Simple Recursive Formula Research Articles

Solving the Power Flow Problem in Bipolar DC Asymmetric Distribution Networks Using Broyden's Method.

This research addresses the power flow analysis in bipolar asymmetric direct current (DC) networks by applying Broyden's numerical method. This general successive approximations method allows for a simple Newton-based recursive formula to reach the roots of multiple nonlinear equations. The main advantage of Broyden's approach is its simple but efficient structure which can be applied to real complex nonlinear equations.The power flow problem in bipolar DC networks is still challenging, as multiple operating options must be considered, e.g., the possibility of having a solidly grounded or floating neutral wire. The main goal of this research is to contribute with a generalization of Broyden's method for the power flow solution in bipolar DC networks, with the main advantage that, under well-defined conditions, this is a numerical method equivalent to the matricial backward/forward power flow, which is equivalent to the successive approximations power flow method. Numerical results in the 21-, 33-, and 85-bus grids while considering two connections for the neutral wire (i.e., solidly grounded at any node or floating) show the effectiveness of Broyden's method in the power flow solution for bipolar asymmetric DC networks. All numerical simulations were carried out in the MATLAB programming environment.

Proximal Decoding for LDPC Codes

This paper presents a novel optimization-based decoding algorithm for LDPC codes. The proposed decoding algorithm is based on a proximal gradient method for solving an approximate maximum a posteriori (MAP) decoding problem. The key idea of the proposed algorithm is the use of a code-constraint polynomial to penalize a vector far from a codeword as a regularizer in the approximate MAP objective function. A code proximal operator is naturally derived from a code-constraint polynomial. The proposed algorithm, called proximal decoding, can be described by a simple recursive formula consisting of the gradient descent step for a negative log-likelihood function corresponding to the channel conditional probability density function and the code proximal operation regarding the code-constraint polynomial. Proximal decoding is experimentally shown to be applicable to several non-trivial channel models such as LDPC-coded massive MIMO channels, correlated Gaussian noise channels, and nonlinear vector channels. In particular, in MIMO channels, proximal decoding outperforms known massive MIMO detection algorithms, such as an MMSE detector with belief propagation decoding. The simple optimization-based formulation of proximal decoding allows a way for developing novel signal processing algorithms involving LDPC codes.

Fast calculation of gravitational effects using tesseroids with a polynomial density of arbitrary degree in depth

Fast and accurate calculation of gravitational effects on a regional or global scale with complex density environment is a critical issue in gravitational forward modelling. Most existing significant developments with tessroid-based modelling are limited to homogeneous density models or polynomial ones of a limited order. Moreover, the total gravitational effects of tesseroids are often calculated by pure summation in these methods, which makes the calculation extremely time-consuming. A new efficient and accurate method based on tesseroids with a polynomial density up to an arbitrary order in depth is developed for 3D large-scale gravitational forward modelling. The method divides the source region into a number of tesseroids, and the density in each tesseroid is assumed to be a polynomial function of arbitrary degree. To guarantee the computational accuracy and efficiency, two key points are involved: (1) the volume Newton’s integral is decomposed into a one-dimensional integral with a polynomial density in the radial direction, for which a simple analytical recursive formula is derived for efficient calculation, and a surface integral over the horizontal directions evaluated by the Gauss–Legendre quadrature (GLQ) combined with a 2D adaptive discretization strategy; (2) a fast and flexible discrete convolution algorithm based on 1D fast Fourier transform (FFT) and a general Toepritz form of weight coefficient matrices is adopted in the longitudinal dimension to speed up the computation of the cumulative contributions from all tesseroids. Numerical examples show that the gravitational fields predicted by the new method have a good agreement with the corresponding analytical solutions for spherical shell models with both polynomial and non-polynomial density variations in depth. Compared with the 3D GLQ methods, the new algorithm is computationally more accurate and efficient. The calculation time is significantly reduced by 3 orders of magnitude as compared with the traditional 3D GLQ methods. Application of the new algorithm in the global crustal CRUST1.0 model further verifies its reliability and practicability in real cases. The proposed method will provide a powerful numerical tool for large-scale gravity modelling and also an efficient forward engine for inversion and continuation problems.

