Recursive Computation Research Articles

Object-Oriented Fixpoint Programming with Datalog

Modern usages of Datalog exceed its original design purpose in scale and complexity. In particular, Datalog lacks abstractions for code organization and reuse, making programs hard to maintain. Is it possible to exploit abstractions and design patterns from object-oriented programming (OOP) while retaining a Datalog-like fixpoint semantics? To answer this question, we design a new OOP language called OODL with common OOP features: dynamic object allocation, object identity, dynamic dispatch, and mutation. However, OODL has a Datalog-like fixpoint semantics, such that recursive computations iterate until their result becomes stable. We develop two semantics for OODL: a fixpoint interpreter and a compiler that translates OODL to Datalog. Although the side effects found in OOP (object allocation and mutation) conflict with Datalog's fixpoint semantics, we can mostly resolve these incompatibilities through extensions of OODL. Within fixpoint computations, we employ immutable algebraic data structures (e.g. case classes in Scala), rather than relying on object allocation, and we introduce monotonically mutable data types (mono types) to enable a relaxed form of mutation. Our performance evaluation shows that the interpreter fails to solve fixpoint problems efficiently, whereas the compiled code exploits Datalog's semi-naïve evaluation.

SBFEM with perturbation method for solving the Reynolds equation

AbstractRotordynamic simulations with nonlinear hydrodynamic bearing forces require a solution of the Reynolds equation at every time step. As a computationally efficient alternative to the standard numerical methods, a semi‐analytical solution based on the scaled boundary finite element method (SBFEM) was developed recently. Through a discretization of the hydrodynamic pressure (dependent variable) along the circumferential but not the axial coordinate, the partial differential equation is transformed into a system of ordinary differential equations. This system of differential equations is referred to as SBFEM equation and can be solved exactly if the influence of shaft tilting is neglected. In common numerical models, this influence can be taken into account without difficulties, but as far as this semi‐analytical approach is concerned, shaft tilting complicates the equations substantially. Therefore, previous studies on the SBFEM solution of the Reynolds equation were conducted without consideration of this effect. The formulation presented in the work at hand no longer requires this simplification. The terms representing the influence of shaft tilting in the SBFEM equation are handled by the perturbation method. The pressure field is expressed by a series expansion, where the solution of order correlates to the power of a perturbation parameter chosen proportional to the tilting angle. The differential equation governing the solution contains lower‐order solutions on its right‐hand side, implying a recursive computation of the series from lowest to highest order. A universal expression for the general solution is formulated, where only the coefficients and the maximum power of the axial coordinate differ for every . This allows the implementation of a general algorithm with no inherent limitation regarding the maximum order of perturbation. For verification, the pressure fields computed by the proposed method are compared to a numerical reference solution, showing that the series converges to the correct result for the investigated set of parameters.

Extracting attribute implications from a formal context: Unifying the basic approaches

There have been several pioneering approaches to the extraction of attribute implications from a formal context, dating from the 1980's: the one of Guigues and Duquenne based on so-called non-redundancy nodes, another one proposed by Ganter highlighting the concept of pseudo-closed set, and the best-known one relying on the recursive computation of so-called pseudo-intents in the book by Ganter and Wille. The Guigues and Duquenne approach has never been compared in detail in the literature with the other two, although they turn out to be equivalent. This paper tries to fill this gap, proposing a unified view, hopefully more easy to grasp.

Minimizing distance between distribution functions: discrete counterparts to continuous random variables with applications in non-life insurance and stochastic reliability

In this work, we propose a novel family of procedures for deriving a discrete counterpart to a continuous probability distribution. They are based on a class of distances between cumulative distribution functions, including the Cramér, the Cramér-von Mises, and the Anderson-Darling distances as particular cases. The discrete counterpart is defined and derived as the random variable which minimizes its distance to the assigned continuous probability distribution among all the discrete random variables supported on the set of integers (or positive integers). Applications are provided with reference to the exponential and the normal distributions, among others; the discrete counterparts are derived, and their main properties are discussed, also in comparison with the one obtained through an existing discretization technique based on the preservation of the cumulative distribution function at integer values. Parameter estimation for these discrete analogs is discussed, along with an analysis of two real datasets, where they are compared in terms of goodness-of-fit with some popular discrete distributions. Furthermore, in order to highlight the effectiveness and the benefits derived from the proposed discretization procedures, we illustrate two practical applications in actuarial science and in reliability engineering. In the former case, the problem of determining the distribution of the total claims amount for a non-life insurance portfolio is considered, where the claim sizes can be modelled as iid random variables, and the number of claims is random as well. Actuaries use a recursive calculation method based on Panjer's formula, which requires an appropriate discretization of the individual claim distribution, and therefore the proposed procedures can be used. Since we consider two simple cases where the distribution of the total claims amount is analytically acquirable, the efficacy of the discretization procedures in the final approximation can be easily assessed and turns out to be satisfactory, especially when compared to the existing discretization. The latter case considers the determination of the reliability parameter for a complex stress-strength model. Here, the approximation by discretization is compared to Monte Carlo simulation and shown to be relevant: with a comparable if not smaller computational effort, discretization leads to similar results as simulation. Such discretizations can also naturally be applied to more complex problems such as scenario generation in stochastic programming. R code for this article is provided as supplementary material.

