In this article, we investigate least squares-based weak Galerkin methods for numerically approximating the solution of the Maxwell interface problems having non-homogeneous jumps across the interface, posed in Lipschitz continuous domains in 2D/3D with a C-2 interface. Rigorous convergence analysis has been carried out to obtain super-convergence of the errors in a discrete energy norm. Some numerical experiments are provided to support the theoretical conclusions.
This paper concerns applications of advanced techniques of pseudo-Jacobians for studying a nonsmooth and nonconvex general optimization problem in Banach spaces. To this end, calculus rules for pseudo-Jacobians in infinite-dimensional spaces are presented; in particular, thwee chain rule for pseudo-Jacobians is developed. Then, using the idea of pseudo-Jacobian, the Robinson, Mangasarian- Fromovitz and Abadie constraint qualifications are generalized. The relations between these constraint qualifications and the local error bound property are investigated. Furthermore, a necessary optimality condition is derived under the Abadie constraint qualification. As an application, new constraint qualifications and necessary optimality conditions are obtained for nonsmooth and nonconvex infinite programming problems. In this way, a pseudo-Jacobian for the pointwise supremum of an arbitrary family of real-valued functions defined on a Banach space is provided. Some examples are given to clarify the results.
In this article, we introduce a subclass of involutions in Banach spaces that satisfy a specific condition. We investigate the existence of fixed points and the convergence of Mann iteration process for this subclass. Specifically, we show that a fixed point can be obtained through this process. Additionally, we provide conditions under which the iteration sequence converges weakly or strongly to a fixed point.
In this paper, we study the Cholesky and Cholesky-QR decompositions of dual quaternion matrices. Firstly, the dual quaternion Gauss transformation matrix is given as the basic tool to realize these two decompositions. Secondly, it is proved that a dual quaternion Hermitian matrix with positive definite standard part can be performed the Cholesky decomposition, and the corresponding direct algorithm is given. Moreover, if the standard part of a dual quaternion matrix is column full rank, the Cholesky-QR decomposition can be realized. Inspired by the real structure-preserving algorithm of quaternion matrices, we design a real calculation-preserving algorithm to realize the Cholesky decomposition of dual quaternion matrices. Numerical experiments show that the real calculation-preserving algorithm is more effective than the direct algorithm. Finally, a concrete example of applying the Cholesky decomposition of dual quaternion matrices to the Cholesky-QR decomposition of dual quaternion matrices is given.
We study robust optimization problems which are counterparts of semi-infinite multiobjective nonsmooth and nonconvex optimization problems with data uncertainty. We propose new types of Mangasarian-Fromovitz and Pshenichnyi-Levin-Valadier constraint qualifications and apply them to establish robust Karush-Kuhn-Tucker (KKT) conditions for weak, Pareto, isolated, and positive proper solutions in terms of the Michel-Penot subdifferential. Robust sufficient optimality conditions are also obtained under generalized convexity assumptions. Finally, we discuss the relationship between constraint qualification conditions and the existence and boundedness of KKT multiplier sets.
In this paper, we explore advanced generalizations of Jensen's inequality specifically for geometrically-arithmetic (GA)-convex functions. Our primary objective is to extend the classical results associated with convex functions to a broader category, thus contributing to the deeper understanding of GA-convexity. To achieve this, we introduce a new function, denoted by Phi, which is directly related to a given GA-convex function phi. This function Phi plays a pivotal role in deriving more intricate inequalities, particularly those that involve both Jensen's inequality and the Jensen-Mercer inequality. These inequalities provide new insights into the behavior of GA-convex functions, offering stronger and more generalized versions of the classical results. Moreover, we demonstrate the utility of these new inequalities by presenting their applications in the context of mean theory. In this setting, the generalized inequalities help to establish novel relationships among various types of means, enriching the existing body of knowledge in this domain. Our findings have the potential to advance research in mathematical inequalities, particularly in the study of convexity and its applications to different areas of mathematical analysis.
In this paper, we define a nonlinear extension of the fractional sum-type neural network operators of Kadak et al. [1] using the max-product approach. Also, we introduce multivariate generalized max fractional neural network-type operators of order alpha>0. The quantitative estimates of operators are analyzed using the multivariate modulus of continuity and absolute moments. Finally, we provide illustrative examples, along with graphs and error tables, that show the excellent accuracy of the operators when using the chosen activation functions.
The primary focus of this article is to investigate the optimal feedback control for second-order damping stochastic neutral impulsive evolution systems in separable Hilbert spaces. To establish the necessary conditions for the proposed problem, the theory of strongly continuous cosine families and Bohnenblast-Karlin's fixed point theorem for multivalued maps is utilized to demonstrate a mild solution for second-order evolution inclusions. Subsequently, we introduce a novel set of sufficient assumptions, employing the Cesari property and the Filippove theorem, to guarantee the existence of feasible pairs in feedback control systems associated with Lagrange control problems. Finally, we illustrate our main results through a application.
In this paper, we present a novel approach to derivatives of scalar fields in Hilbert spaces, which extends the concept of Lp-derivatives introduced by Calder & oacute;n and Zygmund for the case of p=2. We investigate the properties of these generalized derivatives, provide characterizations, and derive explicit formulas to compute them. As a consequence, we establish a necessary and sufficient condition for the existence of the best local approximation in Hilbert spaces by employing these new concepts of derivatives. Furthermore, we show that the local best approximation can be expressed as a generalized Taylor polynomial involving the generalized derivatives. This result provides a practical tool to determine when the best local approximation exists and presents an explicit formula for computing it based on the Hilbert partial derivatives.
