L1-norm Research Articles

Seismic wavelet shape-oriented reflectivity inversion method

Abstract Reflectivity inversion plays a pivotal role in reservoir prediction. Conventional sparse-spike deconvolution assumes that the reflectivity (reflection coefficient) is sparse, which is solved based on the l1 norm. However, the restricted isometry property (RIP) of wavelet matrix and seismic effective bandwidth limits the accuracy of the sparse-spike reflectivity inversion. Consequently, we investigate the connection between seismic amplitude shape and reflectivity. When the reflectivity contains more non-zero values, the wavelet bandwidth within the effective seismic data bandwidth approaches a limit corresponding to the Sinc wavelet, where the main-lobe amplitude closely approximates the reflectivity. Conversely, when the reflectivity has fewer non-zero values, a wavelet with a smaller sidelobe provides a more accurate approximation of the reflectivity. In this paper, we propose a high-resolution inversion optimization method based on joint l2 norm and l1 norm constraints. By parameter tuning, we construct the Sinc wavelet or the wavelet with a weak-sidelobe corresponding to the seismic spectrum. Subsequently, we determine the extremum to approximate the reflectivity. To mitigate the RIP condition's constraints, we employ the l2 norm to balance the l1 norm (joint constraint) by introducing l2 norm with low-pass filtering characteristics. This approach yields more accurate reflectivity estimates. By taking the extremum, this approach yields more accurate reflectivity estimates. The synthetic test demonstrates that our method achieves better reflectivity inversion accuracy compared to sparse-spike inversion with l1–l2 norm constraint. Furthermore, field tests indicate that the proposed reflectivity inversion method not only can better match the well curve, but also exhibits excellent resolution.

Properties of eigenfunctions of a boundary value problem for ordinary differential equations of fourth-order with boundary conditions depending on the spectral parameter

In this paper, we consider an eigenvalue problem for fourth-order ordinary differential equations with a spectral parameter contained in all boundary conditions. We characterize the location of eigenvalues on the real axis, find their multiplicities, and study the oscillatory properties of eigenfunctions. Moreover, we obtain refined asymptotic formulas for the eigenvalues, as well as for the values at the endpoints of the interval of eigenfunctions and their derivatives. Then, using these results and the operator-theoretic formulation, we establish sufficient conditions for the system of eigenfunctions of this problem to form a basis in the space Lp,1<p<∞, after removing four functions.

An effective smoothing Newton projection algorithm for finding sparse solutions to NP-hard tensor complementarity problems

In this paper, we study the sparse solution of the tensor complementarity problem (TCP), which is an NP hard problem due to the nonconvexity and noncontinuity of the ℓ0 norm. We transform the complementarity constraints into a fixed point equation with projection type and use 0/1-loss function to relax ℓ0 norm. We develop a smoothing Newton projection (SNP) algorithm to solve the transformed problem. The subproblem with 0/1-loss function is solved by Newton method. Finally, numerical results show that the SNP algorithm can effectively get the sparse solution to a TCP.

Two-grid methods for nonlinear pseudo-parabolic integro-differential equations by finite element method

In this paper, two efficient two-grid finite element algorithms are proposed for solving two-dimensional nonlinear pseudo-parabolic integro-differential equations. Firstly, we obtain optimal error estimates of the fully discrete finite element method using a temporal-spatial error splitting technique in H1 and Lp norms. Then the two-grid technique to improve computation efficiency of the proposed finite element method. Error estimates in H1 and Lp norms of two-grid solutions are presented. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Finally, numerical examples are provided to support our theoretical results and demonstrate the effectiveness of these methods.

Machine Learning Algorithms for Predicting Wear Rates on the Basis of Friction Noise

Under varying operational conditions, the contact and relative movement of a polymer and metal result in surface wear, accompanied by the emission of noise. The relationship between friction noise and wear is inherently complex and nonlinear. In light of these tribological characteristics, this paper introduces the implementation of a random forest algorithm and generalized regression neural network algorithm to establish a mathematical model for predicting the wear rate based on friction noise. To enhance the accuracy of wear rate regression, this study incorporates L2 norm feature selection and the sparrow search algorithm, which are tailored toward the friction characteristics. These techniques optimize the machine learning-based friction model, ultimately improving the regression accuracy of the wear rate.

