L1-norm Research Articles

Particle Size Inversion Based on L1,∞-Constrained Regularization Model in Dynamic Light Scattering

Dynamic light scattering (DLS) is a highly efficient approach for extracting particle size distributions (PSDs) from autocorrelation functions (ACFs) to measure nanoparticle particles. However, it is a technical challenge to get an exact inversion of the PSD in DLS. Generally, Tikhonov regularization is widely used to address this issue; it uses the L2 norm for both the data fitting term (DFT) and the regularization constraint term. However, the L2 norm’s DFT has poor robustness, and its regularization term lacks sparsity, making the solution susceptible to noise and a reduction in accuracy. To solve this problem, the Lp,q norm restrictive model is formulated to examine the impact of various norms in the DFT and regularization term on the inversion results. On this basis, combined with the robustness of DFT and the sparsity of regularization terms, an L1,∞-constrained Tikhonov regularization model was constructed. This model improves the inversion accuracy of PSD and offers a better noise-resistance performance. Simulation tests reveal that the L1,∞ model has strong noise resistance, exceptional inversion precision, and excellent bimodal resolution. The inversion outcomes for the 33 nm unimodal particles, the 55 nm unimodal, and the 33 nm/203 nm bimodal experimental particles show that L1,∞ reduces peak errors by at most 6.06%, 5.46%, and 12.12%/3.94% compared to L2,2, L1,2, and L2,∞ models, respectively. These simulations are validated by experimental data.

Qualitative analysis of fourth-order hyperbolic equations

We investigate the qualitative properties of weak solutions to the boundary value problems for fourth-order linear hyperbolic equations with constant coefficients in a plane bounded domain convex with respect to characteristics. Our main scope is to prove some analog of the maximum principle, solvability, uniqueness and regularity results for weak solutions of initial and boundary value problems in the space L2. The main novelty of this paper is to establish some analog of the maximum principle for fourth-order hyperbolic equations. This question is very important due to natural physical interpretation and helps to establish the qualitative properties for solutions (uniqueness and existence results for weak solutions). The challenge to prove the maximum principle for weak solutions remains more complicated and at that time becomes more interesting in the case of fourth-order hyperbolic equations, especially, in the case of non-classical boundary value problems with data of weak regularity. Unlike second-order equations, qualitative analysis of solutions to fourth-order equations is not a trivial problem, since not only a solution is involved in boundary or initial conditions, but also its high- order derivatives. Other difficulty concerns the concept of weak solution of the boundary value problems with L2 – data. Such solutions do not have usual traces, thus, we have to use a special notion for traces to poss correctly the boundary value problems. This notion is traces associated with operator L or L-traces. We also derive an interesting interpretation (as periodicity of characteristic billiard or the John's mapping) of the Fredholm's property violation. Finally, we discuss some potential challenges in applying the results and proposed methods.

Numerical Study of Multi-Term Time-Fractional Sub-Diffusion Equation Using Hybrid L1 Scheme with Quintic Hermite Splines

Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and space discretization using the quintic Hermite spline collocation method. The hybrid technique reduces the problem to an iterative scheme of an algebraic system of equations. The stability analysis of the proposed numerical scheme and the optimal error bounds for the approximate solution are also studied. A comparative study of the obtained results and an error analysis of approximation show the efficiency, accuracy, and effectiveness of the technique.

Long time behavior for the two-dimensional magnetohydrodynamic flows in a general domain

By utilizing spectral analysis methods and properties of fractional order operators, some new estimates have been established for the nonlinear terms appearing from the two-dimensional non-stationary magnetohydrodynamic flows in a general domain. Based on these, a series of energy decay properties are given. Furthermore, the large time decay behavior of the solution and that of its gradient have also been addressed in the L∞ and L2 spaces, respectively.

Imaging of permeability defect distribution by electromagnetic tomography with hybrid L1 norm and nuclear norm penalty terms

Electromagnetic tomography (EMT), with the advantages of being non-contact, non-invasiveness, low cost, simple structure, and fast imaging speed, is a multi-functional tomography technique based on boundary measurement voltages to image the conductivity distribution within the sensing field. EMT is widely used in industrial and biomedical fields. Currently, there are few studies on the application of EMT in magnetic permeability materials, which makes it difficult to obtain high-quality reconstructed images due to its own properties that lead to obvious attenuation of electromagnetic waves during propagation, as well as the ill-posed and ill-conditioned characteristics of EMT. In this paper, a multi-feature objective function integrating L2 norm regularization, L1 norm regularization, and low-rank norm regularization is proposed to solve the challenge of magnetic permeability material imaging. This approach emphasizes the smoothness and sparsity. The split Bregman algorithm is introduced to efficiently solve the proposed objective function by decomposing the complex optimization problem into several simple sub-task iterative schemes. In addition, a nine-coil planar array electromagnetic sensor was developed and a flexible modular EMT system was constructed. We use correlation coefficient and error coefficient as indicators to evaluate the performance of the proposed image reconstruction algorithm. The effectiveness of the proposed method in improving the reconstruction accuracy and robustness is verified through numerical simulations and experiments.

