Steepest Descent Method Research Articles

A central limit theorem for integer partitions into small powers

The study of the well-known partition function p(n) counting the number of solutions to n=a1+⋯+aℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = a_{1} + \\dots + a_{\\ell }$$\\end{document} with integers 1≤a1≤⋯≤aℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 \\le a_{1} \\le \\dots \\le a_{\\ell }$$\\end{document} has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into n=a1α+⋯+aℓα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} n=\\left\\lfloor a_1^\\alpha \\right\\rfloor +\\cdots +\\left\\lfloor a_\\ell ^\\alpha \\right\\rfloor \\end{aligned}$$\\end{document}with 1≤a1<⋯<aℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le a_1< \\cdots < a_\\ell $$\\end{document} and some fixed 0<α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0< \\alpha < 1$$\\end{document}. In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.

Boosting with fine-tuning for deep image denoising

While deep learning is ruling the image denoising domain in recent years, earlier works primarily focused on design of network architecture or training strategy. In this paper, we raise two questions: how to combine the advantages of traditional iterative methods and deep learning-based approaches, and how to avoid performance degradation caused by inaccurate modeling and estimation of noise. To answer the questions, we integrate recursive strategies and fine-tuning schemes to boost existing deep denoisers in a plug-and-play fashion. Specifically, based on the framework of plug-and-play priors, the image denoising problem is solved with the half quadratic splitting (HQS) algorithm to achieve iterative denoising. Different from the standard solving process, we develop a joint optimization scheme with regard to image restoration and network fine-tuning, realizing the matching between network and noise, thereby enabling better adaptation to the images contaminated by complex non-Gaussian noise. As such, two types of adaptive denoising boosters with convergence guarantee based on the fixed-point strategy and steepest-descent method are obtained. It is demonstrated in the experiments that the proposed schemes provide promising performance on additive white Gaussian noise (AWGN) and real-noise denoising for both supervised and self-supervised deep learning-based image denoising algorithms.

Study on the design method of an automatic outrigger layout control system for variable outrigger layout trimaran

During the navigation of a trimaran, its wave resistance is influenced by both sailing speed and outrigger layout. In the case of a variable outrigger layout trimaran, due to the adjustability of its outrigger layout according to different speeds, its wave resistance will be lower than that of traditional trimarans. In this paper, an automatic outrigger layout control system is figured out for a variable outrigger layout trimaran. The system can control and regulate the position of outriggers according to the traveling speed in order to reduce the wave resistance during navigation. The slender ship theory based resistance calculation program is applied as the solver in this system to evaluate the wave resistance of the trimaran and further determine the optimal outrigger layout under a specific speed. In order to improve the calculation accuracy, NURBS (Non-Uniform Rational B-Splines) method is introduced to modeling the hull surface. Model tests for trimaran with 6 different outrigger layouts are conducted to obtain the resistance of the trimaran in order to validate the numerical method. The steepest descent method is applied to solve the problem of searching the outrigger layout with minimum wave resistance efficiently. On this basis, an open-loop control system for trimaran outrigger layout is designed based on PID (Proportional Integral Derivative) algorithm with the control objective derived by the solver. Finally, a outrigger layout adjustment simulation (when speed changes from Fr=0.4 to Fr=0.35) is conducted on the experimental trimaran model. By comparing the resistance results obtained by numerical method utilized in this paper to the results obtained by CFD (Computational Fluid Dynamics) simulations, the effectiveness of the proposed system is validated.

Fast phase retrieval for broadband attosecond pulse characterization.

Efficient characterization method for broadband attosecond pulses has become more and more essential, since attosecond pulses with bandwidth spanning few-hundreds electron-volts have been generated. Here we propose a fast phase retrieval algorithm for broadband attosecond pulse characterization with an omega oscillation filtering technique. We introduce a new error function to improve the accuracy of the retrieved phases. More importantly, it can be solved by the steepest descent methods with iterative algorithm, which is much faster than genetic algorithm adopted previously. An experimental spectrogram for isolated attosecond pulses with photon energy covering 52-127 eV and a pulse width of 71 as was successfully retrieved with this method as demonstrated. The proposed technique will help provide real-time feedback on atto-chirp compensation for ultrashort isolated attosecond pulse generation.

An Efficient Threshold Determination Algorithm for DP-TBD Based on Structural Analogy and Saddle-Point Approximation

An Efficient Threshold Determination Algorithm for DP-TBD Based on Structural Analogy and Saddle-Point Approximation

A bound for Stieltjes constants

The main goal of this note is to improve the best known bounds for the Stieltjes constants, using the method of steepest descent that was applied in 2011 by Coffey and Knessl in order to approximate these constants.

