Steepest Descent Algorithm Research Articles

Bounding polynomial entanglement measures for mixed states

We generalize the notion of the best separable approximation (BSA) and best $W$-class approximation (BWA) to arbitrary pure-state entanglement measures, defining the best zero-$E$ approximation (BEA). We show that for any polynomial entanglement measure $E$, any mixed state $\ensuremath{\rho}$ admits at least one ``$S$ decomposition,'' i.e., a decomposition in terms of a mixed state on which $E$ is equal to zero, and a single additional pure state with (possibly) nonzero $E$. We show that the BEA is not, in general, the optimal $S$ decomposition from the point of view of bounding the entanglement of $\ensuremath{\rho}$ and describe an algorithm to construct the entanglement-minimizing $S$ decomposition for $\ensuremath{\rho}$ and place an upper bound on $E(\ensuremath{\rho})$. When applied to the three-tangle, the cost of the algorithm is linear in the rank $d$ of the density matrix and has an accuracy comparable to a steepest-descent algorithm whose cost scales as ${d}^{8}logd$. We compare the upper bound to a lower-bound algorithm given by C. Eltschka and J. Siewert [Phys. Rev. Lett. 108, 020502 (2012)] for the three-tangle and find that on random rank-2 three-qubit density matrices, the difference between the upper and lower bounds is $0.14$ on average. We also find that the three-tangle of random full-rank three-qubit density matrices is less than $0.023$ on average.

Optimal edge preserving restoration with efficient regularisation

Deblurring is an ill-posed inverse problem naturally and becomes more complex if the noise is also tainting the image. Classical approaches of inverse filtering and linear algebraic restorations are now obsolete due to the additive noise, as the problem is very sensitive to the small perturbation in the data. To cater the sensitivity of solution for small perturbations, smoothness constraints are generally added in the classical approaches. Previously neural networks and gradient based approaches have widely been used for optimisation; however, due to improper regularisation and computational cost, the problem is still an active research problem. In this paper, a new simple, efficient and robust method of regularisation based on the difference of grey scale average of image and each element of the estimated image is proposed. Constrained least square error is considered for optimisation and gradient based steepest descent algorithm is designed to estimate the optimal solution for restoration iteratively. The visual results and statistical measures of the experiments are presented in the paper which shows the effectiveness of the approach as compared to the state of art and recently proposed techniques.

Exact bounds for steepest descent algorithms of [formula omitted]-convex function minimization

We analyze minimization algorithms for L♮-convex functions in discrete convex analysis and establish exact bounds for the number of iterations required by the steepest descent algorithm and its variants.

Maximizing Gaussianity using kurtosis measurement in the kernel space for kernel linear discriminant analysis

Kernel-linear discriminant analysis (K-LDA) is the notable breakthrough among the dimensionality reduction techniques for supervised classification. The parameter used in the kernel function is usually tuned using k-fold cross-validation. Recently, a technique is proposed to optimize the bandwidth parameter of the Gaussian kernel by simultaneously maximizing the homoscedasticity (identical co-variance matrices) and the separation of various classes in the higher dimensional space ( HDS ). In this technique, it is assumed that the individual classes are Gaussian distributed in the HDS , which are not usually true in practical applications. In this paper, we propose a technique that maximizes the Guassianity of the individual classes and the separation of various classes in the HDS using “kernel-trick”. This is determined by the tuning parameters used in the kernel function. The steepest-descent algorithm is used to obtain the optimal value of the tuning parameters. The experiments are performed with the synthetic datasets and real datasets to demonstrate the improved results obtained using the proposed technique. The proposed technique can also be adopted for other kernel functions used in K-LDA.

On the discrete variant of the traction method in parameter-free shape optimization

One of the major challenges in the parameter-free approach to computational shape optimization is the avoidance of oscillating (i.e. non-smooth) boundaries in the optimal design trials. In order to achieve this, we investigate a method for regularization that corresponds to the discrete variant of the so-called traction method. In this approach, the design updates are generated in terms of a displacement field, which is obtained as the solution to an auxiliary boundary value problem that is defined on the actual design domain. The main idea herein is to apply fictitious nodal forces corresponding to the discrete sensitivity of the objective function. We propose an algorithm in which constraint functions will be taken into account by using an augmented Lagrangian formulation and a step-length control ensures a sufficient decrease condition in terms of the objective function within each iteration. We examine the benefits of the proposed regularization method on the basis of some numerical examples in comparison to an unregularized steepest-descent algorithm.

