In this paper, we propose a numerical method to construct an approximate solution to Fredholm integral equations of the first kind. These equations are common in image processing, spectroscopy, and other engineering sciences but are often ill-posed. Moreover, the right hand side of such equations could be error-contaminated. To effectively tackle the mentioned challenges, we propose suitable quadrature rules to discretize the continuous problem.The resulting discrete system is solved in a regularized least-squares framework by adopting the Deep-QLP factorization: a novel algorithm that automatically computes an approximate truncated singular value decomposition based on a user-defined tolerance. Additionally, two regularization approaches are introduced. In the first one, given a perturbed right-hand side, a Hankel matrix is constructed and then, the so-called de-Hankelization is performed by adopting the same Deep-QLP algorithm. In the second approach, we rely on locally refined discretization spaces to increase the accuracy of the computed approximate solution, by keeping low the number of the used degrees of freedom. Numerical experiments are conducted on some famous Fredholm integral equations available in the literature.
This paper develops a matrix analysis for recursive compact operators applied to the ninth-order Rosenau-Kawahara-RLW (nRK-RLW) equation with Novikov perturbations. Under periodic boundary conditions, two eighth-order difference schemes are constructed: two-level nonlinear Crank-Nicolson scheme (Scheme A) and three-level linearized scheme (Scheme B). Both schemes achieve accuracy of order O(τ2+h8) and preserve discrete energy. Combining matrix decomposition with discrete energy techniques, the existence, uniqueness, stability, and convergence of the numerical solutions are rigorously established. Numerical experiments confirm the theoretical results, demonstrating high accuracy and energy conservation over long integration times.
In this paper, we consider the variable selection for a class of partially linear single-index varying coefficient spatial autoregressive models. By combining bias correction and penalized estimation method, we propose a calibrated penalty based variable selection method. Under the appropriate conditions, we demonstrate that the proposed variable selection method can consistently select the true important covariates, and can also identify the significance of spatial effects. The asymptotic normality of the estimators for nonzero parameters is also derived under some regularity conditions. Several simulation analyses are conducted for further illustrating the performance of the proposed variable selection method, and the simulation results demonstrate that the proposed bias correction mechanism is effective and proposed variable selection method is workable under finite sample sizes.
The blind identification problem in MIMO (Multiple-Input Multiple-Output) technology involves determining the channel filter vectors based on the signals received by the receiver antennas. Covariance matrices computed from the received signals are utilized to find the filter vectors, requiring the computation of the inverse or pseudo-inverse of these matrices. In the context of MIMO, the covariance matrices are positive semi-definite Toeplitz or block Toeplitz matrices. In this article, we compute the generating function associated with the various covariance matrices and we determine their joint spectral properties in terms of localization, distribution, asymptotic conditioning, size of the ill-conditioned subspaces, by applying (asymptotic) linear algebra tools. Furthermore, the pseudo-inverse of the generalized preconditioners can be used as an approximation for the pseudo-inverse of the covariance matrices. In particular, we analyze the properties of Frobenius optimal matrix-algebra preconditioners, block-diagonal preconditioners as well as their generalized counterparts, for these singular covariance matrices. An estimate for the convergence rate of the generalized preconditioners is given and numerical experiments are presented for visualizing the theoretical findings.
Classical nonmonotone algorithms for continuous optimization rely on a fixed memory length, meaning they always consider a specific number of recent iterations. Our approach, in contrast, explores the dynamic adjustment of this storage factor in the context of a trust region method. Specifically, in our algorithm, the nonmonotonicity memory length is directly adapted based on the local noisy behavior of the cost function, quantified using the arithmetic–geometric mean inequality. We also propose adaptive strategies for setting the trust region radius by leveraging several recent scalar estimations of the second-order information of the model. These estimations are based on ellipsoid norm least-squares approximations of the secant equation, rather than the traditional Euclidean norm-based schemes. Incorporating such algorithmic improvements, we introduce a dynamically nonmonotone trust region algorithm and analyze its theoretical properties, specifically the global convergence.To validate our theoretical developments, we conduct numerical experiments evaluating the performance of the proposed algorithm on both benchmark test problems from the literature and real-world binary classification tasks in the presence of outliers. The computational results confirm the practical efficiency of the proposed trust region algorithm.