LP norm regularized deep CNN classifier based on biwolf optimization for mitosis detection in histopathology images

LP norm regularized deep CNN classifier based on biwolf optimization for mitosis detection in histopathology images

Semi-blind sparse channel estimation using regularized expectation maximization

In Massive multiple-input multiple-output (MIMO) systems, channel estimation is crucial. The large size of the antennas causes a significant pilot and feedback overhead, making it challenging to estimate channels in massive MIMO systems. Besides, the studies have shown that mmWave channels are sparse due to the limited number of dominant propagation paths. Therefore, the motivation of this paper is to exploit the inherent sparsity of the massive mmWave MIMO channels to develop a semi-blind channel estimation with reduced number of pilots. To this goal, an expectation maximization (EM) based technique has been developed which leverages the sparsity of the underlying channel for its better estimation. An iterative approach is proposed to solve the modeled problem which simultaneously updates the channel coefficients and data symbols using available data and system structure at each iteration. The proposed method imposes sparsity with the aid of Smoothed L0 norm (SL0) in the M-step. The simulation results demonstrate the proposed method have quick convergence and lower channel estimation error compared to the existing methods. As a quantitative evaluation, the proposed method attains the normalized mean square error of 9×10−2 and 7×10−3 at SNR=5dB and SNR=15dB, respectively.

Differentially private federated learning with local momentum updates and gradients filtering

Differential Privacy (DP) is applied in Federated Learning (FL) for defending against various privacy attacks. Existing methods based on Gaussian mechanism require the operations of clipping and adding noise, leading to significant accuracy degradation. In this paper, we propose a novel FL scheme named DPFL-LMG to provide user-level DP guarantee while maintaining a high model accuracy. Our main idea is to mitigate the negative effects of the clipping on the model convergence by decreasing the L2 norm of local updates and the cross-client update variance. Specifically, our method includes two techniques, Local Momentum Updates (LMU) and Gradients Filtering (GF). LMU combines local updates of different rounds in a momentum way. It can significantly decrease the cross-client update variance by weakening the gradient noise in local updates caused by stochastic gradient descent (SGD) algorithm. GF estimates the gradient noise in each element of local updates by observing the element-wise variance. Elements with large noise are considered unnecessary and are zeroed out for the reduction of local update norms. We theoretically analyze the privacy guarantee and the convergence of our method. Experiments demonstrate that DPFL-LMG can effectively mitigate the accuracy degradation caused by clipping and outperform previous DPFL methods in the accuracy.

Time two-grid fitted scheme for the nonlinear time fractional Schrödinger equation with nonsmooth solutions

In this paper, we delve into the regularity properties and the development of an efficient numerical strategy for addressing the nonlinear time-fractional Schrödinger equation. Initially, we embark on an examination of the solution’s regularity, followed by an investigation into regularity enhancement through the application of the Laplace and finite Fourier sine transforms. Subsequently, after the decomposition of u-u0, we introduce an innovative time two-grid fitted scheme designed for the equation’s resolution, which demonstrates unconditional stability and convergence in the L2 norm, achieving the global convergence order of O(τF2+τC4+h2). Here, τF and τC denote the temporal step sizes on the fine and coarse grids, respectively, while h represents the spatial step size. Notably, this is the inaugural application of such an approach for solving the nonlinear time fractional Schrödinger equation. Ultimately, our numerical experiments validate that this method not only retains computational efficiency but also significantly enhances convergence accuracy.

On the long-time behavior of the continuous and discrete solutions of a nonlocal Cahn–Hilliard type inpainting model

In this paper, we study the analytical and numerical long-time stability of a nonlocal Cahn–Hilliard model with a fidelity term for image inpainting introduced in our previous work (Jiang et al., 2024). First, we establish the uniform boundedness of the continuous problem in both L2 and H1 spaces, which is obtained by using the Gagliardo–Nirenberg inequality and the uniform Grönwall lemma. Then, for the temporal semi-discrete scheme, the uniform estimates in L2 and H1 spaces are derived with the aid of the discrete uniform Grönwall lemma under a suitable assumption on the nonlinear potential. This demonstrates the long-time stability of the proposed scheme in L2 and H1 spaces. Finally, we validate the long-time stability and the applicability of our method in signal reconstruction and image inpainting. These numerical experiments demonstrate the high effectiveness of our proposed model.

