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Abstract: Abstract Our main focus of this study is based on modified variational inclusion problems with mixed equilibrium problems in real Hilbert space. An iterative algorithm which is the combination of inertial technique and subgradient extragradient algorithm is presented, and under some reasonable assumptions on the control parameters, the strong convergence theorem of the suggested algorithm is proved. In addition, the efficiency of our suggested algorithm is established by a numerical example. The algorithm and analysis provided in this paper are new and generalize some related results in this regard. PubDate: 2023-03-05
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Abstract: Abstract In this paper we study the Brinkman system and the Darcy-Forchheimer-Brinkman system with the boundary condition of the Navier’s type \( {\textbf{u}}_{{\mathbf {\mathcal {T}}}} = {\textbf{g}}_{{\mathbf {\mathcal {T}}}} \) , \(\rho =h\) on \(\partial \Omega \) for a bounded planar domain \(\Omega \) with connected boundary. Solutions are looked for in the Sobolev spaces \(W^{s+1,q}(\Omega ,{\mathbb R}^2)\times W^{s,q}(\Omega )\) and in the Besov spaces \(B_{s+1}^{p,r}(\Omega ,{\mathbb R}^2)\times B_s^{q,r}(\Omega )\) . Classical solutions are from the spaces \({\mathcal C}^{k+1,\gamma }(\overline{\Omega },{\mathbb R}^2) \times {\mathcal C}^{k,\gamma }(\overline{\Omega })\) . For the Brinkman system we show the unique solvability of the problem. Then we study the Navier problem for the Darcy-Forchheimer-Brinkman system and small boundary conditions. PubDate: 2023-02-25
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Abstract: Abstract In this article, we study the following problem $$\begin{aligned} -div (\omega (x) \nabla u ^{N-2} \nabla u) = \lambda \ f(x,u) \quad \text{ in } \quad B, \quad u=0 \quad \text{ on } \quad \partial B, \end{aligned}$$ where B is the unit ball in \(\mathbb {R^{N}}\) , \(N\ge 2\) and w(x) a singular weight of logarithm type. The reaction source f(x, u) is a radial function with respect to x and is subcritical or critical with respect to a maximal growth of exponential type. By using the constrained minimization in Nehari set coupled with the quantitative deformation lemma and degree theory, we prove the existence of nodal solutions. PubDate: 2023-01-28
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Abstract: Abstract In this article, we introduce the multiple-sets split equality variational inequality problem which includes the split feasibility problem, split variational inequality problem, split equality problem and multiple-sets split variational inequality problem to mention a few. Also, we prove a strong convergence theorem for approximation the solution of multiple-sets split equality variational inequality problem and fixed point problem of multi-valued quasi-nonepansive mappings in real Hilbert spaces using a modified Halpern iterative algorithm. The iterative algorithm employed in this paper is designed in such a way that its implementation does not require the estimation of the operator norms. Lastly, we present some consequences and a numerical example to illustrate the performance of our main result. Our result extends and complements many related results in the literature. PubDate: 2023-01-10 DOI: 10.1007/s11565-022-00455-0
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Abstract: Abstract This paper focuses on an inexact block coordinate method designed for nonsmooth optimization, where each block-subproblem is solved by performing a bounded number of steps of a variable metric proximal–gradient method with linesearch. We improve on the existing analysis for this algorithm in the nonconvex setting, showing that the iterates converge to a stationary point of the objective function even when the proximal operator is computed inexactly, according to an implementable inexactness condition. The result is obtained by introducing an appropriate surrogate function that takes into account the inexact evaluation of the proximal operator, and assuming that such function satisfies the Kurdyka–Łojasiewicz inequality. The proof technique employed here may be applied to other new or existing block coordinate methods suited for the same class of optimization problems. PubDate: 2022-12-26 DOI: 10.1007/s11565-022-00456-z
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Abstract: Abstract We establish a set of conditions for the uniform-ultimate boundedness of solutions to a certain system of second order differential equations with variable delay using Lypunov–Krasovskii functional as a basic tool. This result is an addition to the body of literature in many ways. In addition, we provide an example to demonstrate the correctness of our result. PubDate: 2022-11-05 DOI: 10.1007/s11565-022-00454-1
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Abstract: Abstract Gibbs ringing is a feature of MR images caused by the finite sampling of the acquisition space (k-space). It manifests itself with ringing patterns around sharp edges which become increasingly significant for low-resolution acquisitions. In this paper, we model the Gibbs artefact removal as a constrained variational problem where the data discrepancy, represented in denoising and convolutive form, is balanced to sparsity-promoting regularization functions such as Total Variation, Total Generalized Variation and \(L_1\) norm of the Wavelet transform. The efficacy of such models is evaluated by running a set of numerical experiments both on synthetic data and real acquisitions of brain images. The Total Generalized Variation penalty coupled with the convolutive data discrepancy term yields, in general, the best results both on synthetic and real data. PubDate: 2022-11-01 DOI: 10.1007/s11565-022-00431-8
