Differential Approximation Research Articles

On the accuracy of calculating invariants in centered rarefaction waves and in their influence area

We perform a comparative analysis of the accuracy of second-order TVD (Total Variation Diminishing), third-order RBM (Rusanov-Burstein-Mirin), and fifth-order in space and third-order in time A-WENO (Alternative Weighted Essentially Non-Oscillatory) difference schemes for solving a special Cauchy problem for shallow water equations with discontinuous initial data. The exact solution of this problem contains a centered rarefaction wave and does not contain a shock wave. It is shown that in the centered rarefaction wave and its influence area, the solutions of these three schemes with different orders converge to different invariants of the exact solution. This leads to a decrease in the accuracy of these schemes when calculating the vector of base variables of the considered Cauchy problem. The P-form of the first differential approximation of the difference schemes is used for the theoretical justification of these numerical results.

Design and implementation of dynamic s-boxes based on non-degenerate discrete chaotic systems

In recent years, digital chaotic systems have received considerable attention in the field of secure communications. However, during the digitalization of the system, the original properties of the chaotic system may change, resulting in the degradation of the dynamics. To address this problem, this paper designs a novel simplicial non-degenerate discrete chaotic system based on the inverse hyperbolic tangent function, and selects a three-dimensional discrete system as the object of analysis. Through the research conducted, it is found that the chaotic system exhibits high Lyapunov exponents under certain conditions. Furthermore, the excellent randomness of the system has been further validated by NIST SP800–22 tests. At the same time, this paper also proposes a dynamic S-box construction method based on the chaotic sequence generated from the three-dimensional chaotic mapping. By performing a series of basic operations and permutation treatment, many dynamic S-boxes can be generated. This paper comprehensively analyses the performance of S-boxes from two perspectives: single S-boxes and multiple S-boxes. The analysis covers issues such as bijective property, nonlinearity, strict avalanche criterion, differential approximation probability and bit independence criteria. The results of the performance analysis show that the dynamically generated S-boxes have excellent cryptographic properties, making them suitable for the design and application of cryptographic algorithms.

Trajectory optimization for aerodynamically controlled missiles by chance-constrained sequential convex programming

The flight environment of aerodynamically controlled missiles is full of complexity and uncertainty. To cope with the uncertainty more effectively and enhance the convergence performance in trajectory optimization problems for aerodynamically controlled missiles simultaneously, the chance-constrained sequential convex programming (CC-SCP) algorithm is proposed in this paper. The uncertainty is regarded as the chance constraint, and a smooth and differential approximation function is designed to transform this chance constraint into the constraint that the convex optimization method can handle. Subsequently, the originally non-convex trajectory optimization problem is reformulated into a series of convex optimization subproblems, in which an initial reference trajectory guess generation strategy is proposed, and a theoretical proof of the exact convex relaxation is given to enhance the algorithm's convergence performance and theoretical value, respectively. Numerical simulations are provided to verify the convergence and effectiveness of the CC-SCP algorithm, and the advantages of using the CC-SCP algorithm to cope with the uncertainty are illustrated. Furthermore, comparative simulation examples show that the proposed algorithm possesses a low conservatism, which means the proposed algorithm can obtain a bigger convergence region and a better solution than other current methods when handling the same chance constraints. Finally, the robustness of the algorithm is discussed.

A boundary strip indicator for material distribution-based topology optimization

This article proposes a boundary strip indicator for density-based topology optimization that can be used to estimate the design’s surface area (perimeter in 2D) or identify a coating layer. We investigate the theoretical properties of the proposed boundary strip indicator and propose a differentiable approximation that preserves key properties, such as non-negativity. Finally, we use the boundary strip indicator in a heat conduction design optimization problem for a coated structure. The resulting designs show a strong dependence on the properties of the coating.