Generalized nonconvex nonsmooth four-directional total variation with overlapping group sparsity for image restoration

In this paper, we address the challenge of image deblurring in the presence of Gaussian noise. To achieve high-quality image restoration, we introduce an optimization framework that integrates four-directional total variation regularization, overlapping group sparsity, and a range of nonconvex penalties. This novel model effectively mitigates the staircase artifacts associated with total variation regularization and enhances restoration quality by leveraging domain-specific information about image pixels. Compared to the ℓ1 norm, the nonconvex penalty applied to overlapping groups promotes sparsity in image gradients at the group level. To solve this nonconvex optimization problem, we propose a proximal alternating reweighted minimization algorithm, which has a proximal alternating scheme with a reweighted approximation of its subproblem. Theoretically, we establish that the sequences generated by the proposed algorithm converge to a critical point using the Kurdyka–Łojasiewicz property. Experimental results validate the superiority of our approach over competing methods.

Schatten properties of Calderón–Zygmund singular integral commutator on stratified Lie groups

We provide full characterization of the Schatten properties of [Mb,T], the commutator of Calderón–Zygmund singular integral T with symbol b(Mbf(x):=b(x)f(x)) on stratified Lie groups G. We show that, when p is larger than the homogeneous dimension Q of G, the Schatten Lp norm of the commutator is equivalent to the Besov semi-norm BpQ/p of the function b; but when p≤Q, the commutator belongs to Lp if and only if b is a constant. For the endpoint case at the critical index p=Q, we further show that the Schatten LQ,∞ norm of the commutator is equivalent to the Sobolev norm W˙1,Q of b. Our method at the endpoint case differs from existing methods of Fourier transforms or trace formula for Euclidean spaces or Heisenberg groups, respectively, and hence can be applied to various settings beyond.

A multifluid model with chemically reacting components — Construction of weak solutions

We investigate the existence of weak solutions to a multi-component system, consisting of compressible chemically reacting components, coupled with the compressible Stokes equation for the velocity. Specifically, we consider the case of irreversible chemical reactions and assume a nonlinear relation between the pressure and the particular densities. These assumptions cause the additional difficulties in the mathematical analysis, due to the possible presence of vacuum.It is shown that there exists a global weak solution, satisfying the L∞ bounds for all the components. We obtain strong compactness of the sequence of densities in Lp spaces, under the assumption that all components are strictly positive. The applied method captures the properties of models of high generality, which admit an arbitrary number of components. Furthermore, the framework that we develop can handle models that contain both diffusing and non-diffusing elements.

A Study of Some Generalized Results of Neutral Stochastic Differential Equations in the Framework of Caputo–Katugampola Fractional Derivatives

Inequalities serve as fundamental tools for analyzing various important concepts in stochastic differential problems. In this study, we present results on the existence, uniqueness, and averaging principle for fractional neutral stochastic differential equations. We utilize Jensen, Burkholder–Davis–Gundy, Grönwall–Bellman, Hölder, and Chebyshev–Markov inequalities. We generalize results in two ways: first, by extending the existing result for p=2 to results in the Lp space; second, by incorporating the Caputo–Katugampola fractional derivatives, we extend the results established with Caputo fractional derivatives. Additionally, we provide examples to enhance the understanding of the theoretical results we establish.

Magnetic resonance image reconstruction based on image decomposition constrained by total variation and tight frame.

Magnetic resonance imaging (MRI) is a commonly used tool in clinical medicine, but it suffers from the disadvantage of slow imaging speed. To address this, we propose a novel MRI reconstruction algorithm based on image decomposition to realize accurate image reconstruction with undersampled k-space data. In our algorithm, the MR images to be recovered are split into cartoon and texture components utilizing image decomposition theory. Different sparse transform constraints are applied to each component based on their morphological structure characteristics. The total variation transform constraint is used for the smooth cartoon component, while the L0 norm constraint of tight frame redundant transform is used for the oscillatory texture component. Finally, an alternating iterative minimization approach is adopted to complete the reconstruction. Numerous numerical experiments are conducted on several MR images and the results consistently show that, compared with the existing classical compressed sensing algorithm, our algorithm significantly improves the peak signal-to-noise ratio of the reconstructed images and preserves more image details. Our algorithm harnesses the sparse characteristics of different image components to reconstruct MR images accurately with highly undersampled data. It can greatly accelerate MRI speed and be extended to other imaging reconstruction fields.