Fast computation of local dips using the one-lag correlation method

We adapt the one-lag correlation (OLC) method to estimate the local dips of seismic events or velocity contrasts. The resulting local dips have extensive applications in exploration geophysics, including seismic inversion regularization, computation of seismic attributes like coherence-cube, and facilitating structural-oriented smoothing and extrapolation techniques. Compared to the widely adopted structure tensor (ST) method, the proposed OLC algorithm exhibits notable advantages. The OLC method is characterized by a recursive calculation process that yields significant computational efficiency, boasting an approximately 100 times faster performance for 3D scenarios. The fundamental principle of this novel approach lies in the computation of local dips by using the ratio of two inner products, i.e., OLC, performed on a 2D/3D array along the vertical and horizontal axes, respectively. Crucially, the elements of this array consist of complex numbers derived from the application of the Hilbert transform on the corresponding real-number inputs. Furthermore, the implementation of this algorithm is remarkably straightforward, as exemplified by the concise nature of the code, which comprises a mere ten lines of programming instructions. The OLC method excels in accuracy, surpassing the performance of the ST method, and exhibits superior resistance to random noise. Compelling demonstrations are provided herein to corroborate these desirable properties of the OLC approach, using both synthetic and field data examples.

Multi-Point Seawall Settlement Prediction with Limited Data Volume Using an Improved Fractional-Order Grey Model

Settlement prediction based on monitoring data holds significant importance for engineering maintenance of seawalls. In practical engineering, the volume of the collected monitoring data is often limited due to the restrictions of devices and engineering budgets. Previous studies have applied the fractional-order grey model to time series prediction under the situation of limited data volume. However, the performance of the fractional-order grey model is easily affected by the inappropriate settings of fractional order. Also, the model cannot make dynamic predictions due to the characteristic of fixed step size. To solve the above problems, in this paper, the genetic algorithm with enhanced search capabilities was employed to solve the premature convergence problem. Additionally, to solve the problem of the fractional-order grey model associated with fixed step size, the real-time tracing algorithm was introduced to conduct equal-dimensionally recursive calculation. The proposed model was validated using monitoring data of four monitoring points at Haiyan seawall in Zhejiang province, China. The prediction performance of the proposed model was then compared with those of the fractional-order GM(1,1), integer-order GM(1,1), and fractal theory model. Results indicate that the proposed model significantly improves the prediction performance compared to other models.

Errorless robust JPEG steganography using steganographic polar codes

Recently, a robust steganographic algorithm that achieves errorless robustness against JPEG recompression has been proposed. The method employs a lattice embedding scheme and utilizes the syndrome-trellis code (STC) for practical embedding. However, we have noticed that errorless robust embedding with STC may encounter failures due to modifications on wet coefficients, especially when a high quality factor is used by the compression channel. To solve this problem, we have discovered that using steganographic polar code (SPC) for embedding has better performance in avoiding modifications on wet coefficients. In this paper, we conduct theoretical analysis to prove the better performance of SPC in wet paper embedding. We establish the condition of avoiding modifications on wet coefficients, followed by presenting a recursive calculation method for determining the distribution of columns in the generator matrix of SPC. The findings reveal that SPC can avoid modifications on wet coefficients under a larger number of wet coefficients compared with STC, and therefore we propose a better errorless robust embedding method employing SPC. The experimental results demonstrate that under close security performance, the proposed method achieves a higher success rate compared with embedding with STC. Specifically, when the quality factor of the compressor is 95 and the payload size is 0.4 bpnzac, our method achieves a success rate of 99.85%, surpassing the 91.95% success rate of the embedding with STC.

A non-hierarchical multi-layer multi-configurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces.

