Error, Confidence, and Cost in Engineering Computations

The questions that one should answer in engineering computations - deterministic, probabilistic/randomized, as well as heuristic - are (i) how good the computed results/outputs are and (ii) how much the cost in terms of amount of computation and the amount of storage utilized in getting the outputs is. The absolutely errorfree quantities as well as the completely errorless computations done in a natural process can never be captured by any means that we have at our disposal. While the computations including the input real quantities in nature/natural processes are exact, all the computations that we do using a digital computer or are carried out in an embedded form are never exact. The input data for such computations are also never exact because any measuring instrument has inherent error of a fixed order associated with it and this error, as a matter of hypothesis and not as a matter of assumption, is not less than 0.005 per cent. Here by error we imply relative error bounds. The fact that exact error is never known under any circumstances and any context implies that the term error is nothing but error-bounds. Further, in engineering computations, it is the relative error or, equivalently, the relative error-bounds (and not the absolute error) which is supremely important in providing us the information regarding the quality of the results/outputs. Another important fact is that inconsistency and/or near-consistency in nature, i.e., in problems created from nature is completely nonexistent while in our modelling of the natural problems we may introduce inconsistency or near-inconsistency due to human error or due to inherent non-removable error associated with any measuring device or due to assumptions introduced to make the problem solvable or more easily solvable in practice. Thus if we discover any inconsistency or possibly any near-inconsistency in a mathematical model, it is certainly due to any or all of the three foregoing factors. We do, however, go ahead to solve such inconsistent/near-consistent problems and do get results that could be useful in real-world situations. The talk considers several deterministic, probabilistic, and heuristic algorithms in numerical optimisation, other numerical and statistical computations, and in PAC (probably approximately correct) learning models. It highlights the quality of the results/outputs through specifying relative error-bounds along with the associated confidence level, and the cost, viz., amount of computations and that of storage through complexity. It points out the limitation in error-free computations (wherever possible, i.e., where the number of arithmetic operations is finite and is known a priori) as well as in the usage of interval arithmetic. Further, the interdependence among the error, the confidence, and the cost is discussed.

Pre-emptive queue processing scheme for handovers in mobile cellular communication systems

A new scheme for minimizing handover failure probability in mobile cellular communication systems is presented. The scheme involves a reassignment of priorities for handover requests enqueued in adjacent cells to release a channel for a handover request which is about to fail. Performance evaluation of the new scheme carried out by computer simulation of a four-cell highway cellular system has shown a considerable reduction in handover failure probability

Structural Transformations, Composition Anomalies and a Dramatic Collapse of Linear Polymer Chains in Dilute Ethanol-Water Mixtures

Water-ethanol mixtures exhibit many interesting anomalies, such as negative excess partial molar volume of ethanol, excess sound absorption coefficient at low concentrations, and positive deviation from Raoult's law for vapor pressure, to mention a few. These anomalies have been attributed to different, often contradictory origins, but a quantitative understanding is still lacking. We show by computer simulation and theoretical analyses that these anomalies arise from the sudden emergence of a bicontinuous phase that occurs at a relatively low ethanol concentration of x(eth) approximate to 0.06-0.10 (that amounts to a volume fraction of 0.17-0.26, which is a significant range!). The bicontinuous phase is formed by aggregation of ethanol molecules, resulting in a weak phase transition whose nature is elucidated. We find that the microheterogeneous structure of the mixture gives rise to a pronounced nonmonotonic composition dependence of local compressibility and nonmonotonic dependence in the peak value of the radial distribution function of ethyl groups. A multidimensional free energy surface of pair association is shown to provide a molecular explanation of the known negative excess partial volume of ethanol in terms of parallel orientation and hence better packing of the ethyl groups in the mixture due to hydrophobic interactions. The energy distribution of the ethanol molecules indicates additional energy decay channels that explain the excess sound attenuation coefficient in aqueous alcohol mixtures. We studied the dependence of the solvation of a linear polymer chain on the composition of the water-ethanol solvent. We find that there is a sudden collapse of the polymer at x(eth) approximate to 0.05-a phenomenon which we attribute to the formation of the microheterogeneous structures in the binary mixture at low ethanol concentrations. Together with recent single molecule pulling experiments, these results provide new insight into the behavior of polymer chain and foreign solutes, such as enzymes, in aqueous binary mixtures.