Electron-phonon coupling is a fundamental energy transfer process in fields such as laser detection, transistor heat generation, condensed matter physics, and thermoelectric materials, and it has extensive industrial applications and significant value for frontier scientific research. To study it, a semi-implicit Lax-Wendroff scheme was developed to simulate the electron-phonon coupling process in metals based on the two-temperature kinetic equation. The core of this method lies in integrating the transport and coupling processes of electrons/phonons into the numerical modeling process, thereby allowing the time step or cell size to not be limited by relaxation time and mean free path. Specifically, the kinetic model is solved again using the finite difference method when reconstructing the interfacial distribution function. Through this approach, the free migration of electrons/phonons, electron-electron scattering, phonon-phonon scattering and electron-phonon coupling processes are coupled together within a single time step. Numerical tests show that this method can effectively capture the electron-phonon coupling or heat conduction processes from ballistic to diffusive regimes. It provides a new tool for describing electron-phonon coupling or thermal management in microelectronic devices.
We introduce a constructive framework for generating compactly supported, infinitely smooth radial kernels with explicit and spectrally accurate cosine representations. The approach is based on infinite dyadic convolutions of Radon probability measures on R whose Fourier transforms are non-negative and non-vanishing, yielding a smooth, compactly supported limiting kernel whose Fourier transform belongs to the Schwartz class.Under a mild symmetry assumption, the resulting univariate kernel admits a cosine Fourier expansion that converges super-algebraically in the uniform norm. Crucially, this property is stable under repeated applications of Matheron’s dimension walk operator, without spoiling the super-algebraic convergence. This construction generates symmetric, positive definite, compactly supported multivariate kernels of class C∞ in arbitrary dimensions. Up to multiplicative factors, the expansion coefficients remain unchanged, while the univariate cosine expansions are replaced by explicit expansions in terms of confluent hypergeometric limit functions. For each dimension walk, we also establish uniform truncation error estimates that ensure spectral accuracy.The resulting kernels differ fundamentally from classical compactly supported polynomial constructions in that their smoothness and spectral decay allow for efficient numerical truncation and precomputation. Several explicit examples of the proposed kernels are constructed in the final section, illustrating the flexibility of the dyadic approach and its relevance to kernel-based approximation methods.
In this paper, based on the MINMOD scheme and the construction criterion of high-resolution format, a modified high-resolution combination scheme MMINMOD is presented, which theoretically can achieve at least second-order accuracy and meets the boundedness requirements. Secondly, in order to ensure that the stability and convergence for solving the discrete algebraic equations given by MMINMOD, the sufficient conditions for bounded and convergence of numerical computation are given in the form of theorem. Finally, taking into account the delay correction technique and the constraints on γ, the performance of the MMINMOD is tested and analyzed by classical numerical examples, the test results show that the modified format MMINMOD has relatively better performance in calculation accuracy and high-resolution and can effectively balance the contradiction between accuracy and boundedness. Meanwhile, the MMINMOD format can also better simulate the abrupt changes of the step flow field with large gradient. Compared with the FUD, MINMOD and QUICK numerical formats, the MMINMOD on a relatively sparse calculation grids can achieve the calculation accuracy and implementation effect of the FUD, MINMOD and QUICK formats on relatively dense grids. And under the same calculation conditions, the residuals RE and AE calculated by the four numerical formats are monotonically decreasing in the order of FUD, MINMOD, QUICK and MMINMOD. And the convergence order tested in space by the MMINMOD also is relatively higher. All of those further verify the feasibility and effectiveness of the improved format MMINMOD.
Two closely related scenarios of bicycle wheel track design by free-form curves are investigated. In the first, the track of the bicycle front wheel is prescribed and the rear wheel track is obtained from the classical no-slip kinematics. For a rational Bézier front wheel track we write the bicycle equation as a Riccati equation with rational coefficients. We describe a solution that produces rear wheel tracks as piecewise rational Bézier/NURBS curves with guaranteed error bounds. We also show that for Pythagorean-hodograph curve as front wheel track, several expressions are further simplified. In the second, complementary viewpoint the rear wheel track is prescribed and the front wheel track is to be computed. It is proved that if the rear wheel track is given as a Pythagorean–hodograph curve, then the front wheel track can be described as a rational curve and hence admits an exact rational Bézier/NURBS representation. We give the track data explicitly in terms of the algebraic data of the rear wheel track. It is also proved that the Pythagorean-hodograph condition is genuinely special, since this is the only exact case for a CAD setting: for any other standard polynomial, rational or even trigonometric Bézier or spline curve type as rear wheel track, the associated front wheel track is not a polynomial/rational/trigonometric curve and therefore cannot be represented exactly as a standard CAD curve.