In this paper, we revisit a supervised learning approach based on unrolling first introduced in [1] and called Psi DONet, by providing a deeper microlocal interpretation for its theoretical analysis, and extending its study to the case of sparse-angle tomography. Furthermore, we refine the implementation of the original Psi DONet considering special filters whose structure is specifically inspired by the streak artifact singularities characterizing tomographic reconstructions from incomplete data. This allows to considerably lower the number of (learnable) parameters while preserving (or even slightly improving) the same quality for the reconstructions from limited-angle data and providing a proof-of-concept for the case of sparse-angle tomographic data.
In this work we solve, for given bounded operators B,C and Hilbert-Schmidt operator M acting on potentially infinite-dimensional separable Hilbert spaces, the reduced rank approximation problem, min {& Vert;M-BXC & Vert;(L2 ): dim ranX <= r}. This extends the result of Sondermann (Statistische Hefte, 1986) and Friedland and Torokhti (SIAM J. Matrix Analysis and Applications, 2007), which studies this problem in the case of matrices M, B, C, X, and the analysis involves the Moore-Penrose inverse. In classical approximation problems that can be solved by the singular value decomposition or Moore-Penrose inverse, the solution satisfies a minimal norm property. Friedland and Torokhti state such a minimal norm property of the solution. We show that this minimal norm property does not hold in general and give a modified minimality property that does hold. We show that the solution may be discontinuous in infinite-dimensional settings. We give conditions for continuity of the solutions and construct continuous approximations when such conditions are not met. Finally, we study problems from signal processing, reduced rank regression and linear operator learning under a rank constraint. Our theoretical results enable us to explicitly find solutions to these problems and to characterize their existence, uniqueness and minimality property.
In this paper, we propose a viscosity-type CQ algorithm, solving the split equality problem by using two adaptive step sizes and combining the alternated inertial technique in Hilbert space. The strong convergence of the sequences generated by the proposed algorithm is proved under some mild conditions. Finally, numerical experiments are conducted to verify the effectiveness and superiority of the method. The research results are innovative and provide new expansion and supplement to the relevant research.
In this paper, we obtain some sufficient conditions for the existence of a best proximity pair and introduce a new iterative algorithm, which converges to a best proximity pair for a class of noncyclic relatively enriched nonexpansive mappings in uniformly convex Banach spaces. We also provide numerical examples to support our claims. As an application, we find that our algorithm converges to the best proximity pair for a system of ordinary differential equations.
The paper is devoted to the existence of robust optimal solutions for convex polynomial optimization problems in the face of data uncertainty both in the objective function and the constraints. Employing adequate tools of real semi-algebraic geometry, we first introduce the concept called robust tangency variety and investigate its properties. Then under the robust Mangasarian-Fromovitz constraint qualification, we establish connections between robust Palais-Smale condition, robust weak Palais-Smale condition, robust M-tameness, and condition (A) formulated for the restriction of the objective function on the constraint set and uncertain set associated with the objective function. Finally, based on obtained relationships, we derive some sufficient conditions for the existence of robust optimal solutions of the considered problem.
In this paper, a nonlinear least squares problem is studied on Riemannian manifolds. To tackle this problem, we propose the Gauss-Newton and Levenberg-Marquardt algorithm to solve nonlinear least squares problems defined on Riemannian manifolds. Under some reasonable conditions, the global convergence result is established, and the local convergence rate for nonlinear least squares problems is presented on Riemannian manifolds.
We introduce an unconstrained optimization approach to solving mixed split feasibility problems in Hilbert spaces. We use a method of selecting the control parameters so that the implementation of our algorithms does not require prior information on the transfer mapping norms. This avoids the difficult task of estimating these norms. Besides, we also present applications to solving the split feasibility problem with multiple output sets and to solving split equality problems. Finally, we exhibit three numerical examples which bring out the efficiency of our proposed algorithms when compared with some existing ones.
In this paper, we propose a numerical approach to solve the inverse problem of determining the right-hand side function of a nonlinear fractional-order differential equation using over-measured data. In this paper, we use a hybrid function-referred to as the fractional-order hybrid function of block-pulse functions and Fibonacci polynomials-which combines the predictive capabilities of Fibonacci polynomials with the effectiveness of block-pulse functions in modeling discontinuous behavior, allowing it to represent both continuous and discontinuous solutions. The term fractional-order refers to the transformation x -> x alpha applied to the Fibonacci polynomials, where alpha is a real parameter. This hybrid function is used for the first time in the context of inverse problems. We also introduce an exact Riemann-Liouville integral operator, expressed using the regularized beta function, and require that the solutions be represented in a basis of hybrid functions, reducing the problem to an algebraic system of equations with an unknown right-hand side. Discretizing the system using Newton-Cotes nodes yields a homogeneous system of algebraic equations, from which the coefficients of the basis vectors can be determined. Substituting these coefficients into the solutions of the nonlinear fractional differential equation allows us to recover the unknown right-hand side function. Finally, we present several illustrative examples to demonstrate the accuracy of the proposed method, showing that the exact nature of the integral operator results in a more accurate approximation compared to other methods.