Approximating the first passage time density from data using generalized Laguerre polynomials

This paper analyzes a method to approximate the first passage time probability density function which turns to be particularly useful if only sample data are available. The method relies on a Laguerre-Gamma polynomial approximation and iteratively looks for the best degree of the polynomial such that a normalization condition is preserved. The proposed iterative algorithm relies on simple and new recursion formulae involving first passage time moments. These moments can be computed recursively from cumulants, if they are known. In such a case, the approximated density can be used also for the maximum likelihood estimates of the parameters of the underlying stochastic process. If cumulants are not known, suitable unbiased estimators relying on κ-statistics might be employed. To check the feasibility of the method both in fitting the density and in estimating the parameters, the first passage time problem of a geometric Brownian motion is considered.

Virasoro Versus Superintegrability. Gaussian Hermitian Model

Relation between the Virasoro constraints and KP integrability (determinant formulas) for matrix models is a lasting mystery. We elaborate on the claim that the situation is improved when integrability is enhanced to super-integrability, i.e. to explicit formulas for Gaussian averages of characters. In this case, the Virasoro constraints are equivalent to simple recursive formulas, which have appropriate combinations of characters as their solutions. Moreover, one can easily separate dependence on the size of matrix, and deduce superintegrability from the Virasoro constraints. We describe one of the ways to do so for the Gaussian Hermitian matrix model. The result is a spectacularly elegant reformulation of Virasoro constraints as identities for the Schur functions evaluated at appropriate loci in the space of time-variables.

Shape of Numbers and Calculation Formula of Stirling Numbers

This article investigated the idea of Shape of numbers and introduced new operators. The idea divides all products of k distinct integers in [1, n - 1] into 2K-1 catalogs and derives the calculation formula of every catalog. As a simple deduction, a direct formula of the Stirling numbers of the first kind S1(n, n - k) and a simple recursive formula of Stirling numbers of the second kind S2(n, n - k) are obtained. By analyzing the Shape of numbers, new congruence formulas are obtained.

Probabilistic Models for Vehicle Interactions in Fatigue Assessment of Bridges

Fatigue assessments of bridges depend on vehicle interactions, occurring when several vehicles travel simultaneously on the bridge or when two individual stress histories, caused by vehicles traveling in different times, generate a more damaging combined stress history. When interactions are significant, stress histories cannot be directly determined using conventional fatigue load models provided in Codes, unless suitable theoretical models for vehicle interactions are available. In the paper, original approaches are proposed to study different aspects of the problem. Concerning interactions due to simultaneity, the novelty is to consider the bridge a service system. Since the process of vehicle arrivals is a Markov process, vehicle interactions can be studied in the framework of the queuing theory. In this way, in the appropriate context, interacting vehicles are equivalent to queued requests (vehicles) in the service system. The method considers two subcases, to be tackled in the given sequence, so that the solution is noticeably simplified. The first subcase refers to vehicles traveling simultaneously in one lane; the second subcase to vehicle and vehicle convoys traveling simultaneously on two or more lanes. In the first subcase the problem is solved considering each lane as a single channel system with a waiting queue, where the number of vehicles in the queue and the waiting time, depending on the number of vehicles in the queue, are limited. A modified vehicle flow on each bridge lane is thus obtained, composed by vehicles and vehicle convoys separately traveling the lane, which is, if relevant, the input for the second subcase. In the second subcase the multilane bridge is modeled as a multichannel system without the waiting queue. When the number of requests exceeds the number of channels, r , the surplus is lost and cannot reenter the system. The results regarding simultaneity are much more relevant than it appears at the first sight: Two relevant examples demonstrated that they can be fruitfully used to implement optimized Monte Carlo algorithms for artificial traffic generation, as well for adaptation of traffic measurements, when flows are modified. Finally, a “non-interacting traffic flow” is obtained, whose elements (vehicle, vehicle convoy, cluster of vehicles) travel individually on the bridge. The global stress history results thus a mere random assembly of the stress histories induced by each element of the non-interacting traffic flow. These stress histories can only combine, as simultaneity interactions are excluded for them. Combination of stress histories is a complex issue, especially when, as in the Eurocodes, fatigue load models are composed by a set of standardized lorries. In fact, questions concerning: Conditions for the combination; stress history which can combine; expected number of occurrences of combined stress histories and of the remaining individual ones; are still open. Really, they can be tackled resorting to sophisticated and time-consuming simulations based on Monte Carlo methods, but elementary solutions have not been proposed so far. The original method proposed here, whose practical application is illustrated referring to an important case study, allows to solve the problem providing simple recursive formulae. Finally, two relevant examples illustrate, with specific reference to the Eurocodes, some important implications of the study.