Depth imaging of multicomponent seismic data through the application of 2D full‐waveform inversion to P‐ and SH‐wave data: SEAM II Barrett model study

AbstractWe examine the value of the nine‐component seismic survey by generating the Kirchhoff depth migration images of compressional wave (P‐wave), converted wave (PS‐wave) and horizontally polarized shear wave (SH shear wave) data simulated from the SEAM II Barrett unconventional model. We first utilize full waveform inversion to obtain a P‐wave velocity model from P‐wave data and an S‐wave velocity model from SH‐wave data. Both P‐wave and SH‐wave data are generated with the maximum frequency of 20 Hz while assuming that the subsurface is isotropic. To implement full waveform inversion, we use a two‐dimensional time‐domain finite‐difference method and the L2 norm to measure the data misfit. We use both refractions and reflections in P‐ and SH‐wave data to reconstruct the P‐ and S‐wave velocity models from the surface to the reservoir. The inverted P‐ and S‐wave velocities contain the main features of the model (e.g., major faults and channels) but have some difficulties in estimating high‐frequency velocity variation within the first 300‐m depth of the model due to the frequency constraint. We then use the inverted P‐ and S‐wave velocities to generate Kirchhoff depth migration gathers and images from the P‐, PS‐ and SH‐wave data. The flat P‐ and SH‐wave common‐image offset gathers suggest that SH‐ and P‐wave full waveform inversion can generate adequate S‐ and P‐wave velocities for migration. Flat PS‐wave gathers and the clear PS‐wave migration image are also obtained using the inverted P‐ and S‐wave velocities simultaneously. This result indicates that obtaining S‐wave velocities from SH‐wave data can aid PS‐wave data processing and imaging. Moreover, the SH‐wave images and S‐wave images of the radial component provide better delineation of fault planes and small‐scale geobodies within the reservoir since the wavelength of the S‐wave is smaller compared to P‐wave when similar frequency ranges are recorded. Therefore, our study shows that S‐wave velocities can be successfully constructed by the two‐dimensional full waveform inversion application of the SH‐wave data. The subsequent imaging of multicomponent seismic data improves the delineation of certain unconventional reservoirs compared to the traditional P‐wave imaging.

Spectral structure of Moran Sierpinski-type measure on R2

Let Mn=diag[3pn,3qn] with pn,qn⩾1 for all n⩾1 and let D={(0,0)t,(1,0)t,(0,1)t} . One can generate a Borel probability measure μ{Mn},D=δM1−1D∗δ(M2M1)−1D∗δ(M3M2M1)−1D∗⋯. Such measure μ{Mn},D is called a Moran Sierpinski-type measure. It is known Deng et al (Acta Math. Sin. submitted) that the associated Hilbert space L2(μ{Mn},D) has an exponential orthonormal basis. In this paper, we first characterize all the maximal exponential orthogonal sets for L2(μ{Mn},D) . For such a maximal orthogonal set, we then give some sufficient conditions to determine whether it is an orthonormal basis of L2(μ{Mn},D) or not.

Maximum bound principle preserving and mass conservative projection method for the conservative Allen–Cahn equation

In this paper, we propose and analyze an efficient maximum bound principle (MBP) preserving and mass conservative projection method for the conservative Allen–Cahn equation. The proposed projection operator can be proven contractive in the discrete L2 norm. Discrete MBP and mass conservation are rigorously presented for a second-order exponential time differencing scheme with a central difference discretization in space. Various numerical examples are performed to verify these theoretical results and demonstrate the accuracy and robustness of the proposed scheme.

Virtual element discretization method to optimal control problem governed by Stokes equations with pointwise control constraint on arbitrary polygonal meshes

In this paper we investigate virtual element discretization of optimal control problem governed by Stokes equations. Based on the strategy of first-discretize-then-optimize we build up the virtual element discrete scheme and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L2 and H1 norms are derived. Numerical experiments are presented to verify the theoretical findings.

Exponential H∞ Output Control for Switching Fuzzy Systems via Event-Triggered Mechanism and Logarithmic Quantization

This paper investigates the problem of exponential H∞ output control for switching fuzzy systems, considering both impulse and non-impulse scenarios. Unlike previous research, where the average dwell time (ADT: τa) and the upper bound of inter-event intervals (IEIs: T) satisfy the condition τa≥lnμ+(α+β)Tα=lnμ+βTα+T, implying that frequent switching is difficult to achieve, this paper demonstrates that by adopting the mode-dependent event-triggered mechanism (ETM) and a switching law, frequent switching is indeed achieved. Moreover, the question of deriving the normal L2 norm constraint is solved through the ADT method, although only a weighted L2 norm constraint was obtained previously. Additionally, by constructing a controller-mode-dependent Lyapunov function and adopting logarithmic quantizers, the sufficient criteria of exponential H∞ output control problem are presented. The validity of established results is demonstrated by a given numerical simulation.