Hydration dynamics of protein molecules in aqueous solution: Unity among diversity

Dielectric dispersion and NMRD experiments have revealed that a significant fraction of water molecules in the hydration shell of various proteins do not exhibit any slowing down of dynamics. This is usually attributed to the presence of the hydrophobic residues (HBR) on the surface, although HBRs alone cannot account for the large amplitude of the fast component. Solvation dynamics experiments and also computer simulation studies, on the other hand, repeatedly observed the presence of a non-negligible slow component. Here we show, by considering three well-known proteins (lysozyme, myoglobin and adelynate kinase), that the fast component arises partly from the response of those water molecules that are hydrogen bonded with the backbone oxygen (BBO) atoms. These are structurally and energetically less stable than those with the side chain oxygen (SCO) atoms. In addition, the electrostatic interaction energy distribution (EIED) of individual water molecules (hydrogen bonded to SCO) with side chain oxygen atoms shows a surprising two peak character with the lower energy peak almost coincident with the energy distribution of water hydrogen bonded to backbone oxygen atoms (BBO). This two peak contribution appears to be quite general as we find it for lysozyme, myoglobin and adenylate kinase (ADK). The sharp peak of EIED at small energy (at less than 2 k(B)T) for the BBO atoms, together with the first peak of EIED of SCO and the HBRs on the protein surface, explain why a large fraction (similar to 80%) of water in the protein hydration layer remains almost as mobile as bulk water Significant slowness arises only from the hydrogen bonds that populate the second peak of EIED at larger energy (at about 4 k(B)T). Thus, if we consider hydrogen bond interaction alone, only 15-20% of water molecules in the protein hydration layer can exhibit slow dynamics, resulting in an average relaxation time of about 5-10 ps. The latter estimate assumes a time constant of 20-100 ps for the slow component. Interestingly, relaxation of water molecules hydrogen bonded to back bone oxygen exhibit an initial component faster than the bulk, suggesting that hydrogen bonding of these water molecules remains frustrated. This explanation of the heterogeneous and non-exponential dynamics of water in the hydration layer is quantitatively consistent with all the available experimental results, and provides unification among diverse features.

Adam-Gibbs Relation for Glass-Forming Liquids in Two, Three, and Four Dimensions

The Adam-Gibbs relation between relaxation times and the configurational entropy has been tested extensively for glass formers using experimental data and computer simulation results. Although the form of the relation contains no dependence on the spatial dimensionality in the original formulation, subsequent derivations of the Adam-Gibbs relation allow for such a possibility. We test the Adam-Gibbs relation in two, three, and four spatial dimensions using computer simulations of model glass formers. We find that the relation is valid in three and four dimensions. But in two dimensions, the relation does not hold, and interestingly, no single alternate relation describes the results for the different model systems we study.

Anisotropy induced crossover from weakly to strongly first order melting of two dimensional solids

Melting and freezing transitions in two dimensional (2D) systems are known to show highly unusual characteristics. Most of the earlier studies considered atomic systems: the melting of 2D molecular solids is still largely unexplored. In order to understand the role of anisotropy as well as multiple energy and length scales present in molecular systems, here we report computer simulation studies of melting of 2D molecular systems. We computed a limited portion of the solid-liquid phase diagram. We find that the interplay between the strength of isotropic and anisotropic interactions can give rise to rich phase diagram consisting of isotropic liquid and two crystalline phases-honeycomb and oblique. The nature of the transition depends on the relative strength of the anisotropic interaction and a strongly first order melting turns into a weakly first order transition on increasing the strength of the isotropic interaction. This crossover can be attributed to an increase in stiffness of the solid phase free energy minimum on increasing the strength of the anisotropic interaction. The defects involved in melting of molecular systems are quite different from those known for the atomic systems.