A new recursive formulation of the Tau method for solving linear Abel–Volterra integral equations and its application to fractional differential equations

In this paper, the recursive approach of the Tau method is developed for numerical solution of Abel–Volterra type integral equations. Due to the singular behavior of solutions of these equations, the existing spectral approaches suffer from low accuracy. To overcome this drawback we use Muntz–Legendre polynomials as basis functions which have remarkable approximation to functions with singular behavior at origin and express Tau approximation of the exact solution based on a sequence of basis canonical polynomials that is generated by a simple recursive formula. We also provide a convergence analysis for the proposed method and obtain an exponential rate of convergence regardless of singularity behavior of the exact solution. Some examples are given to demonstrate the effectiveness of the proposed method. The results are compared with those obtained by existing numerical methods, thereby confirming the superiority of our scheme. The paper is closed by providing application of this method to approximate solution of a linear fractional integro-differential equation.

New fractional Lanczos vector polynomials and their application to system of Abel–Volterra integral equations and fractional differential equations

In this paper, the recursive approach of the τ-method is developed to construct new fractional order canonical polynomials for solving systems of Abel–Volterra integral equations. Due to the singular behavior of solutions of these equations, the existing spectral approaches suffer from low accuracy. To overcome this drawback we use Müntz polynomials as basis functions which give remarkable approximation to functions with singular behavior at origin and state Tau approximation to the exact solution based on a new sequence of basis vector canonical polynomials that is generated by a simple recursive formula in terms of fractional order Müntz polynomials. The efficiency and simplicity of the proposed method are illustrated by some examples. Convergence analysis of the method is also discussed. The paper is closed by providing application of this method to a linear multi-term fractional differential equations.

Solution procedures for lost sales base-stock inventory systems with compound Poisson demand

The continuous review base-stock policy under the lost sales assumption has been extensively studied in the inventory literature. However, most of the research that deals with the optimal policy parameter determination under compound Poisson demand focuses on the stuttering Poisson case. In this paper, we extend earlier research by providing a method that calculates the best cost-driven base-stock policy under more general compound Poisson demand processes and for both the complete and the partial rejection case. This is done by developing a simple recursive formula that computes the steady-state probabilities of the number of outstanding orders for general demand and lead-time distributions. Based on this, we are able to obtain the optimal base-stock level under the complete rejection case and an approximate base-stock level under the partial rejection policy. Through a numerical investigation, we show that our approach outperforms existing methods with regard to computational requirements.

A new approach for Robot selection in manufacturing using the ellipsoid algorithm

The choice of suitable robots in manufacturing, to improve product quality and to increase productivity, is a complicated decision due to the increase in robot manufacturers and configurations. In this article, a novel approach is proposed to choose among alternatives, differently assessed by decision makers on different criteria, to make the final evaluation for decision-making. The approach is based on the ellipsoid algorithm for systems of linear inequalities. Most of the ranking methods depend on integration that becomes complicated for nonlinear membership functions, which is the case in robot selection. The method simply uses the membership function or its derivative. It takes the decision maker’s attitude in ranking. It effectively ranks fuzzy numbers and their images, preserving symmetry. It is a simple recursive algebraic formula that can be easily programmed. The performance of the algorithm is compared with the performance of some existing methods through several numerical examples to illustrate its advantages in ranking, and a robot selection problem is solved.

Nonparametric estimation of the multivariate survivor function: the multivariate Kaplan–Meier estimator

The Dabrowska (Ann Stat 16:1475-1489, 1988) product integral representation of the multivariate survivor function is extended, leading to a nonparametric survivor function estimator for an arbitrary number of failure time variates that has a simple recursive formula for its calculation. Empirical process methods are used to sketch proofs for this estimator's strong consistency and weak convergence properties. Summary measures of pairwise and higher-order dependencies are also defined and nonparametrically estimated. Simulation evaluation is given for the special case of three failure time variates.