Learning in Mean-Field Games

The purpose of this paper is to show how insight obtained from a mean-field model can be used to create an architecture for approximate dynamic programming (ADP) for a certain class of games comprising of a large number of agents. The general technique is illustrated with the aid of a mean-field oscillator game model introduced in our prior work. The states of the model are interpreted as the phase angles for a collection of nonhomogeneous oscillators, and in this way the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto. The paper introduces ADP techniques for design and adaptation (learning) of approximately optimal control laws for this model. For this purpose, a parameterization is proposed, based on an analysis of the mean-field PDE model for the game. In an offline setting, a Galerkin procedure is introduced to choose the optimal parameters while in an online setting, a steepest descent algorithm is proposed. The paper provides a detailed analysis of the optimal parameter values as well as the Bellman error with both the Galerkin approximation and the online algorithm. Finally, a phase transition result is described for the large population limit when each oscillator uses the approximately optimal control law. A critical value of the control penalty parameter is identified: above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap) the oscillators synchronize. These conclusions are illustrated with results from numerical experiments.

Approximation of Irregular Geometric Data by Locally Calculated Univariate Cubic $$L^1$$ L 1 Spline Fits

\(L^1\) splines have been under development for interpolation and approximation of irregular geometric data. We investigate the advantages in terms of shape preservation and computational efficiency of calculating univariate cubic \(L^1\) spline fits using a steepest-descent algorithm to minimize a global data-fitting functional under a constraint implemented by a local analysis-based interpolating-spline algorithm on 5-node windows. Comparison of these locally calculated \(L^1\) spline fits with globally calculated \(L^1\) spline fits previously reported in the literature indicates that the locally calculated spline fits preserve shape on the average slightly better than the globally calculated spline fits and are computationally more efficient because the locally-calculated-spline-fit algorithm can be parallelized.

A Fast Learning Algorithm for Rainfall Prediction

A PC based application is developed using 51 years of Indian rainfall data for long range prediction of average rain fall. This learning algorithm iteratively estimates 96 coefficients of a 5th order polynomial in few minutes. Proposed prediction model is based on modelling of time series rainfall data using 5th order non-linear predicting code. Steepest descent algorithm is used for extraction of appropriate coefficient of rainfall time series data. These coefficients are reinforced in model learning process. The rainfall data of 1960 to 2010 is used for the development of the model. Model has been tested on rainfall data for different training sets. The proposed model is capable of forecasting yearly rainfall 1 year in advance. Rainfall estimation accuracy is above 85%.

A combined compact genetic algorithm and local search method for optimizing the ARMA(1,1) model of a likelihood estimator

In this paper, a compact genetic algorithm (CGA) is enhanced by integrating its selection strategy with a steepest descent algorithm (SDA) as a local search method to give I-CGA-SDA. This system is an attempt to avoid the large CPU time and computational complexity of the standard genetic algorithm. Here, CGA dramatically reduces the number of bits required to store the population and has a faster convergence. Consequently, this integrated system is used to optimize the maximum likelihood function lnL( 1; 1) of the mixed model. Simulation results based on MSE were compared with those obtained from the SDA and showed that the hybrid genetic algorithm (HGA) and I-CGA-SDA can give a good estimator of ( 1; 1) for the ARMA(1,1) model. Another comparison has been conducted to show that the I-CGA-SDA has fewer function evaluations, minimum search space percentage, faster convergence speed and has a higher optimal precision than that of the HGA.

Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions

In this paper we accomplish the development of the fast rank-adaptive solver for tensor-structured symmetric positive definite linear systems in higher dimensions. In [arXiv:1301.6068] this problem is approached by alternating minimization of the energy function, which we combine with steps of the basis expansion in accordance with the steepest descent algorithm. In this paper we combine the same steps in such a way that the resulted algorithm works with one or two neighboring cores at a time. The recurrent interpretation of the algorithm allows to prove the global convergence and to estimate the convergence rate. We also propose several strategies, both rigorous and heuristic, to compute new subspaces for the basis enrichment in a more efficient way. We test the algorithm on a number of high-dimensional problems, including the non-symmetrical Fokker-Planck and chemical master equations, for which the efficiency of the method is not fully supported by the theory. In all examples we observe a convincing fast convergence and high efficiency of the proposed method.

A New Interval Type-2 Fuzzy Clustering Algorithm for Interval Type-2 Fuzzy Modelling with Application to Heat Treatment of Steel

Abstract In this paper, a new hierarchical data-driven modelling strategy based on Interval Type-2 Fuzzy Clustering is elicited for the Interval Type-2 Takagi-Sugeno-Kang (TSK) Fuzzy Logic System. This framework which we have called the IT2-Squared framework uses interval type-2 fuzzy clustering for initial antecedent parameters and structures determination and least-squares algorithm for deriving initial consequent parameters. To improve the accuracy of the system, we show how the steepest descent algorithm is used to tune the parameters of both the consequent and antecedent parameters. To test the efficacy of this proposed system, the model is used on a real-life engineering project for the prediction of the ultimate tensile strength (UTS) of steel. Results show excellent generalization properties of the IT2-Squared modelling framework when compared to previously elicited models of the same system.

