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Partial data is a common challenge in many applied problems, where we can measure only the part of the solution. This paper considers the poroelasticity problem in fractured media, where only partial measurements of fracture pressures are available. The poroelasticity models have applications in many engineering and scientific fields, such as geomechanics, hydrology, and reservoir simulation. This paper aims to develop a Bayesian decision making approach using partial data and consider its application to the poroelasticity problem in fractured media. The methodology of the proposed approach combines Proper Orthogonal Decomposition, Discrete Empirical Interpolation Method, Bayesian Inversion Approach, and Bayesian Decision Theory. It allows us to compute posterior probabilities and decide according to the optimal decision rule based on the partial data. Moreover, we apply partially explicit temporal discretization to solve forward problems efficiently. For testing the proposed approach, we consider a two-dimensional model problem. We simulate the cases when we need to make a single decision about the reservoir or decisions about the elements of the set of reservoirs. The results show that the proposed approach can efficiently compute the posterior probabilities and make a decision in the presence of only partial data for both cases. [ABSTRACT FROM AUTHOR]

In this paper, we propose an extended Ulm-like method for solving the inverse singular value problem (ISVP for short) with multiple and/or zero singular values. Compared with the Ulm-like method, the proposed method reduces the amount of calculations and is well-defined even when multiple and/or zero singular values appear. Under the nonsingularity assumption for the relative generalized Jacobian at a solution and by using a new technique, a convergence analysis is provided and the quadratic convergence is proved. Our results in the present paper improve and extend significantly the corresponding ones of Vong et al. (2011) and Shen et al. (2018) for the ISVP with distinct and positive singular values and/or with square matrices. In particular, we solve the interesting problem raised in Vong et al. (2011): whether the Ulm-like method and its convergence result can be extended to the cases of multiple singular values and of zero singular values. Numerical results reveal that the extended Ulm-like method is an efficient algorithm for the ISVPs even when multiple and/or zero singular values appear. [ABSTRACT FROM AUTHOR]

In this paper, we consider an optimal control problem whose dynamic system consists of control-dependent time-delayed arguments in the control and discrete time-delayed arguments in the state. Recently, control parameterization methods using hybrid time-scale transformation techniques have been developed to optimize both the control parameters and their switching times when the dynamic system consists of discrete time-delayed arguments only. This transformation maps the variable switching times in the original time horizon to fixed switching times in the new time horizon. Thus, the purpose of this paper is to extend the above technique to solve problems with both control-dependent time-delayed arguments and discrete time-delayed arguments. Using the above technique, we convert both the delayed state vector and the delayed control vector into non-delayed vectors in the new time horizon; hence both the control parameters and their switching times can be optimized. The gradients of the objective functions and the constraint functions with respect to the switching times and the controls are obtained. Two numerical examples, one of which is a practical problem involving the optimal transmission of messages in an internet, are solved to demonstrate the efficiency of our method. [ABSTRACT FROM AUTHOR]

Fourier phase retrieval (FPR) is an inverse problem that aims to recover a signal from its Fourier magnitude measurement. It is an ill-posed problem, especially when the number of measurements is limited. In this paper, we introduce an untrained generative network that learns image priors without training on a large dataset to alleviate the ill-posedness. Based on the alternating direction method of multipliers (ADMM), an algorithm called Net-ADM that utilizes the untrained generative network is proposed to solve the FPR problem. Firstly, the dimension of the variable is increased and the objective function is modified to facilitate calculation. Then, an untrained generative network is embedded in the iterative process of ADMM to project an estimated signal into the generative space that captures some image characteristics. We theoretically prove the existence of a solution to the established model and analyze the convergence of the algorithm. Furthermore, one projection of Net-ADM makes the objective function non-increasing, while the other can bring the estimation closer to the optimal solution (or at least maintain the same distance) under certain conditions. Numerical experiments show that the proposed algorithm achieves superior accuracy and robustness compared to prior works, especially when the number of measurements is limited. • Net-ADM is proposed to solve the Fourier phase retrieval problem. • Net-ADM combines the Alternating Direction method of Multipliers with an untrained generative Network. • The descent of the objective function and the convergence of the iterations are analyzed theoretically. • The accuracy and robustness are superior to prior works, especially when the sampling ratios are low. [ABSTRACT FROM AUTHOR]

