Fast Iterative Shrinkage-thresholding Algorithm Research Articles

Probability-Based Complex-Valued Fast Iterative Shrinkage-Thresholding Algorithm for Deconvolution Beamforming

Probability-Based Complex-Valued Fast Iterative Shrinkage-Thresholding Algorithm for Deconvolution Beamforming

Robust treatment planning for small animal radio-neuromodulation using focused kV x-ray beams.

In preclinical radio-neuromodulation research, small animal experiments are pivotal for unraveling radiobiological mechanism, investigating prescription and planning techniques, and assessing treatment effects and toxicities. However, the target size inside a rat brain is typically in the order of sub-millimeters. The small target inside the visual cortex neural region in rat brain with a diameter of around 1mm was focused in this work to observe the physiological change of this region. Delivering uniform doses to the small target while sparing health tissues is challenging. Focused kV x-ray technique based on modern x-ray polycapillary focusing lens is a promising modality for small animal radio-neuromodulation. The current manual planning method could lead to sub-optimal plans, and the positioning uncertainties due to mechanical accuracy limitations, animal immobilization, and robotic arm motion are not considered. This work aims to design a robust inverse planning method to optimize the intensities of focused kV x-ray beams located in beam trajectories to irradiate small mm-sized targets in rat brains for radio-neuromodulation. Focused kV x-ray beams were generated through polycapillary x-ray focusing lenses on achieving small (≤0.3mm) focus perpendicular to the beam. The beam trajectories were manually designed in 3D space in scanning-while-rotating mode. Geant4 Monte Carlo (MC) simulation generated a dose calculation matrix for each focused kV x-ray beam located in beam trajectories. In the proposed robust inverse planning method, an objective function combining a voxel-wise stochastic programming approach and L1 norm regularization was established to overcome the positioning uncertainties and obtain a high-quality plan. The fast iterative shrinkage thresholding algorithm (FISTA) was utilized to solve the objective function and obtain the optimal intensities. Four cases were employed to validate the feasibility and effectiveness of the proposed method. The manual and non-robust inverse planning methods were also implemented for comparison. The proposed robust inverse planning method achieved superior dose homogeneity and higher robustness against positioning uncertainties. On average, the clinical target volume (CTV) homogeneity index (HI) of robust inverse plan improved to 13.3 from 22.9 in non-robust inverse plan and 53.8 in manual plan if positioning uncertainties were also present. The average bandwidth at D90 was reduced by 6.5Gy in the robust inverse plan, compared to 9.6Gy in non-robust inverse plan and 12.5Gy in manual plan. The average bandwidth at D80 was reduced by 3.4Gy in robust inverse plan, compared to 5.5Gy in non-robust inverse plan and 8.5Gy in manual plan. Moreover, the dose delivery time of manual plan was reduced by an average reduction of 54.7% with robust inverse plan and 29.0% with non-robust inverse plan. Compared to manual and non-robust inverse planning methods, the robust inverse planning method improved the dose homogeneity and delivery efficiency and was resistant to the uncertainties, which are crucial for radio-neuromodulation utilizing focused kV x-rays.

Dynamic Matrix Recovery

Matrix recovery from sparse observations is an extensively studied topic emerging in various applications, such as recommendation system and signal processing, which includes the matrix completion and compressed sensing models as special cases. In this article, we propose a general framework for dynamic matrix recovery of low-rank matrices that evolve smoothly over time. We start from the setting that the observations are independent across time, then extend to the setting that both the design matrix and noise possess certain temporal correlation via modified concentration inequalities. By pooling neighboring observations, we obtain sharp estimation error bounds of both settings, showing the influence of the underlying smoothness, the dependence and effective samples. We propose a dynamic fast iterative shrinkage-thresholding algorithm that is computationally efficient, and characterize the interplay between algorithmic and statistical convergence. Simulated and real data examples are provided to support such findings. Supplementary materials for this article are available online.