Overlapping domain decomposition methods for finite volume discretizations

Two-level additive overlapping domain decomposition methods are applied to solve the linear system arising from the cell-centered finite volume discretization methods (FVMs) for the elliptic problems. The conjugate gradient (CG) methods are used to accelerate the convergence. To analyze the preconditioned CG algorithm, a discrete L2 norm, an H1 norm, and an H1 semi-norm are introduced to connect the matrices resulting from the FVMs and related bilinear forms. It has been proved that, with a small overlap, the condition number of the preconditioned systems does not depend on the number of the subdomains. The result is similar to that for the conforming finite element. Numerical experiments confirm the theory.

Improving the performance of self-organizing map using reweighted zero-attracting method

In this paper, we introduce a novel approach to enhance the accuracy and convergence behavior of Self-Organizing Maps (SOM) by incorporating a reweighted zero-attracting term into the loss function. We evaluated two SOM versions: conventional SOM and robust adaptive SOM (RASOM). The enhanced versions, reweighted zero-attracting SOM (RZA-SOM) and reweighted zero-attracting RASOM (RZA-RASOM), include an l1 norm in the error function to add a zero-attractor term, which improves weight coefficient adjustments while preserving topology. The models were assessed for convergence speed and misadjustment under sparsity assumptions of the true coefficient matrix, and their robustness was tested under conditions of increased non-zero taps. Using six different datasets, we compared the performance of RZA-SOM and RZA-RASOM against conventional SOM and RA-SOM in terms of accuracy, quantization error, and topology preservation. Experimental results consistently demonstrated that RZA-SOM and RZA-RASOM surpassed the performance of conventional SOM and RA-SOM.

A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation

Alternating direction implicit (ADI) difference method for solving a 2D reaction-subdiffusion equation whose solution behaves a weak singularity at t=0 is studied in this paper. A Grünwald-Letnikov (GL) approximation is used for the discretization of Caputo fractional derivative (of order α, with 0<α<1) on a uniform mesh. Stability and convergence of the fully discrete ADI scheme are rigorously established. With the help of a discrete fractional Gronwall inequality, we get the sharp error estimate. The stability in L2 norm and the convergence of the GL-ADI scheme are strictly proved, where the convergent order is O(τtsα−1+τ2α+h12+h22). Numerical experiments are given to verify the theoretical analysis.

Revised and Generalized Results of Averaging Principles for the Fractional Case

The averaging principle involves approximating the original system with a simpler system whose behavior can be analyzed more easily. Recently, numerous scholars have begun exploring averaging principles for fractional stochastic differential equations. However, many previous studies incorrectly defined the standard form of these equations by placing ε in front of the drift term and ε in front of the diffusion term. This mistake results in incorrect estimates of the convergence rate. In this research work, we explain the correct process for determining the standard form for the fractional case, and we also generalize the result of the averaging principle and the existence and uniqueness of solutions to fractional stochastic delay differential equations in two significant ways. First, we establish the result in Lp space, generalizing the case of p=2. Second, we establish the result using the Caputo–Katugampola operator, which generalizes the results of the Caputo and Caputo–Hadamard derivatives.

Hybrid Reconstruction Approach for Polychromatic Computed Tomography in Highly Limited-Data Scenarios

Conventional strategies aimed at mitigating beam-hardening artifacts in computed tomography (CT) can be categorized into two main approaches: (1) postprocessing following conventional reconstruction and (2) iterative reconstruction incorporating a beam-hardening model. While the former fails in low-dose and/or limited-data cases, the latter substantially increases computational cost. Although deep learning-based methods have been proposed for several cases of limited-data CT, few works in the literature have dealt with beam-hardening artifacts, and none have addressed the problems caused by randomly selected projections and a highly limited span. We propose the deep learning-based prior image constrained (PICDL) framework, a hybrid method used to yield CT images free from beam-hardening artifacts in different limited-data scenarios based on the combination of a modified version of the Prior Image Constrained Compressed Sensing (PICCS) algorithm that incorporates the L2 norm (L2-PICCS) with a prior image generated from a preliminary FDK reconstruction with a deep learning (DL) algorithm. The model is based on a modification of the U-Net architecture, incorporating ResNet-34 as a replacement of the original encoder. Evaluation with rodent head studies in a small-animal CT scanner showed that the proposed method was able to correct beam-hardening artifacts, recover patient contours, and compensate streak and deformation artifacts in scenarios with a limited span and a limited number of projections randomly selected. Hallucinations present in the prior image caused by the deep learning model were eliminated, while the target information was effectively recovered by the L2-PICCS algorithm.