Evidence of Ternary Complex Formation in Trypanosoma cruzi trans-Sialidase Catalysis

Trypanosoma cruzi trans-sialidase (TcTS) is a key target protein for Chagas disease chemotherapy. In this study, we investigated the implications of active site flexibility on the biochemical mechanism of TcTS. Molecular dynamics studies revealed remarkable plasticity in the TcTS catalytic site, demonstrating, for the first time, how donor substrate engagement with the enzyme induces an acceptor binding site in the catalytic pocket that was not previously captured in crystal structures. Furthermore, NMR data showed cooperative binding between donor and acceptor substrates, supporting theoretical results. In summary, our data put forward a coherent dynamic framework to understand how a glycosidase evolved its highly efficient trans-glycosidase activity.

Steepest-Descent Approach to Triple Hierarchical Constrained Optimization Problems

We introduce and analyze a hybrid steepest-descent algorithm by combining Korpelevich’s extragradient method, the steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to the unique solution of a triple hierarchical constrained optimization problem (THCOP) over the common fixed point set of finitely many nonexpansive mappings, with constraints of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and a convex minimization problem (CMP) in a real Hilbert space.

Identifying Key Amino Acid Residues That Affect α-Conotoxin AuIB Inhibition of α3β4 Nicotinic Acetylcholine Receptors

α-Conotoxin AuIB is a selective α3β4 nicotinic acetylcholine receptor (nAChR) subtype inhibitor. Its analgesic properties are believed to result from it activating GABAB receptors and subsequently inhibiting CaV2.2 voltage-gated calcium channels. The structural determinants that mediate diverging AuIB activity at these targets are unknown. We performed alanine scanning mutagenesis of AuIB and α3β4 nAChR, homology modeling, and molecular dynamics simulations to identify the structural determinants of the AuIB·α3β4 nAChR interaction. Two alanine-substituted AuIB analogues, [P6A]AuIB and [F9A]AuIB, did not inhibit the α3β4 nAChR. NMR and CD spectroscopy studies demonstrated that [F9A]AuIB retains its native globular structure, so its activity loss is probably due to loss of specific toxin-receptor residue pairwise contacts. Compared with AuIB, the concentration-response curve for inhibition of α3β4 by [F9A]AuIB shifted rightward more than 10-fold, and its subtype selectivity profile changed. Homology modeling and molecular dynamics simulations suggest that Phe-9 of AuIB interacts with a two-residue binding pocket on the β4 nAChR subunit. This hypothesis was confirmed by site-directed mutagenesis of the β4-Trp-59 and β4-Lys-61 residues of loop D, which form a putative binding pocket. AuIB analogues with Phe-9 substitutions corroborated the finding of a binding pocket on the β4 subunit and gave further insight into how AuIB Phe-9 interacts with the β4 subunit. In summary, we identified critical residues that mediate interactions between AuIB and its cognate nAChR subtype. These findings might help improve the design of analgesic conopeptides that selectively "avoid" nAChR receptors while targeting receptors involved with nociception.

The steepest descent algorithm without line search for p-Laplacian

In this paper, the steepest descent algorithm without line search is proposed for p-Laplacian. Its search direction is the weighted preconditioned steepest descent one, and step length is estimated by a formula except the first iteration. Continuation method is applied for solving the p-Laplacian with very large p. Lots of numerical experiments are carried out on these algorithms. All numerical results show the algorithm without line search can cut down some computational time. Fast convergence of these new algorithms is displayed by their step length figures. These figures show that if search direction is the steepest descent one, exact step lengths can be substituted properly with step lengths obtained by the formula.

Neural Network Model Reference Adaptive System Speed Estimation for Sensorless Control of a Doubly Fed Induction Generator

This article proposes a neural network model reference adaptive system for the rotor angle and speed estimation of the doubly fed induction generator used in wind turbines. The model reference adaptive system reference signal is the measured rotor current. The adaptive neural network adjusts the weights minimizing the rotor current vector squared error using the steepest descent algorithm. The neural network maximum stable learning rate will be determined for this application. The validity of the proposed neural network model reference adaptive system is verified and analyzed in a real prototype of 7.5-kW doubly fed induction generator. To validate the proposed estimator, the estimated rotor angle and speed in the process of connecting the doubly fed induction generator to the grid and the sensorless regulation according to a random wind speed profile are presented.