An enhanced sparse autoencoder for machinery interpretable fault diagnosis

The interpretability of individual components within existing autoencoders remains insufficiently explored. This paper aims to address this gap by delving into the interpretability of the encoding and decoding structures and their correlation with the physical significance of vibrational signals. To achieve this, the Sparse Coding with Multi-layer Decoders (SC-MD) model is proposed, which facilitates fault diagnosis from two perspectives: the working principles of the model itself and the evolving trends of fault features. Specifically, a sparse coding protocol to prevent L1-norm collapse is proposed in the encoding process, regularizing the encoding to ensure that each latent code component possesses variance greater than a fixed threshold on a set of sparse representations given the input data. Subsequently, a multi-layer decoder structure is designed to capture the intricate mapping relationship between features and fault patterns. Finally, the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) is employed as the solver for the SC-MD model, enabling end-to-end updates of all parameters by unfolding FISTA. The coherent theoretical framework ensures the interpretability of SC-MD. Utilizing aeroengine bearing data, we demonstrate the exceptional performance of our proposed approach under both normal conditions and intense noise, as compared to state-of-the-art deep learning methods.

Real-Time Decoding of Snapshot Compressive Imaging Using Tensor FISTA-Net.

Snapshot compressive imaging (SCI) cameras compress high-speed videos or hyperspectral images into measurement frames. However, decoding the data frames from measurement frames is compute-intensive. Existing state-of-the-art decoding algorithms suffer from low decoding quality or heavy running time or both, which are not practical for real-time applications. In this article, we exploit the powerful learning ability of deep neural networks (DNN) and propose a novel tensor fast iterative shrinkage-thresholding algorithm net (Tensor FISTA-Net) as a real-time decoder for SCI cameras. Since SCI cameras have an accurate physical model, we can trade training time for the decoding time by generating abundant synthetic data and training a decoder on the cloud. Tensor FISTA-Net not only learns a sparse representation of the frames through convolution layers but also reduces the decoding time and memory consumption significantly through tensor operations, which makes Tensor FISTA-Net an appropriate approach for a real-time decoder. Our proposed Tensor FISTA-Net obtains an average PSNR improvement of 0.79-2.84 dB (video images) and 2.61-4.43 dB (hyperspectral images) over the state-of-the-art algorithms, along with more clear and detailed visual results on real SCI datasets, Hammer and Wheel, respectively. Our Tensor FISTA-Net reaches 45 frames per second in video datasets and 70 frames per second in hyperspectral datasets, meeting the real-time requirement. Besides, the trained model occupies only a 12 -MB memory footprint, making it applicable to real-time Internet of Things (IoT) applications.

Discretized tensor-based model of total focusing method: A sparse regularization approach for enhanced ultrasonic phased array imaging

The total focusing method (TFM) is considered as the standard in ultrasonic phased array imaging and plays a vital role in industrial non-destructive testing (NDT). By utilizing the full matrix capture (FMC) dataset, the TFM can focus on every point within the specified imaging region, and is more accurate than the traditional ultrasonic phased array imaging methods. However, the TFM is essentially a delay and sum technique that often operates linearly on the time-domain signals and takes no prior information into account, so its image quality remains inadequate when dealing with defects in close proximity or scattering materials. To address this problem, this paper formulates the imaging principle of the TFM as a Boolean matrix and establishes the discretized tensor-based model. Subsequently, the model is addressed by employing the sparse regularization strategy, taking into some characteristics of industrial NDT. Regarding the solution algorithm, due to the generation of negative values by the fast iterative shrinkage threshold algorithm (FISTA), this paper introduces the rectified linear unit (ReLU) function as a non-negative constraint and presents a dedicated solution algorithm (ReLU-FISTA) for acquiring detection results. Through verification of simulation and experiment, the proposed approach exhibits superior capabilities of defect characterization and noise suppression when compared to the TFM, leading to substantial enhancements in image quality.

An Efficient Recognition Method for Orbital Angular Momentum via Adaptive Deep ELM.

For orbital angular momentum (OAM) recognition in atmosphere turbulence, how to design a self-adapted model is a challenging problem. To address this issue, an efficient deep learning framework that uses a derived extreme learning machine (ELM) has been put forward. Different from typical neural network methods, the provided analytical machine learning model can match the different OAM modes automatically. In the model selection phase, a multilayer ELM is adopted to quantify the laser spot characteristics. In the parameter optimization phase, a fast iterative shrinkage-thresholding algorithm makes the model present the analytic expression. After the feature extraction of the received intensity distributions, the proposed method develops a relationship between laser spot and OAM mode, thus building the steady neural network architecture for the new received vortex beam. The whole recognition process avoids the trial and error caused by user intervention, which makes the model suitable for a time-varying atmospheric environment. Numerical simulations are conducted on different experimental datasets. The results demonstrate that the proposed method has a better capacity for OAM recognition.