The correlation discrete variable representation (CDVR) enables multi-layer multi-configurational time-dependent Hartree (MCTDH) quantum dynamics simulations on general potential energy surfaces. In a recent study [R. Ellerbrock and U. Manthe, J. Chem. Phys. 156, 134107 (2022)], an improved CDVR that can account for the symmetry properties of a tree-shaped wavefunction representation has been introduced. This non-hierarchical CDVR drastically reduces the number of grid points required in the time-dependent quadrature used to evaluate all potential energy matrix elements. While the first studies on the non-hierarchical CDVR approach have been restricted to single-layer calculations, here the complete theory required for the implementation of the non-hierarchical CDVR approach in the multi-layer MCTDH context will be presented. Detailed equations facilitating the efficient recursive computation of all matrix elements are derived, and a new notation adapted to the symmetry properties of the tree-shaped representation is introduced. Calculations studying the non-adiabatic quantum dynamics of photoexcited pyrazine in 24 dimensions illustrate the properties of the non-hierarchical multi-layer CDVR.

A recursive calculation method of vibration displacements using blade tip timing in angular domain

Blade tip timing (BTT) is a non-contact measurement technique that can be used to monitor blade vibration in turbomachinery. The raw data measured by BTT is the arrival time of all blades at the same stage. However, most blade vibration monitoring methods based on BTT require the use of blade vibration displacements, which are derived from the raw arrival time data, the blade rotation radius and speed. Conventional BTT methods calculate blade tip displacements by comparing the measured arrival time with the ideal arrival time. The ideal arrival time is obtained by assuming that the rotational speed is constant in a revolution, which is inconsistent with the fact, especially when the rotational speed changes rapidly. In this paper, a recursive calculation method called recursive BTT is proposed to reduce the influence of rotational speed fluctuation on the accuracy of blade tip displacement calculation. Recursive BTT calculates the displacements in the angular domain instead of the time domain. A recursive formula is constructed in the angular domain to calculate the blade vibration displacement by using the angle between the vibrating blades. Compared with the conventional BTT method, which relies on the once-per-revolution (OPR) signal, recursive BTT makes full use of the arrival time of all blades at the same stage to calculate the displacements. The advantage of recursive BTT is illustrated through a simulation, a laboratory test and a full-scale aeroengine test. To evaluate the reliability of the calculated displacements, strain signals are resampled to compare with the calculated displacements in the laboratory test. In the full-scale aeroengine test, due to the lack of strain signals, BTT displacements are used to track the blade natural frequency to qualitatively indicate the influence of displacements on blade vibration monitoring. The results of finite element analysis and modal test are used for verification in the full-scale aeroengine test. Simulation and experimental results demonstrate that compared with existing methods, recursive BTT is robust for calculating blade vibration displacements.

Recursive Filtering Under Probabilistic Encoding-Decoding Schemes: Handling Randomly Occurring Measurement Outliers.

This article focuses on the recursive filtering problem for networked time-varying systems with randomly occurring measurement outliers (ROMOs), where the so-called ROMOs denote a set of large-amplitude perturbations on measurements. A new model is presented to describe the dynamical behaviors of ROMOs by using a set of independent and identically distributed stochastic scalars. A probabilistic encoding-decoding scheme is exploited to convert the measurement signal into the digital format. For the purpose of preserving the filtering process from the performance degradation induced by measurement outliers, a novel recursive filtering algorithm is developed by using the active detection-based method where the "problematic" measurements (i.e., the measurements contaminated by outliers) are removed from the filtering process. A recursive calculation approach is proposed to derive the time-varying filter parameter via minimizing such the upper bound on the filtering error covariance. The uniform boundedness of the resultant time-varying upper bound is analyzed for the filtering error covariance by using the stochastic analysis technique. Two numerical examples are presented to verify the effectiveness and correctness of our developed filter design approach.

Evaluating Datalog over Semirings: A Grounding-based Approach

Datalog is a powerful yet elegant language that allows expressing recursive computation. Although Datalog evaluation has been extensively studied in the literature, so far, only loose upper bounds are known on how fast a Datalog program can be evaluated. In this work, we ask the following question: given a Datalog program over a naturally-ordered semiring σ, what is the tightest possible runtime? To this end, our main contribution is a general two-phase framework for analyzing the data complexity of Datalog over σ: first ground the program into an equivalent system of polynomial equations (i.e. grounding) and then find the least fixpoint of the grounding over σ. We present algorithms that use structure-aware query evaluation techniques to obtain the smallest possible groundings. Next, efficient algorithms for fixpoint evaluation are introduced over two classes of semirings: (1) finite-rank semirings and (2) absorptive semirings of total order. Combining both phases, we obtain state-of-the-art and new algorithmic results. Finally, we complement our results with a matching fine-grained lower bound.