Compressed sensing with smooth L0 constraints for moving force identification from bridge response measurements

Abstract Compressed sensing (CS), as an emerging information sampling technique, has been successfully applied in the field of moving force identification (MFI). However, existing MFI CS models often fail to obtain the optimal sparse solutions and frequently underestimate the amplitude of local impact forces. To effectively address this issue, a new CS method is proposed for MFI based on smooth L0 norm constraints and bridge response measurements. Firstly, a smooth function is used to approximate the L0 norm, establishing a noise CS reconstruction model for MFI. The introduction of the smoothing function can locally convexify the original MFI problem and enhance the smoothness and differentiability of the objective function, making the optimization problem easier to solve. Subsequently, the Polak–Ribiere–Polyak formula is adopted to point the descent direction of the new objective function, and the sparse solution is iteratively advanced through the conjugate gradient algorithm. Finally, the applicability and feasibility of the proposed method is confirmed by numerical simulations and vehicle–bridge interaction tests, respectively. The results show that the proposed method can accurately identify moving forces from limited measurements of bridge responses. Compared with existing methods, it can provide more precise sparse solutions with higher robustness to measurement noises, and address the issue of underestimating on the amplitude of local impact forces, which is expected to enhance the performance and in-situ applicability of MFI.

A Boolean sum interpolation for multivariate functions of bounded variation

This paper deals with the approximation error of trigonometric interpolation for multivariate functions of bounded variation in the sense of Hardy-Krause. We propose interpolation operators related to both the tensor product and sparse grids on the multivariate torus. For these interpolation processes, we investigate the corresponding error estimates in the Lp norm for the class of functions under consideration. In addition, we compare the accuracy with the cardinality of these grids in both approaches.

Functional L1-Lp inequalities in the CAR algebra

In the present paper, we use the semigroup method to investigate various functional inequalities invoking L1 and Lp norms in the framework of canonical anti-commuting relations algebra (CAR algebra for short). As the main results, we obtain the Poincaré inequality for Talagrand type sum, Eldan-Gross inequality for projections, and the Talagrand influence inequality along with its strengthening form in the CAR algebra. All our results strengthen the noncommutative Poincaré inequality of Efraim and Lust-Piquard at several points. We conclude the paper with two applications of our inequalities. In the first application, we apply the noncommutative Eldan-Gross inequality to derive two KKL-type inequalities in the CAR algebra, which are closely related to the quantum KKL conjecture of Montanaro and Osborne. The second application is the CAR algebra counterpart of the superconcentration phenomenon derived from the noncommutative Talagrand influence inequality.

Sifat Kekonvergenan Lemah pada Ruang Bernorma beserta Operator pada Ruang Barisan Konvergen Lemah

The concept of weak convergence is generated by “weakening” the existing convergence properties in normed spaces. The outcome of this article is to prove the basic properties of weak convergence, the space of weakly convergent sequence is a Banach space and the operators from the space of weakly convergent sequence to other sequence spaces (l1(X),lp(X), linfty(X),S(X) ) and vice versa are linear and bounded .

The energy-diminishing weak Galerkin finite element method for the computation of ground state and excited states in Bose-Einstein condensates

In this paper, we employ the weak Galerkin (WG) finite element method and the imaginary time method to compute both the ground state and the excited states in Bose-Einstein condensate (BEC) which is governed by the Gross-Pitaevskii equation (GPE). First, we use the imaginary time method for GPE to get the nonlinear parabolic partial differential equation. Subsequently, we apply the WG method to spatially discretize the parabolic equation. This yields a semi-discrete scheme, in which an energy function is explicitly defined. For the case β⩾0, we demonstrate that the energy is diminishing with respect to time t at each time step. Applying the backward Euler scheme for temporal discretization yields a fully discrete scheme. For the case β=0, we provide a mathematical justification, establishing the convergence analysis for the numerical solution of the ground state. Moreover, based on the theory of solving eigenvalue problems using the WG method, we present the error estimates between the ground state and its numerical solution under the H1 and L2 norms. Numerical experiments are provided to illustrate the effectiveness of the proposed schemes. Moreover, the results indicate that our method also can compute the first excited state, achieving optimal convergence orders.