Application of Sparse Regularization in Spherical Radial Basis Functions-Based Regional Geoid Modeling in Colorado

Spherical radial basis function (SRBF) is an effective method for calculating regional gravity field models. Calculating gravity field models with high accuracy and resolution requires dense basis functions, resulting in complex models. This study investigated the application of sparse regularization in SRBFs-based regional gravity field modeling. L1-norm regularization, also known as the least absolute shrinkage selection operator (LASSO), was employed in the parameter estimation procedure. LASSO differs from L2-norm regularization in that the solution obtained by LASSO is sparse, specifically with a portion of the parameters being zero. A sparse model would be advantageous for improving the numerical efficiency by reducing the number of SRBFs. The optimization problem of the LASSO was solved using the fast iterative shrinkage threshold algorithm, which is known for its high efficiency. The regularization parameter was selected using the Akaike information criterion. It was specifically tailored to the L1-norm regularization problem. An approximate covariance matrix of the estimated parameters in the sparse solution was analytically constructed from a Bayesian viewpoint. Based on the remove–compute–restore technique, a regional geoid model of Colorado (USA) was calculated. The numerical results suggest that the LASSO adopted in this study provided competitive results compared to Tikhonov regularization; however, the number of basis functions in the final model was less than 25% of the Tikhonov regularization. Without significantly reducing model accuracy, the LASSO solution provides a very simple model. This is the first study to apply the LASSO to SRBFs-based modeling of the regional gravity field in real gravity observation data.

A unified path seeking algorithm for IMRT and IMPT beam orientation optimization

Objective. Fully automated beam orientation optimization (BOO) for intensity-modulated radiotherapy and intensity modulated proton therapy (IMPT) is gaining interest, since achieving optimal plan quality for an unknown number of fixed beam arrangements is tedious. Fast group sparsity-based optimization methods have been proposed to find the optimal orientation, but manual tuning is required to eliminate the exact number of beams from a large candidate set. Here, we introduce a fast, automated gradient descent-based path-seeking algorithm (PathGD), which performs fluence map optimization for sequentially added beams, to visualize the dosimetric benefit of one added field at a time. Approach. Several configurations of 2–4 proton and 5–15 photon beams were selected for three head-and-neck patients using PathGD, which was compared to group sparsity-regularized BOO solved with the fast iterative shrinkage-thresholding algorithm (GS-FISTA), and manually selected IMPT beams or one coplanar photon VMAT arc (MAN). Once beams were chosen, all plans were compared on computational efficiency, dosimetry, and for proton plans, robustness. Main results. With each added proton beam, Clinical Target Volume (CTV) and organs at risk (OAR) dosimetric cost improved on average across plans by [1.1%, 13.6%], and for photons, [0.6%, 2.0%]. Comparing algorithms, beam selection for PathGD was faster than GS-FISTA on average by 35%, and PathGD matched the CTV coverage of GS-FISTA plans while reducing OAR mean and maximum dose in all structures by an average of 13.6%. PathGD was able to improve CTV [Dmax, D95%] by [2.6%, 5.2%] and reduced worst-case [max, mean] dose in OARs by [11.1%, 13.1%]. Significance. The benefit of a path-seeking algorithm is the beam-by-beam analysis of dosimetric cost. PathGD was shown to be most efficient and dosimetrically desirable amongst group sparsity and manual BOO methods, and highlights the sensitivity of beam addition for IMPT in particular.

