Heuristics for Finding Sparse Solutions of Linear Inequalities

In this paper, we consider the problem of finding a sparse solution, with a minimal number of nonzero components, for a set of linear inequalities. This optimization problem is combinatorial and arises in various fields such as machine learning and compressed sensing. We present three new heuristics for the problem. The first two are greedy algorithms minimizing the sum of infeasibilities in the primal and dual spaces with different selection rules. The third heuristic is a combination of the greedy heuristic in the dual space and a local search algorithm. In numerical experiments, our proposed heuristics are compared with the weighted-[Formula: see text] algorithm and DCA programming with three different non-convex approximations of the zero norm. The computational results demonstrate the efficiency of our methods.

Sparse Solutions of a Class of Constrained Optimization Problems

In this paper, we consider a well-known sparse optimization problem that aims to find a sparse solution of a possibly noisy underdetermined system of linear equations. Mathematically, it can be modeled in a unified manner by minimizing [Formula: see text] subject to [Formula: see text] for given [Formula: see text] and [Formula: see text]. We then study various properties of the optimal solutions of this problem. Specifically, without any condition on the matrix A, we provide upper bounds in cardinality and infinity norm for the optimal solutions and show that all optimal solutions must be on the boundary of the feasible set when [Formula: see text]. Moreover, for [Formula: see text], we show that the problem with [Formula: see text] has a finite number of optimal solutions and prove that there exists [Formula: see text] such that the solution set of the problem with any [Formula: see text] is contained in the solution set of the problem with p = 0, and there further exists [Formula: see text] such that the solution set of the problem with any [Formula: see text] remains unchanged. An estimation of such [Formula: see text] is also provided. In addition, to solve the constrained nonconvex non-Lipschitz Lp-L1 problem ([Formula: see text] and q = 1), we propose a smoothing penalty method and show that, under some mild conditions, any cluster point of the sequence generated is a stationary point of our problem. Some numerical examples are given to implicitly illustrate the theoretical results and show the efficiency of the proposed algorithm for the constrained Lp-L1 problem under different noises.

Introduction to Finite Elements in Engineering

Thoroughly updated with improved pedagogy, the fifth edition of this classic textbook continues to provide students with a clear and comprehensive introduction the fundamentals of the finite element method. New features include coverage of core topics – including mechanics and heat conduction, energy and Galerkin approaches, convergence and adaptivity, time-dependent problems, and computer implementation – in the context of simple 1D problems, before advancing to 2D and 3D problems; expanded coverage of reduction of bandwidth, profile and fill-in for sparse solutions, time-dependent problems, plate bending, and nonlinearity; over thirty additional solved problems; and downloadable Matlab, Python, C, Javascript, Fortran and Excel VBA code providing students with hands-on experience. Accompanied by online solutions for instructors, this is the definitive text for senior undergraduate and graduate students studying a first course in the finite element method, and for professional engineers keen to shore up their understanding of finite element fundamentals.

An lp-space Matching Pursuit algorithm and its application to robust seismic data denoising via time-domain Radon transforms

Sparse solutions of linear systems of equations are important in many applications of seismic data processing. These systems arise in many denoising algorithms, such as those that use Radon transforms. We propose a robust Matching Pursuit algorithm for the retrieval of sparse Radon domain coefficients. The algorithm is robust to outliers and, hence, applicable for seismic data deblending. The classical Matching Pursuit algorithm is often adopted to approximate data by a small number of basis functions. It performs effectively for data contaminated with well-behaved, typically Gaussian, random noise.On the other hand, Matching Pursuit tends to identify the wrong basis functions when erratic noise contaminates our data. Incorporating a lp space inner product into the Matching Pursuit algorithm significantly increases its robustness to erratic signals. Our work describes a Robust Matching Pursuit algorithm that includes lp space inner products. We also provide a detailed description of steps required to implement the proposed lp space Robust Matching Pursuit algorithm when the basis functions are not given in an explicit form, such as is the case with the time-domain Radon transform. Finally, we test the proposed algorithm with deblending problems. Both synthetic and field data examples show a significant denoising improvement compared to deblending via the standard Matching Pursuit algorithm.

A new hybrid $ l_p $-$ l_2 $ model for sparse solutions with applications to image processing

<p style='text-indent:20px;'>Finding sparse solutions to a linear system has many real-world applications. In this paper, we study a new hybrid of the <inline-formula><tex-math id="M3">\begin{document}$ l_p $\end{document}</tex-math></inline-formula> quasi-norm (<inline-formula><tex-math id="M4">\begin{document}$ 0 &lt;p&lt; 1 $\end{document}</tex-math></inline-formula>) and <inline-formula><tex-math id="M5">\begin{document}$ l_2 $\end{document}</tex-math></inline-formula> norm to approximate the <inline-formula><tex-math id="M6">\begin{document}$ l_0 $\end{document}</tex-math></inline-formula> norm and propose a new model for sparse optimization. The optimality conditions of the proposed model are carefully analyzed for constructing a partial linear approximation fixed-point algorithm. A convergence proof of the algorithm is provided. Computational experiments on image recovery and deblurring problems clearly confirm the superiority of the proposed model over several state-of-the-art models in terms of the signal-to-noise ratio and computational time.</p>