Handling boundaries with the one‐dimensional first‐order recursive filter

In variational data assimilation, a crucial task is to determine the background‐error covariance matrix B. The effect of B on a field is often modelled through a series of operators among which is a correlation operator. The recursive filter, when properly normalized to ensure the maximum of the solution is 1, is a convenient correlation operator and is widely used as such. Often, multi‐dimensional operators are constructed from the product of one‐dimensional operators. When their coefficients are calculated appropriately, the normalized one‐dimensional first‐order recursive filter applied N times models an autoregressive function of order N.In ocean data assimilation, an extra difficulty is to handle, within the correlation operator, the boundaries formed by the different coastlines or the bathymetry. Handling the east/west wrapping or polar folds in global configurations are also part of the issue. The purpose of this article is to set up a mathematical framework for using the recursive filter in ocean data assimilation and properly accounting for boundaries. Generally, these problems are dealt with by calculating proper normalization factors using costly methods, or extending the grid near boundaries. It is shown that the normalization factors can be calculated inexpensively through an analytical formula with a corrective term to handle properly the boundary when Neumann or Dirichlet boundary conditions are used.In its classical formulation however, the recursive filter uses Robin (third type) boundary conditions. This formulation can be slightly modified in order to account for Neumann, Dirichlet or periodic boundary conditions. To do so, extra coefficients are calculated through a simple recursive formula. In a global 1/4° analysis framework, this new formulation is shown to be less expensive and more accurate than the classical formulation associated with an appropriate grid extension.

Pessimistic Optimal Choice for Risk-Averse Agents: The Continuous-Time Limit

We extend Hansen and Sargent's (Discounted linear exponential quadratic gaussian control, 1994, IEEE Trans Autom Control 40:968---971 1995, 2013) analysis of dynamic optimization with risk-averse agents in two directions. Firstly, following Whittle (Risk-sensitive optimal control, 1990), we show that the optimal risk-averse policy is identified via a pessimistic choice mechanism and described by simple recursive formulae. Secondly, we investigate the continuous-time limit and show that sufficient conditions for the existence of optimal solutions coincide with those which apply under risk-neutrality. Our analysis is conducted both under perfect and imperfect state observation. As an illustrative example, we analyze the optimal production policy of an entrepreneur running a monopolistic firm which faces a demand schedule subject to stochastic shocks, showing that risk-aversion induces her to act more aggressively.

Augmented monomials in terms of power sums.

The problem of base changes for the classical symmetric functions has been solved a long time ago and has been incorporated into most computer software packages for symmetric functions. In this paper, we develop a simple recursive formula for the expansion of the augmented monomial symmetric functions into power sum symmetric functions. As corollaries, we present two algorithms that can be used to expressing the augmented monomial symmetric functions in terms of the power sum symmetric functions.

SU‐E‐T‐355: Efficient Scatter Correction for Direct Ray‐Tracing Based Dose Calculation

Purpose:To propose a scatter correction method with linear computational complexity for direct‐ray‐tracing (DRT) based dose calculation. Due to its speed and simplicity, DRT is widely used as a dose engine in the treatment planning system (TPS) and monitor unit (MU) verification software, where heterogeneity correction is applied by radiological distance scaling. However, such correction only accounts for attenuation but not scatter difference, causing the DRT algorithm less accurate than the model‐based algorithms for small field size in heterogeneous media.Methods:Inspired by the convolution formula derived from an exponential kernel as is typically done in the collapsed‐cone‐convolution‐superposition (CCCS) method, we redesigned the ray tracing component as the sum of TERMA scaled by a local deposition factor, which is linear with respect to density, and dose of the previous voxel scaled by a remote deposition factor, D(i)=aρ(i)T(i)+(b+c(ρ(i)‐1))D(i‐1),where T(i)=e^(‐αr(i)+β(r(i))^2) and r(i)=Σ_(j=1,..,i)ρ(j).The two factors together with TERMA can be expressed in terms of 5 parameters, which are subsequently optimized by curve fitting using digital phantoms for each field size and each beam energy.Results:The proposed algorithm was implemented for the Fluence‐Convolution‐Broad‐Beam (FCBB) dose engine and evaluated using digital slab phantoms and clinical CT data. Compared with the gold standard calculation, dose deviations were improved from 20% to 2% in the low density regions of the slab phantoms for the 1‐cm field size, and within 2% for over 95% of the volume with the largest discrepancy at the interface for the clinical lung case.Conclusion:We developed a simple recursive formula for scatter correction for the DRT‐based dose calculation with much improved accuracy, especially for small field size, while still keeping calculation to linear complexity. The proposed calculator is fast, yet accurate, which is crucial for dose updating in IMRT optimization.