Block Cipher Nonlinear Component Generation via Hybrid Pseudo-Random Binary Sequence for Image Encryption

To analyze the security of encryption, an effectual encryption scheme based on colored images utilizing the hybrid pseudo-random binary sequence (HPRBS) and substitution boxes, known as S-boxes, is proposed. The presented work aims to design S-boxes using pseudo-random binary numbers acquired by Linear Feedback Shift Registers (LFSRs) in combination with a modified quadratic chaotic map. Firstly, cryptographically robust S-boxes are constructed by using binary pseudo-random number sequences, and then the cryptographic properties of the presented S-boxes are tested. The suggested S-boxes showed good results. Secondly, an RGB image encryption algorithm utilizing sequences generated by modified quadratic chaotic maps and S-boxes is offered. The new color image encryption techniques comprise two steps, including a permutation and a substitution step. The key association with the content of the image is also addressed. This strategy can result in a “one-time pad” effect and make the algorithm resistant to chosen-plaintext attack (CPA). The proposed scheme has been confirmed to be more valuable than most of the existing schemes. S-boxes are analyzed by the nonlinearity test, bit independence criterion (BIC), linear and differential approximation probabilities (LPs; DPs), and Strict-Avalanche Criterion (SAC) tests. A comparison with different S-boxes presented in the literature is also carried out. The comparison shows encouraging results about the quality of the proposed box. From security and experimental outcomes, the effectiveness of the presented color image encryption technique is verified. The proposed scheme has evident efficiency benefits, which implies that the proposed colored encryption of the image scheme has better potential for application in encryption schemes in real-time.

COBLAH: A chaotic OBL initialized hybrid algebraic-heuristic algorithm for optimal S-box construction

The Substitution box (S-box) is the main nonlinear component responsible for the cryptographic strength of any Substitution-Permutation Network (SPN) based block cipher. Generating the S-box with optimal cryptographic properties is one of cryptography's most challenging combinatorial problems because of its enormous search space, lack of guidance, and conflicting performance criteria. This paper introduces a novel Chaotic Opposition-based Learning Initialized Hybrid Algebraic-Heuristic (COBLAH) algorithm, combining the favorable traits of Algebraic and heuristics methods based on Galois field inversion, affine mapping, and Genetic Algorithm (GA). The Galois field inversion and affine mapping are used to construct the S-box, while the GA guides the algebraic construction to find the best bit-matrix and additive vector based on any irreducible polynomial for GF(28). GA initializes with a random population generated using a newly constructed cosine-cubic map incorporated with binarization and Opposition-based Learning (OBL). Further, Multi-Objective Optimization Ratio Analysis (MOORA) is utilized to identify the best S-box from the final optimized population. The performance of the proposed algorithm is evaluated by comparing the generated COBLAH S-box with more than twenty state-of-the-art S-boxes, including Advanced Encryption Standard (AES), Skipjack, Gray, and Affine Power Affine (APA). The COBLAH S-box has nonlinearity 112, Strict Avalanche Criterion (SAC) offset 0.0202, Distance to SAC (DSAC) 332, Differential Approximation Probability (DP) 0.0625, Linear Approximation Probability (LP) 0.0156, Bit Independence Criterion-Strict Avalanche Criterion (BIC-SAC) 0.50006, and Bit Independence Criterion-Nonlinearity (BIC-NL) 112, which stands as the optimal observed thus far. The absence of fixed and opposite fixed points and the fact that it adheres to a single cycle aligns the COBLAH S-box with an ideal S-box. In addition, an image encryption mechanism is utilized to encrypt and decrypt the different images sourced from the standard USC-SIPI image dataset using COBLAH S-box and compared against different state-of-the-art S-boxes based on various image characteristics.

Differentiable approximation of continuous definable maps that preserves the image

Differentiable approximation of continuous definable maps that preserves the image

Echo-aware room impulse response generation.

In real-time applications, like interactive virtual reality environments, there is a significant need for low-complexity simulation of room impulse responses in highly complex virtual scenes, but this remains a challenging issue. In particular, simulating late reverberation using physically based acoustic modeling requires much computational effort, contrary to the early reflections that can be modeled by simpler techniques, e.g., the image source method. To tackle this computational complexity issue, we propose a neural network-based hybrid artificial reverberation framework (Echo2Reverb) that generates late reverberation from given early reflections. The proposed model can control both temporal texture and frequency-dependent energy decay, i.e., echo density and spectral energy distribution, of the generated reverberations by extracting spectral and echo-related features and filtering sampled sparse sequences and Gaussian noises using estimated features. To support the end-to-end training with controlled echo density, a differentiable approximation of the normalized echo density profile is proposed. We train and test the model not only for nearly diffuse but also distinct echoes prominent in late reverberations, such as with flutter echoes in narrow corridors. Evaluation results demonstrate that the proposed model can accurately reproduce frequency-dependent energy decay and temporal texture of a room impulse response using only early reflections.