Comparative Study of Protein Unfolding in Aqueous Urea and Dimethyl Sulfoxide Solutions: Surface Polarity, Solvent Specificity, and Sequence of Secondary Structure Melting

Elucidation of possible pathways between folded (native) and unfolded states of a protein is a challenging task, as the intermediates are often hard to detect. Here, we alter the solvent environment in a controlled manner by choosing two different cosolvents of water, urea, and dimethyl sulfoxide (DMSO) and study unfolding of four different proteins to understand the respective sequence of melting by computer simulation methods. We indeed find interesting differences in the sequence of melting of alpha helices and beta sheets in these two solvents. For example, in 8 M urea solution, beta-sheet parts of a protein are found to unfold preferentially, followed by the unfolding of alpha helices. In contrast, 8 M DMSO solution unfolds alpha helices first, followed by the separation of beta sheets for the majority of proteins. Sequence of unfolding events in four different alpha/beta proteins and also in chicken villin head piece (HP-36) both in urea and DMSO solutions demonstrate that the unfolding pathways are determined jointly by relative exposure of polar and nonpolar residues of a protein and the mode of molecular action of a solvent on that protein.

Smoothed Functional Algorithms for Stochastic Optimization Using q-Gaussian Distributions

Smoothed functional (SF) schemes for gradient estimation are known to be efficient in stochastic optimization algorithms, especially when the objective is to improve the performance of a stochastic system However, the performance of these methods depends on several parameters, such as the choice of a suitable smoothing kernel. Different kernels have been studied in the literature, which include Gaussian, Cauchy, and uniform distributions, among others. This article studies a new class of kernels based on the q-Gaussian distribution, which has gained popularity in statistical physics over the last decade. Though the importance of this family of distributions is attributed to its ability to generalize the Gaussian distribution, we observe that this class encompasses almost all existing smoothing kernels. This motivates us to study SF schemes for gradient estimation using the q-Gaussian distribution. Using the derived gradient estimates, we propose two-timescale algorithms for optimization of a stochastic objective function in a constrained setting with a projected gradient search approach. We prove the convergence of our algorithms to the set of stationary points of an associated ODE. We also demonstrate their performance numerically through simulations on a queuing model.

Phase Diagram Study of Succinonitrile-Vanillin Organic Alloy System

Phase diagram studies of succinonitrile-vanillin system show the formation of 2:1 congruent melting type compound. Crystallization velocities of pure components, succinonitrile-vanillin complex, and two eutectics have been determined at different undercoolings. On the basis of heat of fusion measurements, excess thermodynamic functions have been calculated. Microstructural studies revealed that impurities modify the morphology. FTIR spectral studies and computer simulation have shown the existence of hydrogen bonding in the eutectics and the congruent melting compound. On the basis of experimental results, the mechanism of formation of eutectics and its solidification behavior are discussed.

Proper orthogonal decomposition of oscillatory Marangoni flow in half-zone liquid bridges of low-Pr fluids

Proper orthogonal decomposition (POD) using method of snapshots was performed on three different types of oscillatory Marangoni flows in half-zone liquid bridges of low-Pr fluid (Pr = 0.01). For each oscillation type, a series of characteristic modes (eigenfunctions) have been extracted from the velocity and temperature disturbances, and the POD provided spatial structures of the eigenfunctions, their oscillation frequencies, amplitudes, and phase shifts between them. The present analyses revealed the common features of the characteristic modes for different oscillation modes: four major velocity eigenfunctions captured more than 99% of the velocity fluctuation energy form two pairs, one of which is the most energetic. Different from the velocity disturbance, one of the major temperature eigenfunctions makes the dominant contribution to the temperature fluctuation energy. On the other hand, within the most energetic velocity eigenfuction pair, the two eigenfunctions have similar spatial structures and were tightly coupled to oscillate with the same frequency, and it was determined that the spatial structures and phase shifts of the eigenfunctions produced the different oscillatory disturbances. The interaction of other major modes only enriches the secondary spatio-temporal structures of the oscillatory disturbances. Moreover, the present analyses imply that the oscillatory disturbance, which is hydrodynamic in nature, primarily originates from the interior of the liquid bridge. (C) 2007 Elsevier B.V. All rights reserved.