Fitting Tweedie's compound Poisson model to pure premium with the EM algorithm

We consider the situation when the number of claims is unavailable, and a Tweedie's compound Poisson model is fitted to the observed pure premium. Currently, there are two different models based on the Tweedie distribution: a single generalized linear model (GLM) for mean and a double generalized linear model (DGLM) for both mean and dispersion. Although the DGLM approach facilitates the heterogeneous dispersion, its soundness relies on the accuracy of the saddlepoint approximation, which is poor when the proportion of zero claims is large. For both models, the power variance parameter is estimated by considering the profile likelihood, which is computationally expensive. We propose a new approach to fit the Tweedie model with the EM algorithm, which is equivalent to an iteratively re-weighted Poisson-gamma model on an augmented data set. The proposed approach addresses the heterogeneous dispersion without needing the saddlepoint approximation, and the power variance parameter is estimated during the model fitting. Numerical examples show that our proposed approach is superior to the two competing models.

Time-dependent reliability analysis of planar mechanisms considering truncated random variables and joint clearances

The dimension variables and joint clearances are key factors affecting the motion accuracy of a mechanism. The existing time-dependent reliability analysis methods usually assume that the dimension variables follow ideal distribution, ignoring the truncation restrictions due to production and manufacturing. In addition, improper handling of the correlation between joint clearance variables leads to inaccurate calculation results. To address the above issues, this paper proposes a time-dependent kinematic reliability analysis method, in which the truncated random variables and joint clearances are comprehensively considered. Firstly, the complex correlation of clearance variables in the motion error function is processed by the dimension reduction method. Then, the truncated variables are converted into truncated standard normal variables. Furthermore, the envelope function method with expansion at the most probable point (MPP) is used to transform time-dependent reliability into time-independent reliability at multiple time points, greatly reducing computational complexity. On this basis, the saddlepoint method is used to fit the motion error cumulative distribution function at each time point, and the upper and lower bounds of failure probability are obtained through the Copula function and second-order narrow reliability bounds. The effectiveness of the proposed method is verified by a typical 4-bar linkage and a single crank double rocker mechanism.

Study on reliability of emergency braking performance of high-speed and heavy-load monorail crane

The reliability of monorail crane braking system has an important influence on braking safety, high-speed and heavy-load operation, which poses great challenges to braking safety, and it is necessary to evaluate its braking reliability accurately and efficiently. Firstly, the dynamic performance and thermal-mechanical coupling characteristics of high-speed and heavy-load monorail crane under different braking parameters were analyzed. Secondly, the random response model of braking distance and braking temperature was established by combining the design of experiment method (DoE) and Dendrite Net (DD). Finally, the high-order moment saddlepoint approximation (SPA) method was used to evaluate the emergency braking reliability of the monorail crane. The results can provide a reference for the selection of key parameters and the evaluation of braking safety of the monorail crane braking system under high-speed and heavy-load conditions.

Assessment of the partial saddle point approximation in field-theoretic polymer simulations.

Field-theoretic simulations are numerical treatments of polymer field theory models that go beyond the mean-field self-consistent field theory level and have successfully captured a range of mesoscopic phenomena. Inherent in molecularly-based field theories is a "signproblem" associated with complex-valued Hamiltonian functionals. One route to field-theoretic simulations utilizes the complex Langevin (CL) method to importance sample complex-valued field configurations to bypass the signproblem. Although CL is exact in principle, it can be difficult to stabilize in strongly fluctuating systems. An alternate approach for blends or block copolymers with two segment species is to make a "partial saddle point approximation" (PSPA) in which the stiff pressure-like field is constrained to its mean-field value, eliminating the signproblem in the remaining field theory, allowing for traditional (real) sampling methods. The consequences of the PSPA are relatively unknown, and direct comparisons between the two methods are limited. Here, we quantitatively compare thermodynamic observables, order-disorder transitions, and periodic domain sizes predicted by the two approaches for a weakly compressible model of AB diblock copolymers. Using Gaussian fluctuation analysis, we validate our simulation observations, finding that the PSPA incorrectly captures trends in fluctuation corrections to certain thermodynamic observables, microdomain spacing, and location of order-disorder transitions. For incompressible models with contact interactions, we find similar discrepancies between the predictions of CL and PSPA, but these can be minimized by regularization procedures such as Morse calibration. These findings mandate caution in applying the PSPA to broader classes of soft-matter models and systems.

