling scheme which improves the accuracy of the Monte Carlo sum for small samples is also given.

Two-Higgs-doublet model from the group-theoretic perspective

In the two-Higgs-doublet model, different Higgs doublets can be viewed as components of a generic “hyperspinor”. We decompose the Higgs potential of this model into irreducible representations of the SU(2) group of transformations of this hyperspinor. We discuss invariant combinations of Higgs potential parameters λi that arise in this decomposition and provide simple and concise sets of conditions for the hidden Z2-symmetry, Peccei–Quinn symmetry, and explicit CP-conservation in 2HDM. We show that some results obtained previously by brute-force calculations are reduced to simple linear algebraic statements in our approach.

Efficient Computation of the Hidden Markov Model Entropy for a Given Observation Sequence

Hidden Markov models (HMMs) are currently employed in a wide variety of applications, including speech recognition, target tracking, and protein sequence analysis. The Viterbi algorithm is perhaps the best known method for tracking the hidden states of a process from a sequence of observations. An important problem when tracking a process with an HMM is estimating the uncertainty present in the solution. In this correspondence, an algorithm for computing at runtime the entropy of the possible hidden state sequences that may have produced a certain sequence of observations is introduced. The brute-force computation of this quantity requires a number of calculations exponential in the length of the observation sequence. This algorithm, however, is based on a trellis structure resembling that of the Viterbi algorithm, and permits the efficient computation of the entropy with a complexity linear in the number of observations.

A Discrete-Time Approach to Analyze Hard Handoff Performance in Cellular Networks

The handoff algorithm employed in a cellular network has a significant impact on overall network performance, but evaluation of handoff performance has for the most part been done using only crude approximations or brute-force computer simulation. We introduce a new discrete-time approach to analyze the performance of handoff algorithms based on pilot signal strength measurements. We derive exact analytical expressions and develop a recursive numerical procedure to evaluate handoff performance metrics for a mobile station moving along a straight-line trajectory in a cellular network employing hard handoff. The numerical procedure provides a computational solution to a level-crossing problem in discrete time. Our discrete-time approach provides valuable analytical insight into the performance impact of handoff algorithms. Moreover, our numerical procedure for discrete-time handoff analysis provides an accurate and efficient tool for the design and dimensioning of high-performance handoff algorithms. The accuracy of the numerical procedure is validated by simulation.

Efficient evaluation of macro‐basis‐function reaction terms in the subdomain multilevel approach

AbstractThis paper presents an improved subdomain multilevel approach (SMA), a technique efficiently used for solving large complex‐shaped antennas printed in multilayered media structure. In a large electromagnetic problem, where the memory occupation and computational time have been already significantly reduced using the SMA, the calculation of the macro basis function (MBF) reaction terms remains the most time‐consuming part of the procedure. In this paper, we present an efficient way of computing MBF reaction terms and reducing them to equivalent moments. The numerical results are compared with brute‐force computations, the SMA, and the measurements. They demonstrate conclusively the efficiency and power of the new proposed technique. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 42: 138–143, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20232