The effect of thermal-activation on dislocation processes at an atomistic crack-tip

The effects of thermal activation on the dislocation emission from an atomistic crack tip are discussed, Molecular dynamics simulations at different constant temperatures are carried out to investigate the thermal effects. The simulated results show that the processes of the partial dislocation generation and emission are temperature dependent. As the temperature increases, the incipient duration of the partial dislocation nucleation becomes longer, the critical stress intensity factor for partial dislocation emission is reduced and, at the same loading level, more dislocations are emitted. The dislocation velocity moving away from the crack tip and the separations of partial dislocations are apparently not temperature dependent. The simulated results also show that, as the temperature increases, the stress distribution along the crack increases slightly. Therefore stress softening at the crack tip induced by thermal activation does not exist in the present simulation. A simple model is proposed to evaluate the relation of the critical stress intensity factor versus temperature. The obtained relation is in good agreement with our molecular dynamics results.

Crack tip behavior and crack-propagation in ductile materials

Dilatational plastic equations, which can include the effects of ductile damage, are derived based on the equivalency in expressions for dissipated plastic work. Void damage developed internally at the large-strain stage is represented by an effective continuum being strain-softened and plastically dilated. Accumulation of this local damage leads to progressive failure in materials. With regard to this microstructural background, the constitutive parameters included for characterizing material behaviour have the sense of internal variables. They are not able to be determined explicitly by macroscopic testing but rather through computer simulation of experimental curves and data. Application of this constitutive model to mode-I cracking examples demonstrates that a huge strain concentration accompanied by a substantial drop of stress does occur near the crack tip. Eventually, crack propagation is simulated by using finite elements in computations. Two numerical examples show good accordance with experimental data. The whole procedure of study serves as a justification of the constitutive formulation proposed in the text.

A Model of Ischemia-Induced Neuroblast Activation in the Adult Subventricular Zone

12 p.

Random response of nonlinear continuous systems

This thesis presents a technique for obtaining the stochastic response of a nonlinear continuous system. First, the general method of nonstationary continuous equivalent linearization is developed. This technique allows replacement of the original nonlinear system with a time-varying linear continuous system. Next, a numerical implementation is described which allows solution of complex problems on a digital computer. In this procedure, the linear replacement system is discretized by the finite element method. Application of this method to systems satisfying the one-dimensional wave equation with two different types of constitutive nonlinearities is described. Results are discussed for nonlinear stress-strain laws of both hardening and softening types.

A mathematical study of finite-amplitude rock-folding

The problem of the finite-amplitude folding of an isolated, linearly viscous layer under compression and imbedded in a medium of lower viscosity is treated theoretically by using a variational method to derive finite difference equations which are solved on a digital computer. The problem depends on a single physical parameter, the ratio of the fold wavelength, L, to the "dominant wavelength" of the infinitesimal-amplitude treatment, L_d. Therefore, the natural range of physical parameters is covered by the computation of three folds, with L/L_d = 0, 1, and 4.6, up to a maximum dip of 90°.

Significant differences in fold shape are found among the three folds; folds with higher L/L_d have sharper crests. Folds with L/L_d = 0 and L/L_d = 1 become fan folds at high amplitude. A description of the shape in terms of a harmonic analysis of inclination as a function of arc length shows this systematic variation with L/L_d and is relatively insensitive to the initial shape of the layer. This method of shape description is proposed as a convenient way of measuring the shape of natural folds.

The infinitesimal-amplitude treatment does not predict fold-shape development satisfactorily beyond a limb-dip of 5°. A proposed extension of the treatment continues the wavelength-selection mechanism of the infinitesimal treatment up to a limb-dip of 15°; after this stage the wavelength-selection mechanism no longer operates and fold shape is mainly determined by L/L_d and limb-dip.

Strain-rates and finite strains in the medium are calculated f or all stages of the L/L_d = 1 and L/L_d = 4.6 folds. At limb-dips greater than 45° the planes of maximum flattening and maximum flattening rat e show the characteristic orientation and fanning of axial-plane cleavage.