Probing supersymmetric black holes with surface defects

It has long been conjectured that the large N deconfinement phase transition of mathcal{N} = 4 SU(N) super-Yang-Mills corresponds via AdS/CFT to the Hawking-Page transition in which black holes dominate the thermal ensemble, and quantitative evidence of this has come through the recent matching of the superconformal index of frac{1}{16} -BPS states to the supersymmetric black hole entropy. We introduce the half-BPS Gukov-Witten surface defect as a probe of the superconformal index, which also serves as an order parameter for the deconfinement transition. This can be studied directly in field theory as a modification of the usual unitary matrix model or in the dual description as a D3-brane probe in the background of a (complex) supersymmetric black hole. Using a saddle point approximation, we determine our defect index in the large N limit as a simple function of the chemical potentials and show independently that it is reproduced by the renormalized action of the brane in the black hole background. Along the way, we also comment on the Cardy limit and the thermodynamics of the D3-brane in the generalized ensemble. The defect index sharply distinguishes between the confining and the deconfining phases of the gauge theory and thus is a supersymmetric non-perturbative order parameter for these large N phase transitions which deserves further investigation. Finally, our work provides an example where the properties of a black hole coupled to an external system can be analyzed precisely.

Rényi negativities in non-equilibrium open free-boson chains

In this paper, we consider the dynamics of Rényi negativities after a quantum quench in the free-boson chain with homogeneous dissipation. Initially we prepare the system in the squeezed thermal state, and then let it evolves under the tight-binding bosonic Hamiltonian with local linear dissipation. We use the Lindblad equation to solve the time evolution of the covariance matrix, from which one can obtain the time dependence of Rényi negativities. We are interested in the weak dissipation hydrodynamic limit where a quasi-particle picture emerges. In this limit, exact results of non-equilibrium dynamics of Rényi negativities can be obtained using the stationary phase method. We consider the Rényi negativities between both adjacent and disjoint regions in a infinite chain. We numerically test our analytical predictions and perfect matches have found.

A review of regularization strategies and solution techniques for ill-posed inverse problems, with application to inverse heat transfer problems

With the presence of a large number of inversion algorithms for inverse heat transfer problems (IHTPs) and non-IHTPs, a need for review to have a holistic view is seen. An exhaustive literature review, with the motivation of selecting the inversion technique best fit for a given problem, was made for a general inverse problem. For ill-posedness, a classification of available regularization algorithms namely Tikhonov’s regularization, Bayesian regularization, mollification method, Beck’s sequential approach and Alifanov’s iterative approach, has been provided. Inversion methods like singular value decomposition, truncated singular value decomposition, Tikhonov regularization and total variation regularization are explained. Optimization methods namely steepest descent method, conjugate gradient method, Newton method, Levenberg–Marquardt method, Lagrange method, adjoint method, function specification method, genetic algorithm, differential evolution and particle swarm optimization (PSO) are reviewed. Further, a technique based on neural networks is studied, and wavelet methods like shrinking and wavelet vaguelette decomposition are reviewed. Associated literature has also been listed, highlighting the gaps. The usability of various algorithms in IHTP, starting from the golden section search method, for retrieval of a single parameter, to the regularized versions of the inversion technique, for retrieval of multiple parameters with uncertainty, demonstrates real-life applications to fins in IHTP. An inversion algorithm capable to handle every kind of nonlinearity is sought in literature, whose absence raises the research question, “Is there a technique that works globally for every inverse problem?”, is asked prior to, “What if the available techniques were not utilized to an extent that they should?” is posed. In lieu of this gap, a general comparative framework is developed, such that an efficient technique is selected, based on the total minimum error, which can be used in any field of interest.

Modified control variates method based on second-order saddle-point approximation for practical reliability analysis

Abstract. A novel method is presented for efficiently analyzing the reliability of engineering components and systems with highly nonlinear complex limit state functions. The proposed method begins by transforming the integral of the limit state function into an integral of a highly correlated limit state function using the control variates method. The second-order reliability method is then employed within the control variates framework to approximate the highly correlated limit state function as a quadratic polynomial. Subsequently, the probability of failure is obtained through the estimation of the saddle-point approximation and a small number of samples generated by Latin hypercube sampling. To demonstrate the effectiveness of the proposed method, four examples involving mathematical functions and mechanical problems are solved. The results are compared with those obtained using the second-order reliability method (SORM), control variates based on Monte Carlo simulation (CVMCS) with second-order saddle-point approximation (SOSPA), importance sampling (IS) and Monte Carlo simulation (MCS). The findings demonstrate that, while maintaining high-precision reliability results, the proposed method significantly reduces the number of evaluations of the limit state function through a small number of initial samples. The method is capable of efficiently and accurately solving complex practical engineering reliability problems.

A Multi-Fidelity Integration Method for Reliability Analysis of Industrial Robots

Abstract A multi-fidelity integration method is proposed to analyze the reliability of multiple performance indicators (MPI) for industrial robots. In order to high-fidelity mapping the performance of industrial robots, a unified multi-domain model (UMDM) is first established. The contribution-degree analysis is then used to classify the input random variables into interacting and non-interacting ones. Thus, the high-dimensional integration of reliability analysis is separated into a low-dimensional integration and multiple one-dimensional integrations in an additive form. Here, the low-dimensional integration consisting of the interacting variables is calculated using the high-precision mixed-degree cubature formula (MDCF), and the computational results are treated as high-fidelity data. The one-dimensional integration consisting of non-interacting variables is then computed by the highly efficient five-point Gaussian Hermite quadrature (FGHQ), and the computational results are named low-fidelity data. A multi-fidelity integration method is constructed by fusing the high-fidelity data and the low-fidelity data to obtain the statistical moments of the MPI. Subsequently, the probability density function and the failure probability of the MPI are estimated using the saddlepoint approximation method. Finally, some representative methods are performed to verify the superiority of the proposed method.