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Abstract: Abstract The approximation of functions and data in one and high dimensions is an important problem in many mathematical and scientific applications. Quasi-interpolation is a general and powerful approximation approach having many advantages. This paper deals with spline quasi-interpolants and its aim is to collect the main results obtained by the authors, also in collaboration with other researchers, in such a topic through spline dimension, i.e. in the 1D, 2D and 3D setting, highlighting the approximation properties and the reconstruction of functions and data, the applications in numerical integration and differentiation and the numerical solution of integral and differential problems. PubDate: 2022-11-01 DOI: 10.1007/s11565-022-00427-4
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Abstract: Abstract We consider 2D transient linear wave propagation problems defined in the exterior of bounded domains. In particular, we first consider the problem represented by the classical wave equation, and then that of the linear elastodynamics in a homogeneous isotropic medium. Both problems will be solved in the time domain. For the solution of these problems several methods are known since long time. However, a novel and interesting numerical approach has been proposed by Ch. Lubich in 1994 for the 3D wave equation Dirichlet problem. This approach is based on the space-time boundary integral equation associated with the PDE problem. It combines a new discrete convolution quadrature, for the discretization of the time integral, with a classical Galerkin boundary element method for the space integral. For this method, convergence has been proved, including also the error estimate. In the last decade we have considered this new approach, with the Galerkin space discretization replaced by the much (computationally) cheaper collocation analogue. We have successfully applied it to 2D problems, and also to some 3D ones. Although until now no convergence results have been proved for this approach, the intensive numerical testing we have performed confirm that its convergence holds with the same order of the Galerkin BEM. In this paper we give an overview of the main applications we have made of the collocation approach, for the solution of the above mentioned wave propagation problems. These also include a case with rotating obstacles. PubDate: 2022-11-01 DOI: 10.1007/s11565-022-00420-x
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Abstract: Abstract Environmental hydrodynamics is typically characterized by free-surface hydrostatic flows. Within such a framework a set of simplified models are derived from the Reynolds averaged Navier–Stokes equations under various simplifying assumptions. Numerically, a simple method for the 1D open channel flow is derived first. This method is then extended to the \(2D_{xz}\) laterally averaged model and to the \(2D_{xy}\) vertically averaged model. Next, a semi-implicit method for the 3D model is described and discussed. PubDate: 2022-11-01 DOI: 10.1007/s11565-022-00406-9
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Abstract: Abstract Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along a suitable orthonormal basis. Interestingly, this approach can be extended to cope with more general differential problems. In this paper we sketch this fact, by considering some relevant examples. PubDate: 2022-11-01 DOI: 10.1007/s11565-022-00409-6
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Abstract: Abstract In this work, three genetic regulatory networks are considered, that model the post–transcriptional regulation of the PTEN onco–suppressor gene, mediated by microRNAs and competitive endogenous RNAs, in glioblastoma multiforme, the most severe of brain tumours. We simulate solutions of the resulting stochastic differential systems and discuss the effects of this miRNA–fashioned regulation on PTEN expression. PubDate: 2022-11-01 DOI: 10.1007/s11565-022-00416-7
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Abstract: Abstract This paper studies the numerical solution of strictly convex unconstrained optimization problems by linesearch Newton-CG methods. We focus on methods employing inexact evaluations of the objective function and inexact and possibly random gradient and Hessian estimates. The derivative estimates are not required to satisfy suitable accuracy requirements at each iteration but with sufficiently high probability. Concerning the evaluation of the objective function we first assume that the noise in the objective function evaluations is bounded in absolute value. Then, we analyze the case where the error satisfies prescribed dynamic accuracy requirements. We provide for both cases a complexity analysis and derive expected iteration complexity bounds. We finally focus on the specific case of finite-sum minimization which is typical of machine learning applications. PubDate: 2022-11-01 DOI: 10.1007/s11565-022-00435-4