Inspired by CCC-Schoenberg operators that converge only quadratically, this paper proposes a generalization that has provably higher order of convergence. Two kinds of proposed approximations can be chosen to have order less or equal than the order of the underlying CCC-spline space. With the intention to compare our approximations with an interpolant, its error bound is also calculated. The proposed methods of maximal order are comparable with the interpolant and usually show more acceptable behavior. As an application of the new methods, certain boundary value problems are solved in the sense of quasi-collocation. [ABSTRACT FROM AUTHOR]

Multi-objective programming problem often contains numerous efficient solutions, which confuses the decision-maker. To assist in selecting the most desirable solution, optimizing a function over the efficient set becomes crucial. In this paper, we present a novel method for optimizing a general quadratic function over the efficient set of a multi-objective integer linear programming problem. To solve this problem, a ranking approach and efficiency test is utilized. The proposed methodology obtains a globally optimal solution by systematically scanning ranked solutions of an integer quadratic programming problem until the efficiency condition is satisfied. For generating ranked solutions, we construct a related integer linear programming problem. Then, ranked solutions of the integer linear programming problem are used for enumerating ranked solutions of the integer quadratic programming problem. The convergence of our algorithm is established theoretically, and its steps are illustrated using a numerical example. Aparticular case of the proposed method for optimizing a linear function over the efficient set of a multi-objective integer linear programming problem is also discussed. Further, extensive computational results demonstrate the effectiveness of our method for solving problems with large number of constraints, variables, and objective functions. Moreover, comparative analysis shows that the developed algorithm came out to be computationally more efficient as compared to the existing state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

The double-step scale splitting and the minimum residual modified HSS iteration method are effective methods for a class of large sparse complex symmetric systems of linear equations. In this paper, we give an efficient method to solve the linear equations related to complex symmetric systems. Firstly, we modify the double-step scale splitting (DSS) iteration methods, obtaining the new double-step scale splitting (NDSS) iteration methods. Furthermore, we improve the algorithm by taking use of minimum residual technique. We call the modified algorithm minimum residual NDSS (MRNDSS) iteration method. For the new presented method, we study some properties of the algorithm carefully. Compared with the NDSS method, MRNDSS has two different parameters in each alternating iteration step and two additional parameters depending on the residual error from previous step. In our numerical experiments, our new method performs more efficient than four other iteration methods which are usually applied to solve the same problems. [ABSTRACT FROM AUTHOR]

The Barzilai–Borwein (BB) method is a popular and efficient gradient method for solving large-scale unconstrained optimization problems. In general, it converges much faster than the steepest descent (Cauchy) method. However, it may not converge, even when the objective function is strongly convex. To overcome this, by virtue of bounding the distance between sequential iterates, [Burdakov et al. (2019)] proposed a stabilized BB method. In this paper, by combining the stabilized BB method with a projection approach, we develop a stabilized BB projection (SBBP) method to solve nonlinear monotone equations with convex constraints. SBBP does not involve computing matrices proving the usefulness for solving large-scale problems. Its global convergence and Q-linear convergence rate are also established. We report numerical experiments on recovering sparse signals and restoring blurred images arising from compressive sensing, demonstrating the usefulness of SBBP. [ABSTRACT FROM AUTHOR]

Fast and accurate solutions are of great importance to dynamic quadratic minimization (DQM) problems in many fields of engineering and science. By making a general survey of existing methods, DQM problems are capable of being solved very validly while there is still a lack of a superior performance method to face a noisy environment. To tolerate the ubiquitous noise, a zeroing neural network with a nonconvex saturated activation function model (ZNN-NCS) is proposed in this paper. The ZNN-NCS model exhibits high tolerance for bounded noise and responds to noise beyond the boundary, thereby monitoring environmental noise to a certain extent. Furthermore, combined with the control theory, an integral enhanced ZNN-NCS (IEZNN-NCS) model is proposed to adapt to the noise with unknown bounded. The IEZNN-NCS model has an effective ability to suppress noise that exceeds predefined boundary and noise that occurs suddenly. Afterwards, rigorous theoretical proofs of proposed models are conducted, and their global convergence and high robustness performance are demonstrated from a mathematical point of view. Finally, a special DQM problem is solved using the proposed models to demonstrate their effectiveness. In addition, the original ZNN (OZNN) and integral enhanced ZNN (IEZNN) models are used for comparison through this problem. Simulation results are consistent with the proposed theorems which validate proposed models. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a singularly perturbed Fredholm integro-differential problem with a discontinuous source term, leading to the formation of interior layers in the solution at the point of discontinuity. We apply the exponentially fitted mesh method to solve the problem. Our analysis demonstrates that the method exhibits almost first-order convergence in the maximum norm, independent of the diffusion parameter. Numerical experiments are conducted to verify the validity of our theoretical results and they support the accuracy of our estimates. [ABSTRACT FROM AUTHOR]