Optimal solution techniques for control problem of evolution equations

A lot of optimal solution techniques exist in literature and have been employed to solve optimization problems. However, in this work, two optimal solution techniques namely the Steepest Descent Method (SDM) and the Extended Conjugate Gradient Method (ECGM) were applied to two evolution equations to determine which best approximates the system to its analytic solution. Comparisons were also carried out between the SDM and the ECGM for optimal solution and convergence. In doing so, the Liberty Basic Programming Language (LBPL) was employed to perform various numerical iterations on the respective algorithms of the techniques. It was discovered that the SDM proved to be a better technique in obtaining an optimal solution than the ECGM.

Formal Stationary Phase for the Mellin Transform of a $\mathcal D$-Module

Given a holonomic \mathbb{C}[z,z^{-1}]\langle \partial_z\rangle -module \mathbb{M} , following Loeser and Sabbah (Comment. Math. Helv. \mathbf{66} (1991), 458–503), one can consider its Mellin transform, which is a difference system on the affine line over \mathbb{C} . In this note we prove a stationary phase formula, which shows that its formal behavior at infinity is determined by the local germs defined by \mathbb{M} at its singular points.

A Curious Identity Arising From Stirling's Formula and Saddle-Point Method on Two Different Contours

We prove the curious identity in the sense of formal power series:\[\int_{-\infty}^{\infty}[y^m]\exp\left(-\frac{t^2}2+\sum_{j\ge3}\frac{(it)^j}{j!}\, y^{j-2}\right)\mathrm{d} t= \int_{-\infty}^{\infty}[y^m]\exp\left(-\frac{t^2}2+\sum_{j\ge3}\frac{(it)^j}{j}\, y^{j-2}\right)\mathrm{d} t,\]for $m=0,1,\dots$, where $[y^m]f(y)$ denotes the coefficient of $y^m$ in the Taylor expansion of $f$, which arises from applying the saddle-point method to derive Stirling's formula. The generality of the same approach (saddle-point method over two different contours) is also examined, together with some applications to asymptotic enumeration.

Predicting the occurrence of rogue waves in short-crested mixed seas

ABSTRACT In this paper we have for the first time proposed a novel transformed Gaussian process method for the estimation of rogue wave occurrence in a short crested mixed sea. The saddle point approximation of a specific joint density is utilised to further improve the computational efficiency of the proposed method. The proposed method has been applied to forecast the rogue wave occurrence in four short-crested mixed sea states, two with directional wave spectra based on measured surface elevation data at the coast of Yura, and another two with typical directional bimodal wave spectra. It is shown in these four cases that the proposed method can offer more accurate forecasting results than those obtained from the linear simulation method, from the transformation based on Hermite polynomials and from using the Rayleigh distribution model. It is also demonstrated that the proposed method is more efficient than the nonlinear simulation method.

Eccentric catastrophes & what to do with them

Analytic modeling of gravitational waves (GWs) from inspiraling eccentric binaries poses an interesting mathematical challenge. When constructing analytic waveforms in the frequency domain, one has to contend with the fact that the phase of the Fourier integral in non-monotonic, resulting in a breakdown of the standard stationary phase approximation (SPA). In this work, we study this breakdown within the context of catastrophe theory. We find that the SPA holds in the context of eccentric Keplerian orbits when the Fourier frequency satisfies , where are integer multiples of the apocenter/pericenter frequencies, respectively. For values outside of this interval, the phase undergoes a fold catastrophe, giving rise to an Airy function approximation of the Fourier integral. Using these two different approximations, we generate a matched asymptotic expansion that approximates generic Fourier integrals of Keplerian motion for bound orbits across all frequency values. This asymptotic expansion is purely analytic and closed-form. We discuss several applications of this investigation and the resulting approximation, specifically: (1) the development and improvement of effective fly-by waveforms for binary black holes, (2) the transition from burst emission in the high eccentricity limit to wave-like emission in the quasi-circular limit, which results in an analogy between eccentric GW bursts and Bose–Einstein condensates, and (3) the calculation of f-mode amplitudes in eccentric binary neutron stars and black hole-neutron star binaries in terms of complex Hansen coefficients. The techniques and approximations developed herein are generic, and will be useful for future studies of GWs from eccentric binaries within the context of post-Newtonian theory.