Appointment scheduling for logistics parks under truck heterogeneity and order fulfillment delay

ABSTRACT Appointment scheduling is critical for logistic parks of manufacturing industry firms to manage inventory turnover and schedule service objects. This work studies the appointment scheduling problem of a logistic park considering truck heterogeneity and order fulfillment delay, where the number of time windows in a day, the number of each type of appointment slot, and the service schedule in each time window are decided. The objective is to minimise the expected total cost of the truck waiting time, inventory, and overtime. We introduce a weighted shortest processing time first scheduling rule to determine the service sequence in each time window and prove that it generates optimal scheduling solutions. Based on our proposed scheduling rule, we develop a recursive calculation to formulate the cost function, which provides a tractable approach to solving combinatorially complex problems. The proposed model is solved by a dual-selection based genetic algorithm (DSGA). The performance of the DSGA is validated by comparing it to a genetic algorithm using a single tournament selection operation. Our experimental results show that the DSGA enlarges the size of subgroups with better fitness and therefore performs satisfactorily.

Design of an English vocabulary e-learning recommendation system based on word bag model and recurrent neural network algorithm

This article takes learning English vocabulary as the foundation, constructs precise classification strategies, and designs a recommendation system for English vocabulary learning. This article constructs a recommendation system for English vocabulary learning based on the word bag calculation model and recursive neural network calculation method, which has great significance, and also conducts in-depth research on this system. Thus, a classification calculation method based on the word bag calculation model was proposed. The calculation method in this paper is different from traditional calculation methods, and the classification accuracy in this paper is higher. Meanwhile, recurrent neural networks themselves have one or more feedback loops that can feed back information from the neural network to other neural networks, resulting in recurrent networks with different structures. After a series of research results, it has been found that the SIFT method takes longer to extract than the SURF method, and the number of extracted features is relatively large. Therefore, the recommendation system for English vocabulary learning proposed in this article can be improved according to the different needs of users, in order to better meet their individual requirements.

Convergence of datalog over (Pre-) Semirings

Recursive queries have been traditionally studied in the framework of datalog, a language that restricts recursion to monotone queries over sets, which is guaranteed to converge in polynomial time in the size of the input. But modern big data systems require recursive computations beyond the Boolean space. In this article, we study the convergence of datalog when it is interpreted over an arbitrary semiring. We consider an ordered semiring, define the semantics of a datalog program as a least fixpoint in this semiring, and study the number of steps required to reach that fixpoint, if ever. We identify algebraic properties of the semiring that correspond to certain convergence properties of datalog programs. Finally, we describe a class of ordered semirings on which one can use the semi-naïve evaluation algorithm on any datalog program.

Finite Element Analysis for Linear Viscoelastic Materials Considering Time-Dependent Poisson’s Ratio: Variable Stiffness Method

For linear viscoelastic materials, this paper proposes a finite element analysis method based on an integral constitutive relationship that can simultaneously consider the relaxation behavior of the elastic modulus and the creep Poisson’s ratio. Firstly, the generalized Maxwell model is employed to depict the relaxation characteristics of the elastic modulus, while the generalized Kelvin model is used to represent the creep Poisson’s ratio. Subsequently, the element relaxation stiffness matrix is established, thereby forming a convolutional finite element equation. Furthermore, the recursive calculation of the convolutional integral is derived, and the calculation steps of the finite element for viscoelasticity considering the time-dependent nature of both the elastic modulus and Poisson’s ratio are established. Finally, the accuracy of the proposed algorithm is verified through two numerical examples with linear viscoelastic material. The results indicate that the proposed variable stiffness method for the finite element analysis of linear viscoelastic materials can simultaneously consider the changes in the elastic modulus and Poisson’s ratio over time, thereby establishing a new path and idea for the more accurate simulation of viscoelastic materials’ mechanical properties. Compared with the initial strain method for linear viscoelastic materials, the variable stiffness method proposed in this paper effectively avoids the assumption of constant stress during the micro time interval, thus significantly enhancing the finite element calculation accuracy of linear viscoelastic materials. The proposed method establishes a simulation algorithm that matches existing commercial software with viscoelastic material experiments by considering the elastic modulus and Poisson’s ratio as material parameters.