Adjoint shape optimization applied to electromagnetic design

We present an adjoint-based optimization for electromagnetic design. It embeds commercial Maxwell solvers within a steepest-descent inverse-design optimization algorithm. The adjoint approach calculates shape derivatives at all points in space, but requires only two "forward" simulations. Geometrical shape parameterization is by the level set method. Our adjoint design optimization is applied to a Silicon photonics Y-junction splitter that had previously been investigated by stochastic methods. Owing to the speed of calculating shape derivatives within the adjoint method, convergence is much faster, within a larger design space. This is an extremely efficient method for the design of complex electromagnetic components.

Analog Circuit Fault Diagnosis Based on Levenberg-Marquardt Learning Algorithm

This paper presents a fault diagnosis method of BP neural network based on Levenberg-Marquardt learning algorithm. First, the use of principal component analysis to reduce the dimension of the fault sample reduced BP neural network input variables. Then use the Levenberg-Marquardt learning algorithm to adjust the network weights. Levenberg-Marquardt learning algorithm is combination of the Gauss - Newton algorithm and steepest descent algorithm. It has Gauss - Newton algorithm of local convergence and gradient descent algorithm of the global characteristic. So it has higher convergence speed, reduces the training time, to a certain extent, overcomes the problem of traditional BP network convergence speed slow and easy to fall into local minimum point. Simulation results demonstrate the correctness and accuracy of this fault diagnosis method.

Molecular Mechanisms, Thermodynamics, and Dissociation Kinetics of Knob-Hole Interactions in Fibrin

Polymerization of fibrin, the primary structural protein of blood clots and thrombi, occurs through binding of knobs 'A' and 'B' in the central nodule of fibrin monomer to complementary holes 'a' and 'b' in the γ- and β-nodules, respectively, of another monomer. We characterized the A:a and B:b knob-hole interactions under varying solution conditions using molecular dynamics simulations of the structural models of fibrin(ogen) fragment D complexed with synthetic peptides GPRP (knob 'A' mimetic) and GHRP (knob 'B' mimetic). The strength of A:a and B:b knob-hole complexes was roughly equal, decreasing with pulling force; however, the dissociation kinetics were sensitive to variations in acidity (pH 5-7) and temperature (T = 25-37 °C). There were similar structural changes in holes 'a' and 'b' during forced dissociation of the knob-hole complexes: elongation of loop I, stretching of the interior region, and translocation of the moveable flap. The disruption of the knob-hole interactions was not an "all-or-none" transition as it occurred through distinct two-step or single step pathways with or without intermediate states. The knob-hole bonds were stronger, tighter, and more brittle at pH 7 than at pH 5. The B:b knob-hole bonds were weaker, looser, and more compliant than the A:a knob-hole bonds at pH 7 but stronger, tighter, and less compliant at pH 5. Surprisingly, the knob-hole bonds were stronger, not weaker, at elevated temperature (T = 37 °C) compared with T = 25 °C due to the helix-to-coil transition in loop I that helps stabilize the bonds. These results provide detailed qualitative and quantitative characteristics underlying the most significant non-covalent interactions involved in fibrin polymerization.

Use of a beat effect for the automatic positioning of flow obstructions to control tonal fan noise: Theory and experiments

Tonal noise generated by axial fans at the Blade Passage Frequency and its harmonics is a source of discomfort for low-speed fans used in many cooling and ventilation applications. The noise control approach presented here is based on the interference between the unsteady aerodynamic blade loads responsible for tonal noise generation and secondary aerodynamic loads generated in the rotor plane by fixed, carefully positioned, small obstructions in the upstream flow. Although not strictly active control, the magnitude and phase of the secondary tonal noise can be adjusted by varying the axial distance between the rotor and the obstruction, and the circumferential position of the obstruction, respectively. An optimal position of the obstruction generally exists, that minimizes the total noise at a given frequency. This paper establishes a practical method for automatic positioning of such control obstructions. In a first step, the method searches for the optimal axial distance between the rotor and the obstruction using a slowly rotating control obstruction. The modulation created by the rotation of the obstruction allows for the primary and secondary noises to be distinguished in the frequency response of the sound field. The steepest descent algorithm is used to find the optimal axial distance, for which the magnitudes of the primary and secondary tonal noise are equal at the error microphone. Then, the optimal angular position of the obstruction is obtained by slowly rotating the obstruction until minimal total noise is achieved. Finally, it is shown that at the optimal axial and angular position, the BPF tone, which produced the largest area in the loudness pattern, has been greatly reduced.