Analysis of a Crank–Nicolson fast element-free Galerkin method for the nonlinear complex Ginzburg–Landau equation

A fast element-free Galerkin (EFG) method is proposed in this paper for solving the nonlinear complex Ginzburg–Landau equation. A second-order accurate time semi-discrete system is presented by using the Crank–Nicolson scheme for the temporal discretization, and then a meshless fully discrete system is established by using the EFG method for the spatial discretization. In the proposed EFG method, Nitsche’s technique is used to impose the essential boundary conditions in a weak sense, and the reproducing kernel gradient smoothing integration is used to accelerate the calculation. Theoretical errors for the time semi-discrete system and the fully discrete EFG system are analyzed in detail. Optimal error estimates of the fully discrete Crank–Nicolson EFG method are obtained in both L2 and H1 norms. Numerical results validate the theoretical results and the effectiveness of the method.

Spectrality and non-spectrality of a class of Moran measures

Let {Mn}n=1∞∈M2(R) be a sequence of real expanding matrices and {Dn}n=1∞ be a sequence of digit sets withDn={(00),(anbn),(cndn)}, where |andn−bncn|=ϕ(n)∈Z, and letμ{Mn},{Dn}=δM1−1D1⁎δ(M1M2)−1D2⁎δ(M1M2M3)−1D3⁎⋯ be the associated Moran measure. This paper establishes a necessary and sufficient condition for μMn,Dn to be a spectral measure when each Mn is a diagonal matrix, and characterizes the structure of the spectrum Λ. Furthermore, we demonstrate that for any sequence of matrices {Mn}n=1∞ ∈ M2(Z) with det⁡(Mn)∉3Z, the space L2(μ{Mn},{Dn}) contains at most 9 mutually orthogonal exponential functions. The precise maximum cardinality of such functions is also determined.

Fan noise radial mode detection based on [formula omitted] regularization method

Radial mode detection has been widely employed in fan noise research. Many radial mode detection algorithms by conducting the convex L1 norm optimization have been proposed by taking advantage of the fact that the fan noise sound field is usually dominated by a limited set of radial modes. Nevertheless, L1 norm optimization has been shown to be suboptimal and cannot enforce further sparsity. A fan noise radial mode detection method based on L1/2 regularization algorithm is proposed in this paper to improve its robustness and dynamic range. This method employs the L1/2 regularization to directly solve the in-duct sound propagation model. Synthesized simulations indicates that the L1/2 regularization method can detect the correct radial mode order and amplitude under low signal-to-noise ratio. Moreover, the new method is insensitive to regularization parameter. An experimental system of radial rake is implemented to validate the new method. It is shown that the L-curve method, which do not require any prior information of the acoustic mode distribution or contaminating measurement noise, is an appropriate choice of the regularization parameter determination method for the L1/2 regularizer. It is demonstrated that the new L1/2 regularization method can enhance the mode sparsity, especially under low signal-to-noise ratio. Moreover, the resolution and dynamic range can be improved by the L1/2 regularization method, which makes it an effective and robust fan noise radial mode detection technique in practical applications.

Monotonicities of Quasi-Normed Orlicz Spaces

In this paper, we introduce a new Orlicz function, namely a b-Orlicz function, which is not necessarily convex. The Orlicz spaces LΦ generated by the b-Orlicz function Φ equipped with a Luxemburg quasi-norm contain both classical spaces Lp(p≥1) and Lp(0&lt;p&lt;1). The Orlicz spaces LΦ are quasi-Banach spaces. Some basic properties in quasi-normed Orlicz spaces are discussed, and the criteria that a quasi-normed Orlicz space is strictly monotonic and lower (upper) locally uniformly monotonic are given.

Enhancing SONAR performance through statistically-informed sub-array processing

Ocean fluctuations affect acoustic propagation and reduce the performance of SONAR arrays. In particular, internal waves can lead to a loss of coherence in incident signals, reducing the efficiency of classical processing. Precise knowledge of the environment is necessary to predict the degradation it causes. However, it is impossible to know the exact state of the ocean, it is therefore necessary to use statistical methods to study the ocean and acoustic propagation. In this work, statistical method, canonical correlation analysis (CCA), is used to find linear relationships between acoustic variables of interest and oceanographic measurements. In particular, the coherence radius is inferred from empirical modes of temperature using the model learned by CCA. A sub-array processing scheme can then be informed by defining the length of the sub-arrays from the coherence radius. To compensate for the loss of angular resolution inherent in sub-arrays, we introduce the use of a Bayesian algorithm exploiting l0 norm regularization. We show on experimental data from the ALMA 2017 campaign that the proposed processing improves detection when signal coherence is low.