Standalone gradient measurement of matrix norm for programmable unitary converters

Programmable unitary converters are powerful tools for realizing unitary transformations, advancing the fields of computing and communication. The accuracy of these unitary transformations is crucial for maintaining high fidelity in such applications. However, various physical artifacts can impair the accuracy of the synthesized transformations. A commonly employed approach uses the system’s gradient to restore accuracy. Matrix norm is used to define error between matrices, and minimization of this norm using the gradient restores the accuracy. Although this gradient can indeed be physically measured using external equipment, it leads to a rather complex optical system. In this study, we propose a standalone method for measuring matrix norm gradients, where “standalone” means that no additional optical equipment is needed. This method is based on the mathematical fact that the central difference, which is generally used for the approximation of differentiation, can yield exact differentiation for any unitary converters. Furthermore, we introduce a new matrix distance that is suitable for optimizing unitary converters that use intensity detectors at the output. This distance also yields the exact differentiation with the central difference. Numerical analysis demonstrates that our method exhibits orders of magnitude higher tolerance to measurement noise than prior similar approaches.

A superconvergence result in the RBF-FD method

Radial Basis Function-generated Finite Differences (RBF-FD) is a meshless method that can be used to numerically solve partial differential equations. The solution procedure consists of two steps. First, the differential operator is discretised on given scattered nodes and afterwards, a global sparse matrix is assembled and inverted to obtain an approximate solution. Focusing on Polyharmonic Splines as our Radial Basis Functions (RBFs) of choice, appropriately augmented with monomials, it is well known that the truncation error of the differential operator approximation is determined by the degree of monomial augmentation. Naively, one might think that the solution error will have the same order of convergence. We present a superconvergence result that shows otherwise - for some augmentation degrees, order of convergence is higher than expected.

Enhancing Demodulation Performance of DCM Algorithm in φ-OTDR System Through Temporal Spline Interpolation

For expanding the amplitude-frequency response range of the differential cross-phase multiply (DCM) algorithm in the φ-OTDR system, a temporal spline interpolation (TSI) method is proposed to pre-process Rayleigh backscattering (RBS) signals. Through the TSI method, the discrete temporal signals characterizing RBS traces are subjected to interpolation, facilitating a reduction in differential approximation errors. This, in turn, establishes a heightened level of precision in phase demodulation, especially relevant across extensive sensing distances. By comparing the recovered time-domain waveforms and the corresponding power spectral densities without and with the TSI, the above improvement effect has been experimentally validated by utilizing the TSI. The results show that, with the TSI, the amplitude-frequency response range of the DCM algorithm is enlarged by 2.78 times, and the new relationship among fpulse, f, and D under the root mean square error (RMSE) tolerance less than 0.1 can be expressed as 1.9(D+1)f ≤ fpulse. This contribution underscores a substantial advancement in the capabilities of the DCM algorithm, holding promise for refined performance in optical fiber sensing applications.

Computer-algebraic approach to first differential approximation: Van der Pol oscillator

First differential approximation has been used to analyze various numerical methods for solving systems of ordinary differential equations. This has made it possible to estimate the stiffness of the ODE system that models the oscillations of the Van der Pol oscillator and the error of the method as well as to propose simple criteria for choosing a calculation step. The presented methods allow one to perform efficient calculations using computer algebra systems.

