L1-norm Research Articles

Strichartz and Smoothing Estimates for Dispersive Semi-Group e−itP(D) in Weighted L2 Spaces and Their Applications

Strichartz and Smoothing Estimates for Dispersive Semi-Group e−itP(D) in Weighted L2 Spaces and Their Applications

An alternating minimization algorithm with trajectory for direct exoplanet detection. The AMAT algorithm

Effective image post-processing algorithms are vital for the successful direct imaging of exoplanets. Standard point spread function (PSF) subtraction methods use techniques based on a low-rank approximation to separate the rotating planet signal from the quasi-static speckles and rely on signal-to-noise ratio maps to detect the planet. These steps do not interact or feed each other, leading to potential limitations in the accuracy and efficiency of exoplanet detection. We aim to develop a novel approach that iteratively finds the flux of the planet and the low-rank approximation of quasi-static signals in an attempt to improve upon current PSF subtraction techniques. In this study, we extend the standard L2 norm minimization paradigm to an L1 norm minimization framework in order to better account for noise statistics in the high contrast images. Then, we propose a new method, referred to as the alternating minimization algorithm with trajectory (AMAT), that makes more advanced use of estimating the low-rank approximation of the speckle field and the planet flux by alternating between them and utilizing both L1 and L2 norms. For the L1 norm minimization, we propose using L1 norm low-rank approximation (L1-LRA), a low-rank approximation computed using an exact block-cyclic coordinate descent method, while we use randomized singular value decomposition for the L2 norm minimization. Additionally, we enhance the visibility of the planet signal using a likelihood ratio as a post-processing step. Numerical experiments performed on a VLT/SPHERE-IRDIS dataset show the potential of AMAT to improve upon the existing approaches in terms of higher S/N, sensitivity limits (contrast curves), and receiver operating characteristic curves. Moreover, for a systematic comparison, we used datasets from the exoplanet data challenge to compare our algorithm with other algorithms in the challenge, and we find AMAT with a likelihood ratio map performs better than most algorithms tested on the exoplanet data challenge.

New Properties of Frames

Let H be a finite-dimentional complex Hilbert space and l2(H) is the space of square summable sequences in H. We will give a new characterization of a frame for H, we give our definition of a frame for the Hilbert space l2(H), we also define and give the properties of the frame operator. We equally show that our definition is equivalent to the definition of a frame for the Hilbert space H. Finally, we give a way to construct frames for l2(Hn) from frames for l2(Hp) such that p<n via fusion frame theory.

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.

Ris-aided integrated sensing and communication: Beamforming design and antenna selection

This work considers an integrated sensing and communication system, where a reconfigurable intelligent surface (RIS) is utilized to manage interference and radar signals. The sensors are attached to the RIS to sense multiple targets. A joint design of the base station transmit beamforming and RIS phase shift matrix is proposed to minimize total interference and maximize the worst received signal power at the RIS sensors. Due to highly coupled transmit beamforming and RIS phase matrices, the optimization problem is decoupled into two subproblems and solved iteratively by semidefinite programming and a manifold-based Riemannian steepest descent algorithm. We further design energy-aware beamforming to eliminate the interference induced by radar probing signals. Antenna selection with the ℓ0 norm is introduced to exclude redundant antennas while maintaining the sufficient multiple beams for multiple users and targets with minimized required antennas. Due to the nonconvexity of the ℓ0-norm, we relax the number of active transmit antennas as a weighted ℓ1-norm and employ a concave approximation for the constraint on the radar beampattern. Numerical results illustrate that the proposed algorithms can effectively reduce interference and strengthen the received signal power for radar sensing, achieving mutual benefit for communication and sensing performance.

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.

Nonsmooth optimization method for determining nonsmooth potential parameter in nonlinear subdiffusion equation

A determination of the nonsmooth potential parameter in a nonlinear subdiffusion model using terminal observation through a nonsmooth optimal control approach is proposed. This is achieved by converting the inverse problem to an optimal control problem with the fidelity and regularization terms in the cost functional represented by the L1 and TV norms, respectively, in order to account for the nonsmoothness of the potential. The existence of a minimizer for the optimal control problem is established and numerical solutions are obtained using the primal–dual method. The effectiveness of the proposed method is demonstrated via numerical results in comparison with the gradient descent method.

Surrogate modeling of multi-dimensional premixed and non-premixed combustion using pseudo-time stepping physics-informed neural networks

Physics-informed neural networks (PINNs) have emerged as a promising alternative to conventional computational fluid dynamics (CFD) approaches for solving and modeling multi-dimensional flow fields. They offer instant inference speed and cost-effectiveness without the need for training datasets. However, compared to common data-driven methods, purely learning the physical constraints of partial differential equations and boundary conditions is much more challenging and prone to convergence issues leading to incorrect local optima. This training robustness issue significantly increases the difficulty of fine-tuning PINNs and limits their widespread adoption. In this work, we present improvements to the prior field-resolving surrogate modeling framework for combustion systems based on PINNs. First, inspired by the time-stepping schemes used in CFD numerical methods, we introduce a pseudo-time stepping loss aggregation algorithm to enhance the convergence robustness of the PINNs training process. This new pseudo-time stepping PINNs (PTS-PINNs) method is then tested in non-reactive convection–diffusion problem, and the results demonstrated its good convergence capability for multi-species transport problems. Second, the effectiveness of the PTS-PINNs method was verified in the case of methane–air premixed combustion, and the results show that the L2 norm relative error of all variables can be reduced within 5%. Finally, we also extend the capability of the PTS-PINNs method to address a more complex methane–air non-premixed combustion problem. The results indicate that the PTS-PINNs method can still achieve commendable accuracy by reducing the relative error to within 10%. Overall, the PTS-PINNs method demonstrates the ability to rapidly and accurately identify the convergence direction of the model, surpassing traditional PINNs methods in this regard.

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.

Enhanced Improved Proportionate Set-Membership Fast NLMS Using the ℓ0 Norm

Enhanced Improved Proportionate Set-Membership Fast NLMS Using the ℓ0 Norm

Super-Resolution Reconstruction of Remote Sensing Images Using Chaotic Mapping to Optimize Sparse Representation.

Current super-resolution algorithms exhibit limitations when processing noisy remote sensing images rich in surface information, as they tend to amplify noise during the recovery of high-frequency signals. To mitigate this issue, this paper presents a novel approach that incorporates the concept of compressed sensing and explores the super-resolution problem of remote sensing images for space cameras, particularly for high-speed imaging systems. The proposed algorithm employs K-singular value decomposition (K-SVD) to jointly train high- and low-resolution image blocks, updating them column by column to obtain overcomplete dictionary pairs. This approach compensates for the deficiency of fixed dictionaries in the original algorithm. In the process of dictionary updating, we innovatively integrate the circle chaotic mapping into the solution process of the dictionary sequence, replacing pseudorandom numbers. This integration facilitates balanced traversal and simplifies the search for global optimal solutions. For the optimization problem of sparse coefficients, we utilize the orthogonal matching pursuit method (OMP) instead of the L1 norm convex optimization method used in most reconstruction techniques, thereby complementing the K-SVD dictionary update algorithm. After upscaling and denoising the image using the dictionary pair mapping relationship, we further emphasize image edge details with local gradients as constraints. When compared with various representative super-resolution algorithms, our algorithm effectively filters out noise and stains in low-resolution images. It not only performs well visually but also stands out in objective evaluation indicators such as the peak signal-to-noise ratio and information entropy. The experimental results validate the effectiveness of the proposed method in super-resolution remote sensing images, yielding high-quality remote sensing image data.

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.