TEFISTA-Net: A learnable method for high-resolution range profile reconstruction with low-frequency ultra-wideband radar

In low-frequency ultra-wideband (LFW) radar, the reconstruction of high-resolution range profile (HRRP) is an important task. The state-of-the-art compressive sensing (CS) method for HRRP reconstruction based on geometrical theory of diffraction (GTD) requires manual algorithm parameter tuning and is computationally expensive. In this paper, we propose a deep learning-based CS method for LFW radar HRRP reconstruction. We design a neural network architecture, i.e., target enhancement-based FISTA-Net (TEFISTA-Net), by unrolling the fast iterative shrinkage thresholding algorithm (FISTA). A new loss function based on the target-to-background ratio (TBR) is introduced for network training with the target enhancement capability in low signal-to-noise ratio (SNR) scenarios. The algorithm parameters in the traditional CS methods are substituted by network parameters learned from training data, getting rid of manual parameter tuning. In addition, simple convolution operations in our new method lead to lower computational complexity compared with existing methods. Experimental results based on diverse data show that the proposed method leads to higher computational efficiency with similar or better performance in comparison with existing state-of-the-art methods.

Non-line-of-sight reconstruction via structure sparsity regularization.

Non-line-of-sight (NLOS) imaging allows for the imaging of objects around a corner, which enables potential applications in various fields, such as autonomous driving, robotic vision, medical imaging, security monitoring, etc. However, the quality of reconstruction is challenged by low signal-to-noise ratio (SNR) measurements. In this study, we present a regularization method, referred to as structure sparsity (SS) regularization, for denoising in NLOS reconstruction. By exploiting the prior knowledge of structure sparseness, we incorporate nuclear norm penalization into the cost function of the directional light-cone transform (DLCT) model for the NLOS imaging system. This incorporation effectively integrates the neighborhood information associated with the directional albedo, thereby facilitating the denoising process. Subsequently, the reconstruction is achieved by optimizing a directional albedo model with SS regularization using the fast iterative shrinkage-thresholding algorithm (FISTA). Notably, the robust reconstruction of occluded objects is observed. Through comprehensive evaluations conducted on both synthetic and experimental datasets, we demonstrate that the proposed approach yields high-quality reconstructions, surpassing the state-of-the-art reconstruction algorithms, especially in scenarios involving short exposure and low-SNR measurements.

Analytical HDR prostate brachytherapy planning with automatic catheter and isotope selection.

High dose rate (HDR) brachytherapy is commonly used to treat prostate cancer. Existing HDR planning systems solve the dwell time problem for predetermined catheters and a single energy source. Additional degrees of freedom can be obtained by relaxing the catheters' pre-designation and introducing more source types, and may have a dosimetric benefit, particularly in improving conformality to spare the urethra. This study presents a novel analytical approach to solving the corresponding HDR planning problem. The catheter and dual-energy source selection problem was formulated as a constrained optimization problem with a non-convex group sparsity regularization. The optimization problem was solved using the fast-iterative shrinkage-thresholding algorithm (FISTA). Two isotopes were considered. The dose rates for the HDR 4140 Ytterbium (Yb-169) source and the Elekta Iridium (Ir-192) HDR Flexisource were modeled according to the TG-43U1 formalism and benchmarked accordingly. Twenty-two retrospective HDR prostate brachytherapy patients treated with Ir-192 were considered. An Ir-192 only (IRO), Yb-169 only (YBO), and dual-source (DS) plan with optimized catheter location was created for each patient with N catheters, where N is the number of catheters used in the clinically delivered plans. The DS plans jointly optimized Yb-169 and Ir-192 dwell times. All plans and the clinical plans were normalized to deliver a 15Gy prescription (Rx) dose to 95% of the clinical treatment volume (CTV) and evaluated for the CTV D90%, V150%, and V200%, urethra D0.1cc and D1cc, bladder V75%, and rectum V75%. Dose-volume histograms (DVHs) were generated for each structure. The DS plans ubiquitously selected Ir-192 as the only treatment source. IRO outperformed YBO in organ at risk (OARs) OAR sparing, reducing the urethra D0.1cc and D1cc by 0.98% ( ) and 1.09% ( ) of the Rx dose, respectively, and reducing the bladder and rectum V75% by 0.09 ( ) and 0.13 cubic centimeters (cc) ( ), respectively. The YBO plans delivered a more homogenous dose to the CTV, with a smaller V150% and V200% by 3.20 ( ) and 1.91 cc ( ), respectively, and a lower CTV D90% by 0.49% ( ) of the prescription dose. The IRO plans reduce the urethral D1cc by 2.82% ( ) of the Rx dose compared to the clinical plans, at the cost of increased bladder and rectal V75% by 0.57 ( ) and 0.21 cc ( ), respectively, and increased CTV V150% by a mean of 1.46 cc ( ) and CTV D90% by an average of 1.40% of the Rx dose ( ). While these differences are statistically significant, the clinical differences between the plans are minimal. The proposed analytical HDR planning algorithm integrates catheter and isotope selection with dwell time optimization for varying clinical goals, including urethra sparing. The planning method can guide HDR implants and identify promising isotopes for specific HDR clinical goals, such as target conformality or OAR sparing.