Direct detection Φ-OTDR based on UWFBG array using linear-phase-modulated double-pulse

We propose what we believe to be a novel direct detection phase-sensitive optical time-domain reflectometry (Φ-OTDR) based on ultra-weak fiber Bragg grating (UWFBG) array to achieve distributed vibration measurements with exceptional sensitivity and remarkable stability. Our system employs a pulse modulator to generate a double pulse and achieves linear phase modulation of one pulse by one cycle through a phase modulator. The phase change can be quantitatively demodulated using our proposed N-step phase-shifted demodulation algorithm. This method effectively mitigates the influence of phase noise of the laser and the pulse modulator, while also eliminating fluctuations in the half-voltage of the phase modulator. Compared with the existing phase modulation methods, our method avoids stringent requirements for the stability and precision of phase modulation. Moreover, we propose a phase-shifted approximation method, breaking the limitation of sensing length on the traditional differential approximation method and improving the accuracy significantly. The technique's effectiveness is experimentally demonstrated on a 1 km UWFBG array with a reflectivity of −40 dB to −45 dB and a spatial resolution of 10 m. Vibrations with different amplitudes are measured quantitatively with good linearity. The low-frequency self-noise is greatly suppressed and the overall self-noise is −54.3 dB rad2/Hz.

Computer-Algebraic Approach to First Differential Approximations: Van der Pol Oscillator

Computer-Algebraic Approach to First Differential Approximations: Van der Pol Oscillator

An innovative design of substitution-box using Trigonometric-Multiplicative Functions Using Square Root Arguments: A Data-driven study.

In the evolving landscape of digital technology, the imperative for robust data security mechanisms has escalated, given the increasing sophistication of cyber threats. This abstract delineates a study focused on enhancing cryptographic defenses through the innovation of a Substitution box (S-Box), which is pivotal in the architecture of modern encryption algorithms. The proposed S-Box, deriving its foundation from chaotic maps integrated with trigonometric-multiplicative functions, represents a novel approach in cryptographic design, utilizing square root arguments to instigate dynamic characteristics. The evaluation of the proposed S-Box was methodically conducted using a comprehensive set of cryptographic benchmarks including Nonlinearity (NL), Strict Avalanche Criterion (SAC), Bit Independence Criterion (BIC), Linear Approximation Probability (LP), and Differential Approximation Probability (DP), to ascertain its defensive robustness against cryptanalytic attacks. The comparative analysis delineated in this study reveals that the cryptographic strength of the proposed S-Box transcends that of other contemporaneously designed S-Boxes thereby underscoring its potential applicability in real-world security scenarios. The findings of this research not only contribute to the theoretical underpinnings of cryptographic security but also have practical implications in the development of more secure digital environments fortifying data against unauthorized access and ensuring the integrity of confidential information in digital communications.

Near-integrable dynamics of the Fermi-Pasta-Ulam-Tsingou problem.

It is well known that the classic Fermi-Pasta-Ulam-Tsingou (FPUT) study of a chain of nonlinear oscillators is closely related to a number of completely integrable systems, including the Toda lattice. Here, we present a method that captures the departure of nonintegrable FPUT dynamics from those of a nearby integrable Toda lattice. Using initial long-wave data, we find that the former depart rather sharply from the latter near the predicted shock time of an asymptotic partial differential equation approximation, at which point energy cascades into higher lattice modes. Our method provides an appropriate frame of reference for one to distinguish the short-term dynamics of the two systems, whose macroscopic trajectories diverge noticeably only on a much longer timescale, when the FPUT dynamics thermalize.

Dynamic S-Box Construction Using Mordell Elliptic Curves over Galois Field and Its Applications in Image Encryption

Elliptic curve cryptography has gained attention due to its strong resilience against current cryptanalysis methods. Inspired by the increasing demand for reliable and secure cryptographic methods, our research investigates the relationship between complex mathematical structures and image encryption. A substitution box (S-box) is the single non-linear component of several well-known security systems. Mordell elliptic curves are used because of their special characteristics and the immense computational capacity of Galois fields. These S-boxes are dynamic, which adds a layer of complexity that raises the encryption process’s security considerably. We suggest an effective technique for creating S-boxes based on a class of elliptic curves over GF(2n),n≥8. We demonstrate our approach’s robustness against a range of cryptographic threats through thorough examination, highlighting its practical applicability. The assessment of resistance of the newly generated S-box to common attack methods including linear, differential, and algebraic attacks involves a thorough analysis. This analysis is conducted by quantifying various metrics such as non-linearity, linear approximation, strict avalanche, bit independence, and differential approximation to gauge the S-box’s robustness against these attacks. A recommended method for image encryption involves the use of built-in S-boxes to quickly perform pixel replacement and shuffling. To evaluate the efficiency of the proposed strategy, we employed various tests. The research holds relevance as it can provide alternative guidelines for image encryption, which could have wider consequences for the area of cryptography as a whole. We believe that our findings will contribute to the development of secure communication and data protection, as digital security is becoming increasingly important.