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Abstract: Abstract Anisotropic diffusion is a well recognized tool in digital image processing, including edge detection and focusing. We present here a particular nonlinear time-dependent operator together with an appropriate high-order discretization for the space variable. In just a single step, the procedure emphasizes the contour lines encircling the objects, paving the way to accurate reconstructions at a very low cost. One of the main features of such an approach is the possibility of relying on a rather large set of invariant discontinuous images, whose edges can be determined without introducing any approximation. PubDate: 2022-11-01 DOI: 10.1007/s11565-022-00419-4
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Abstract: Abstract Images that have been contaminated by various kinds of blur and noise can be restored by the minimization of an \(\ell ^p\) - \(\ell ^q\) functional. The quality of the reconstruction depends on the choice of a regularization parameter. Several approaches to determine this parameter have been described in the literature. This work presents a numerical comparison of known approaches as well as of a new one. PubDate: 2022-11-01 DOI: 10.1007/s11565-022-00430-9
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Abstract: Abstract We consider the discretization of elliptic boundary-value problems by variational physics-informed neural networks (VPINNs), in which test functions are continuous, piecewise linear functions on a triangulation of the domain. We define an a posteriori error estimator, made of a residual-type term, a loss-function term, and data oscillation terms. We prove that the estimator is both reliable and efficient in controlling the energy norm of the error between the exact and VPINN solutions. Numerical results are in excellent agreement with the theoretical predictions. PubDate: 2022-09-19 DOI: 10.1007/s11565-022-00441-6
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Abstract: Abstract In order to solve constrained optimization problems on convex sets, the class of scaled gradient projection methods is often exploited in combination with non-monotone Armijo–like line search strategies. These techniques are adopted for efficiently selecting the steplength parameter and can be realized by means of two different approaches: either the one along the arc or the one along the feasible directions. In this paper we deeply analyze the convergence properties of the scaled gradient projection methods equipped with the non-monotone version of both these Armijo–like line searches. To the best of our knowledge, not all the convergence results proved for either the non-scaled or the monotone gradient projection algorithm have been also stated for the non-monotone and scaled counterpart. The goal of this paper is to fill this gap of knowledge by detailing which hypotheses are needed to guarantee both the stationarity of the limit points and the convergence of the sequence generated by the non-monotone scaled gradient projection schemes. Moreover, in the case of polyhedral constraint set, we discuss the identification of the active set at the solution for the sequence generated by the along the arc approach. Several numerical experiments on quadratic and non-quadratic optimization problems have been carried out in order to compare the behaviour of the considered scaled gradient projection methods. PubDate: 2022-08-25 DOI: 10.1007/s11565-022-00437-2
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Abstract: Abstract In this paper, we address the preconditioned iterative solution of the saddle-point linear systems arising from the (regularized) Interior Point method applied to linear and quadratic convex programming problems, typically of large scale. Starting from the well-studied Constraint Preconditioner, we review a number of inexact variants with the aim to reduce the computational cost of the preconditioner application within the Krylov subspace solver of choice. In all cases we illustrate a spectral analysis showing the conditions under which a good clustering of the eigenvalues of the preconditioned matrix can be obtained, which foreshadows (at least in case PCG/MINRES Krylov solvers are used), a fast convergence of the iterative method. Results on a set of large size optimization problems confirm that the Inexact variants of the Constraint Preconditioner can yield efficient solution methods. PubDate: 2022-08-18 DOI: 10.1007/s11565-022-00422-9
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Abstract: Abstract The aim of this work is to present a general and simple strategy for the construction of compactly supported fundamental spline (piecewise-polynomial) functions for local interpolation, that are defined over quadrangulations of the real plane with extraordinary vertices. The proposed strategy — which extends the univariate framework introduced in Antonelli et al. (Adv Comput Math 40:945–976, 2014) and Beccari et al. (J Comput Appl Math 240:5–19, 2013)—consists in considering a suitable combination of bivariate polynomial interpolants with blending functions that are the natural generalization of odd-degree tensor-product B-splines. These blending functions are constructed as basic limit functions of the bivariate, primal subdivision schemes developed simultaneously in Stam (Comput Aided Geom Des 18:397–427, 2001) and Zorin et al. (Comput Aided Geom Des 18:483–502, 2001). As an application example of our constructive strategy we present the compactly supported \(C^2\) fundamental functions for local interpolation that arise by considering as blending functions the basic limit functions of the celebrated Catmull–Clark subdivision scheme proposed in Catmull and Clark (Comput Aided Des 46:103–124, 2016). PubDate: 2022-08-18 DOI: 10.1007/s11565-022-00423-8
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