Automatic Modulation Recognition Based on a New Deep K-SVD Denoising Algorithm

Automatic modulation recognition has a wide range of applications in wireless communication. To solve the problem that the previous methods convert signal modulation recognition into image recognition, leading to increased time costs and information loss, an automatic modulation recognition approach consisting of the improved deep K-SVD denoising algorithm is suggested. First, the effectiveness of the model for random sine wave denoising is demonstrated by simulation. Second, the original I/Q signals are fed into the modified deep K-SVD model for denoising, skipping the complicated image processing steps. Finally, the noise-reduced signals are input into a multi-channel convolutional long short-term neural network to complete the modulation recognition. To solve the slow convergence problem of Iterative Shrinkage Thresholding Algorithms in sparse coding, the Fast Iterative Shrinkage Thresholding Algorithm is adopted to improve the computational efficiency and obtain a better denoising effect. The experiments show that the improved model has an average recognition accuracy of 91.26% at different SNRs from -2 dB to 18 dB, which is better than the state-of-the-art modulation recognition models.

FISTA-NET: Deep Algorithm Unrolling for Cerenkov luminescence tomography.

Cerenkov luminescence tomography (CLT) is a highly sensitive and promising imaging technique that can be used to reconstruct the three-dimensional distribution of radioactive probes in living animals. However, the accuracy of CLT reconstruction is limited by the simplified radiative transfer equation and ill-conditioned inverse problem. To address this issue, we propose a model-based deep learning network that combines the neural network with a model-based approach to enhance the performance of CLT reconstruction. The Fast Iterative Shrinkage Thresholding Algorithm (FISTA), a traditional model-based approach, is expanded into a deep network (known as FISTA-NET). Each layer in the network represents an iteration of the algorithm steps, and connecting these layers can form a deep neural network. In addition, different from the traditional FISTA, the key parameters in FISTA, such as gradient step size and threshold value, can be learned through training data without manual production. To evaluate the performance of FISTA-NET, numerical simulation experiments were conducted, which demonstrate its excellent positioning and shape recovery abilities.Clinical Relevance-This indicates that FISTA-NET strategy can significantly improve the quality of CLT reconstruction, which is further beneficial to the assessment of disease activity and treatment effect based on CLT.

Novel Approach-Based Sparsity for Damage Localization in Functionally Graded Material

Model-based approaches have been widely employed in damage detection and localization studies. However, alternative techniques, such as built-in online detection methods, hold promise for future advancements in structural health monitoring technologies. In this research paper, we present a dynamic algorithm specifically designed for accurate damage localization in functionally graded plates. The suggested method involves the creation of a grid matrix that captures the dynamic response of the structure over time. Subsequently, an optimization process is performed using a linear equation that incorporates the information contained within the grid, enabling the precise localization of damage. To address the inherent sparsity of the localization nature, we utilize the FISTA (fast iterative shrinkage-thresholding algorithm) as a problem solver. The effectiveness of our approach is evaluated through experimental tests on a functionally graded plate with clamped free boundary conditions. Multiple damage scenarios are investigated, including cases with damage signals on and off-the-grid. The results demonstrate that our proposed approach is capable of accurately predicting the position of damage, indicating its suitability for application in low-size data systems.

Numerical analysis of aerospace plate damage via 3D electrical impedance tomography.