This study develops a Kelvin-Voigt (KV) parameter-scaled Helmholtz-filtered lattice Boltzmann method (LBM) for the incompressible KV momentum equation. The KV stress term is reformulated as a macroscopic force and incorporated into the lattice Boltzmann framework via a consistent forcing scheme. Chapman-Enskog (CE) analysis demonstrates that the proposed method recovers the target KV momentum equation within the low-Mach-number limit. At the fully discrete level, the backward-difference approximation of the KV velocity time derivative yields a two-step amplification system. A Fourier/von Neumann stability analysis is conducted based on an augmented amplification matrix to quantitatively evaluate the numerical stability characteristics. To suppress the high-wavenumber effect introduced by the velocity time-difference term in the KV force, a Helmholtz filter scaled by the KV parameter is applied exclusively to this term. The stability properties of the resulting filtered scheme are investigated through spectral-radius analysis for fixed parameter sets and systematic scans of stability regions over a range of relevant parameters, with comparisons made against the backward-difference formulation. Manufactured-solution tests are conducted to validate the accuracy of the proposed κ-scaled Helmholtz-filtered LBM, demonstrating that the filter introduces only negligible perturbations in smooth and stable regimes while providing effective stabilization in high-frequency-dominated cases. Furthermore, a lid-driven cavity flow is simulated to assess the performance of the method in a physically relevant KV flow problem. The proposed filtering treatment improves the robustness of the fully discrete scheme while maintaining good accuracy within an appropriate parameter range.
This paper explores the finiteness of the solution set of the polynomial complementarity problem (PCP). To achieve this goal, we introduce two new classes of structured tensor tuples, namely the non-degenerate tensor tuple and the strong non-degenerate tensor tuple, as a generalization of non-degenerate tensors, and discuss their properties and interconnections. We investigate the finiteness of the solution set of the PCP in the context of these structured tensor tuples and establish a sufficient condition that guarantees a finite solution set. As a consequence, we establish a result related to the finiteness of the solution set of tensor complementarity problems.
Complex nonsymmetric indefinite linear systems arise in many scientific and engineering applications. However, efficient iterative methods for this class of problems remain much less developed than those for complex symmetric systems. In this paper, we propose an alternating Anderson–accelerated quasi– SSOR–Richardson method, referred to as the A3SRreal method, for solving complex nonsymmetric indefinite linear systems. The proposed method employs a novel quasi-SSOR preconditioner within the Richardson framework to effectively exploit the underlying problem structure, while Anderson acceleration is incorporated to further improve convergence. A rigorous convergence analysis shows that the A3SRreal method converges unconditionally when the optimal parameters ω*=σ*=1 are used. Numerical experiments demonstrate that the proposed method is efficient and robust, outperforming several state-of-the-art solvers, including PGSOR, PRESB–GMRES, HSS–GMRES, and PMHSS–GMRES. Furthermore, the influence of parameter selection and the acceleration effect of Anderson acceleration are investigated in detail.
As a classic first-order optimization technique, the stochastic conjugate gradient method stands as one of the most effective approaches for addressing large-scale and noisy data-driven tasks in fields like machine learning and signal processing. In this work, we propose a novel stochastic conjugate gradient method tailored for machine learning problems, integrating a custom-designed conjugate parameter and a structured restart procedure to construct efficient search directions, coupled with the weak Wolfe line search for adaptive step size selection. For smooth and strongly convex objective functions, we establish that the proposed algorithm achieves a linear convergence rate without invoking the restrictive assumption of non-increasing gradient estimate norms. Notably, this convergence guarantee holds independently of the adopted line search strategy, enhancing the algorithm’s robustness and practical applicability. Extensive experimental evaluations on four representative machine learning models (encompassing convex, nonconvex, and nonsmooth scenarios) demonstrate that the proposed algorithm outperforms both non-randomized conjugate gradient counterparts and state-of-the-art stochastic gradient-type algorithms in terms of convergence speed and numerical stability.
This paper focuses on the fast numerical solution of initial-boundary value problems for space-time fractional advection-diffusion equations. The forward difference formula and Grünwald formula are used to discretize the equations, resulting in a structured linear system. To accelerate the convergence of the GMRES method for solving this system, an approximate inverse preconditioner is constructed and applied. The proposed preconditioning method is first based on the splitting approximation of the coefficient matrix. Then the inverse is approximated row by row and the involved Toeplitz matrix is approximated by the τ matrix. The spectral distribution of the preconditioned matrix is theoretically analyzed, demonstrating that its eigenvalues cluster around 1. Numerical results compared with other preconditioning methods show that the proposed preconditioner exhibits higher computational efficiency.