On some compound distributions with Borel summands

The generalized Poisson distribution is well known to be a compound Poisson distribution with Borel summands. As a generalization we present closed formulas for compound Bartlett and Delaporte distributions with Borel summands and a recursive structure for certain compound shifted Delaporte mixtures with Borel summands. Our models are introduced in an actuarial context as claim number distributions and are derived only with probabilistic arguments and elementary combinatorial identities. In the actuarial context related compound distributions are of importance as models for the total size of insurance claims for which we present simple recursion formulas of Panjer type.

Two-way communication retrial queues with multiple types of outgoing calls

In this paper, we start with single server Markovian retrial queues with multiple types of outgoing calls. Incoming calls arrive at system according to a Poisson process. Service times of incoming calls follow the exponential distribution. Incoming calls that find the server busy upon arrival join an orbit and retry after some exponentially distributed time. On the other hand, the server makes an outgoing call after some exponentially distributed idle time. We assume that there are multiple types of outgoing calls whose durations follow distinct exponential distributions. For this model, we obtain explicit expressions for the joint stationary distribution of the number of calls in the orbit and the state of the server via the generating function approach. We also obtain simple asymptotic and recursive formulae for the joint stationary distribution. We show a stochastic decomposition property where we prove that the number of incoming calls in the system (server and orbit) can be decomposed into the sum of three independent random variables which have a clear physical meaning. We then consider the multiserver model for which we obtain the stability condition and derive some exact formulae by mean value analysis. Finally, we extend the single server model to the case where service time distribution of incoming calls and that of each type of outgoing calls are arbitrary. For this case, we obtain explicit expressions for the partial generating functions and recursive formulae for the joint stationary distribution of the server’s state and the number of calls in the orbit.

Pessimistic Optimal Choice for Risk-Averse Agents

We propose a general framework for the analysis of dynamic optimization with risk-averse agents, extending Whittle's (Whittle, 1990) formulation of risk-sensitive optimal control problems to accommodate time-discounting. We show how, within a Markovian set-up, optimal risk-averse behavior is identified via a pessimistic choice mechanism and described by simple recursive formulae. We apply this methodology to two distinct problems formulated respectively in discrete- and continuous-time. In the former, we extend Svensson's (Svensson, 1997) analysis of optimal monetary policy, showing that with a pessimistic central bank the inflation forecast is not longer an explicit intermediate target, the monetary authorities do not expect the inflation rate to mean revert to its target level and apply a more aggressive Taylor rule, while the inflation rate is less volatile. In the latter, we investigate the optimal production policy of a monopolistic entrepreneur which faces a demand schedule subject to stochastic shocks, once again showing that risk-aversion induces her to act more aggressively.

Analytical Model of Four-Wave Mixing in Semiconductor Optical Amplifier

The four-wave mixing process in pump-probe configuration requires the solutions of propagation equations of the three pulses: pump, probe and conjugate simultaneously. Since the pump power is much larger than the other pulses, this makes the differential equation of pump not coupled with the other equations, hence it can be solved alone. In this paper, we consider that the differential equation of pump is linear by assuming that the amplifier is divided into small sections to be the power change through each section is too small. Solving the resulting pump equation requires calculating the integrated gain, where the integrated gain is calculated after each section numerically by an iterative equation that was derived. Simultaneously, the solution of the probe and conjugate equations were also calculated after each section, where a simple recursive formula was found. Finally, the conversion efficiency will be calculated, where the present analysis agrees very well with the numerical solution.