Optimal investment-disinvestment choices in health-dependent variable annuity

This paper exploits the influence of the policyholder's health status on the optimal time at which the policyholder decides to stop paying health-dependent premiums and starts withdrawing health-dependent benefits from a variable annuity (VA) contract accompanied by a guaranteed lifelong withdrawal benefit (GLWB). A mixed continuous-discrete time model is developed to find the optimal time for withdrawal regime initiation. The model determines the investment and disinvestment triggers according to the market conditions for both dynamic and static cases. In the static case, the optimal time is computed at the policy's inception time. In contrast, in the dynamic case, the optimal initiation time is achieved by recursive calculation of the exercise frontier of a real deferral option. Another finding is the sensitivity analysis of the contract concerning the insurance fee and the age of the policyholder.

Distributed Energy-Based Estimation Over Harvesting-Constrained Sensor Networks.

This article investigates the distributed joint state and fault estimation issue for a class of nonlinear time-varying systems over sensor networks constrained by energy harvesting. It is assumed that data transmission between sensors requires energy consumption, and each sensor can harvest energy from the external environment. A Poisson process models the energy harvested by each sensor, and the sensor's transmission decision depends on its current energy level. One can obtain the sensor transmission probability through a recursive calculation of the probability distribution of the energy level. Under such energy harvesting constraints, the proposed estimator only uses local and neighbor data to simultaneously estimate the system state and the fault, thereby establishing a distributed estimation framework. Moreover, the estimation error covariance is determined to possess an upper bound, which is minimized by devising energy-based filtering parameters. The convergence performance of the proposed estimator is analyzed. Finally, a practical example is presented to verify the usefulness of the main results.

Geometry of Enumerable Class of Surfaces Associated with Mylar Balloons

In this paper, the very fundamental geometrical characteristics of the Mylar balloon like the profile curve, height, volume, arclength, surface area, crimping factor, etc. are recognized as geometrical moments In(x) and In and this observation has been used to introduce an infinite family of surfaces Sn specified by the natural numbers n=0,1,2,…. These surfaces are presented via explicit formulas (through the incomplete Euler’s beta function) and can be identified as an interesting family of balloons. Their parameterizations is achieved relying on the well-known relationships among elliptic integrals, beta and gamma functions. The final results are expressed via the fundamental mathematical constants, such as π and the lemniscate constant ϖ. Quite interesting formulas for recursive calculations of various quantities related to associated figures modulo four are derived. The most principal results are summarized in a table, illustrated via a few graphics, and some direct relationships with other fundamental areas in mathematics, physics, and geometry are pointed out.

A Hardware Accelerator for the Semi-Global Matching Stereo Algorithm: An Efficient Implementation for the Stratix V and Zynq UltraScale+ FPGA Technology

The semi-global matching stereo algorithm is a top performing algorithm in stereo vision. The recursive nature of the computations involved in this algorithm introduces an inherent data dependency problem, hindering the progressive computations of disparities at pixel clock. In this work, a novel hardware implementation of the semi-global matching algorithm is presented. A hardware structure of parallel comparators is proposed for the fast computation of the minima among large cost arrays in one clock cycle. Also, a hardware-friendly algorithm is proposed for the computation of the minima among far-indexed disparity costs, shortening the length of computations in the datapath. As a result, the recursive path cost computation is accelerated considerably. The system is implemented in a Stratix V device and in a Zynq UltraScale+ device. A throughput of 55,1 million disparities per second is achieved with maximum disparity 128 pixels and frame resolution 1280 × 720. The proposed architecture is less elaborate and more resource efficient than other systems in the literature and its performance compares favorably to them. An implementation on an actual FPGA board is also presented and serves as a real-world verification of the proposed system.

Fast evaluation of generalized associated linear equations (GALEs) for nonlinear systems characterization and compensation

Nonlinear Output Frequency Response Functions (NOFRFs), which are a one-dimensional extension of the linear Frequency Response Function (FRF) to nonlinear cases, have been introduced for nonlinear system characterization and compensation. The evaluation of NOFRFs is accomplished by solving a series of linear difference or differential equations, known as Generalized Associated Linear Equations (GALEs). However, the derivation of GALEs often includes numerous recursive and symbolic calculations, posing significant challenges in both practical applications and computer programming. In the present study, a fast-computing algorithm, based on combinatorial calculations as opposed to recursive and symbolic calculations, is proposed for fast evaluation of GALEs. This allows for the evaluation of NOFRFs in a more streamlined and straightforward manner. The applications of the proposed fast-computing algorithm for nonlinear system characterization and compensation are demonstrated through several case studies including a High Power Amplifier (HPA) and a Duffing system. The results of the case studies show the effectiveness of applying the GALEs and NOFRFs for system design and safety control across a wide range of engineering applications.