Construction of Piano Performing Arts Talent Cultivation and Creative Thinking Based on the Constant Differential Error Approximation Method

Abstract Piano talent cultivation mostly focuses on repetitive training only, and lacks a clear cultivation program for theoretical knowledge and skill points. In this paper, we adopt the curriculum knowledge graph to extract the knowledge point subgraph of piano learning and provide an initialized knowledge graph structure for the learning path under the cultivation scheme. A multi-objective optimization model of the path containing constraints such as the difficulty and mastery of knowledge points is set, and the relevant parameters of the model are defined and calculated to establish a multi-optimization objective function. A solution method that uses constant differential error approximation is proposed as a means of approaching the Pareto optimal solution. To develop the LSPIA algorithm with reciprocal weights, the adjustment vectors are modified using a least squares asymptotic iterative approximation with minimum squares values. Following the discovery of the most suitable route, a teaching experiment was carried out. The results show that the posttest P-value of the total creative thinking score of the experimental and control classes is 0.022 less than 0.05, which indicates that there is a significant difference between the creative thinking of the students in the two classes in terms of affective characteristics. The experimental group achieved a mean score of 122.69 on the post-test, which was a significant improvement from the mean score on the pre-test. An innovative approach to cultivating piano playing talents involves optimizing learning paths.

Parameter estimation of a model describing the human fingers.

The goal of this paper is twofold: firstly, to provide a novel mathematical model that describes the kinematic chain of motion of the human fingers based on Lagrangian mechanics with four degrees of freedom and secondly, to estimate the model parameters using data from able-bodied individuals. In the literature there are a variety of mathematical models that have been developed to describe the motion of the human finger. These models offer little to no information on the underlying mechanisms or corresponding equations of motion. Furthermore, these models do not provide information as to how they scale with different anthropometries. The data used here is generated using an experimental procedure that considers the free response motion of each finger segment with data captured via a motion capture system. The angular data collected are then filtered and fitted to a linear second-order differential approximation of the equations of motion. The results of the study show that the free response motion of the segments is underdamped across flexion/extension and ad/abduction.

Robust and Refined Salient Object Detection Based on Diffusion Model

Salient object detection (SOD) networks are vulnerable to adversarial attacks. As adversarial training is computationally expensive for SOD, existing defense methods instead adopt a noise-against-noise strategy that disrupts adversarial perturbation and restores the image either in input or feature space. However, their limited learning capacity and the need for network modifications limit their applicability. In recent years, the popular diffusion model coincides with the existing defense idea and exhibits excellent purification performance, but there still remains an accuracy gap between the saliency results generated from the purified images and the benign images. In this paper, we propose a Robust and Refined (RoRe) SOD defense framework based on the diffusion model to simultaneously achieve adversarial robustness as well as improved accuracy for benign and purified images. Our proposed RoRe defense consists of three modules: purification, adversarial detection, and refinement. The purification module leverages the powerful generation capability of the diffusion model to purify perturbed input images to achieve robustness. The adversarial detection module utilizes the guidance classifier in the diffusion model for multi-step voting classification. By combining this classifier with a similarity condition, precise adversarial detection can be achieved, providing the possibility of regaining the original accuracy for benign images. The refinement module uses a simple and effective UNet to enhance the accuracy of purified images. The experiments demonstrate that RoRe achieves superior robustness over state-of-the-art methods while maintaining high accuracy for benign images. Moreover, RoRe shows good results against backward pass differentiable approximation (BPDA) attacks.