The carbon fiber reinforced plastic (CFRP) is widely used in the aerospace industry due to its high strength and lightweight characteristics, making it crucial to ensure the reliability of these materials. This has led to an increasing focus on research on the health monitoring technologies of aerospace materials. Electrical impedance tomography (EIT) is a non-invasive and cost-effective technology that has the potential to realize real-time health monitoring of materials by measuring changes in electrical parameters. This paper investigates the application of EIT for direct 3D reconstruction of damage in CFRP laminates with significant conductivity anisotropy distribution. Based on the corrected sensitivity matrix formula, the direct 3D image reconstruction method combined with the fast iterative shrinkage-thresholding algorithm (FISTA) is proposed to achieve damage imaging of CFRP laminates in the inverse problem. The fast convergence of the FISTA makes it possible to solve complex inverse problems. The numerical simulation results indicate that, compared with 2D EIT, the proposed method is more capable of providing damage information, especially in the depth direction. This research plays a constructive role in realizing 3D image reconstruction of CFRP material damage and has significant implications for improving the reliability and safety of CFRP materials in aerospace applications.

Robust RFI Excision for Pulsar Signals by a Novel Nonlinear M-type Estimator with an Application to Pulsar Timing

Radio frequency interference (RFI) mitigation for pulsar signals is a long perplexing issue in astrophysical measurements. Linear mitigation methods are often criticized for limited RFI excision range and weakness of RFI modeling. Meanwhile, thresholding methods (e.g., the SumThreshold) suffer from empirical factors. In our opinion, the main defect of the current status is the lack of a concise definition to distinguish signals from RFI with the aid of certain techniques, e.g., sparse representation. This point is the root cause of these problems and also forms our motivation. This paper aims to expand the excision range (e.g., the on-pulse and quasiperiodic RFI) and cut down some empirical factors. The main contribution is that we give a definition and derive a widely practicable nonlinear framework for RFI excision. This framework can overcome the susceptibility of the least-square criterion to RFI, and excise almost all types of RFI once and for all. A robust LnCosh criterion based nonlinear maximum likelihood-type (M-type) penalized smoothing estimator is introduced. The novelty is that this estimator is first embedded into the iterative shrinkage-thresholding algorithm (ISTA) and the fast ISTA. Nonlinearity highlights this method. Curvelet sparsity gives satisfying approximation for pulsar signals containing dispersion feature. Finally, useful signal details will be retrieved from the excision residual by a morphological component analysis. This method is applied to the time-frequency signals collected by the Nanshan 26 m Radio Telescope. The numerical experiments can persuasively prove that it has desired application prospects.

An accelerated proximal gradient method for multiobjective optimization

This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm, for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method ( $$O(1 / k^2)$$ ), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments.

Broadband Acoustic Modal Identification by Array Nonsynchronous Measurements with One Reference

This paper proposes a broadband acoustic modal identification method based on nonsynchronous measurements from a microphone array, which overcomes the acoustic modal aliasing problem caused by a limited number of microphones. In the proposed method, the number of measurement points is augmented by sequentially rotating a circular microphone array that is circumferentially mounted in the wall of the compressor duct. This technique employs only one reference microphone to enhance the robustness of the cross-spectral matrix completion for nonsynchronous measurements. The fast iterative shrinkage threshold algorithm is adopted to iteratively complete the cross-spectral matrix of all measurement points for acoustic modal identification. The in-duct acoustic modal coefficients of the broadband noise are then determined using the least-squares method with the sound pressure vector recovered from the completed cross-spectral matrix. The accuracy of the cross-spectral matrix and the identified acoustic modals is verified through numerical simulations. The proposed method is validated by experimental results from an axial flow compressor. The results show that it can effectively suppress the modal aliasing problem and identify the azimuthal modes of compressor broadband noise.

The variant of the iterative shrinkage-thresholding algorithm for minimization of the ℓ1 over ℓ∞ norms

In this paper, we study minimization of the ratio of ℓ1 and ℓ∞ norms (ℓ1/ℓ∞) as a nonconvex and sparsity-promoting metric for solving the unconstrained compressed sensing problems. To design some efficient algorithms, we derive a closed-form solution to the proximal operator of the ℓ1/ℓ∞ function. With the newly variant of the shrinkage operator, we propose a novel variant of fast iterative shrinkage-thresholding algorithm (FISTA) and a specific variable-splitting scheme of the alternating direction method of multipliers (ADMM) which is guaranteed to have sub-sequential convergence. Experimentally, we construct extensive numerical simulations to demonstrate the efficiency of these two proposed approaches over the state-of-the-art algorithms in sparse recovery.