This paper considers the regularization parameter determination of l 1 -regularized minimization problem. We solve the l 1 -regularized problem using iterative reweighted least squares (IRLS) which involves solving a linear system whose coefficient matrix has the form α M + ( 1 − α ) N ( α ∈ ( 0 , 1 ) ). The aim of this paper is to find an efficient and computationally inexpensive algorithm to both choose the regularization parameter and solve the l 1 -regularized problem. In order to achieve this, we propose an IRLS algorithm with adaptive regularization parameter selection based on a heuristic parameter determination rule—de Boor’s parameter selection criterion. Compared with some of the state-of-the-art algorithms and parameter selection rules, the numerical experiments show the efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

An inverse source problem of a time–space fractional diffusion equation is considered in this paper. Due to the ill-posedness of the inverse problem, we propose a novel nonstationary iterated quasi-boundary value regularization method for reconstructing the source function, and show that the regularization problem is well-posed. The convergence rates are established under a priori and a posterior choice rules of regularization parameters, respectively. A numerical scheme for solving the regularization problem in one-dimensional case is derived from a finite difference method. Moreover, various of numerical examples are performed to test the efficiency of our method. • A new iterated regularization method for solving ISP is proposed. • A numerical scheme for implementing the proposed method is presented. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce new iterative algorithms for approximating a solution to a class of monotone operator equations. More precisely, we study the split common solution problem with multiple output sets for monotone operator equations in Hilbert spaces. In order to solve this problem, we propose three new algorithms and establish strong convergence theorems for them. [ABSTRACT FROM AUTHOR]

Box-constrained optimization problems in the real m × n matrix space have been widely applied in big data mining. However, efficient solution of them is still a challenge. In this paper, a new nonmonotone line search rule is first proposed by extending the well-known ones and inheriting their advantages. Then, by analyzing and exploiting properties of this rule, a new nonmonotone spectral projected gradient algorithm is developed to solve the box-constrained optimization problems in the matrix space. Global convergence of the developed algorithm is also established. Numerical tests are conducted on a series of randomly generated test problems and those in the set of benchmark test problems. Compared with other existing nonmonotone line search rules, our rule shows its advantages in terms of the significantly reduced number of function evaluations and significantly reduced number of iterations. To further validate applicability of this research, we apply the studied optimization problem and the developed algorithm to solve the problems of image clustering. Numerical results demonstrate that the proposed method can generate better clustering results and is more robust than the similar ones available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a variational inequality problem which is defined over the intersection of the solution set of variational inequality problem and the solution set of split common fixed point problem for averaged operators in Hilbert spaces. We present a new and efficient iterative method for solving this problem and establish its strong convergence. The proposed algorithm does not require knowledge of the Lipschitz constants of the operators and does not depend on the prior information of the bounded linear operator norms. Finally, we apply our result to study certain classes of optimization problems, and present some numerical experiments to demonstrate the applicability of our proposed algorithm. [ABSTRACT FROM AUTHOR]

Nonnegative matrix factorization (NMF) is a dimension reduction and clustering technique for data analysis which has been widely used in image processing, text analysis and hyperspectral decomposition because of its stronger practical significance and better interpretability. Approximate matrix factorization techniques with both nonnegativity and orthogonality constraints, referred to as orthogonal NMF (ONMF), have been shown to work remarkably better for clustering tasks than NMF. At present, a large number of algorithms have been used to solve the ONMF problems, but these methods usually cannot take into account the classification accuracy and calculation speed. In this paper, we propose a new form of penalized NMF with orthogonal regularization that combines the decomposition residual minimization based on the Euclidean distance and the orthogonality maximization based on the Kullback–Leibler divergence. This paper uses Majorization–Minimization (MM) method by minimizing a majorization function of the original problem and obtains a new iterative scheme (MM-ONMF). Comparing with several traditional ONMF methods on eight datasets, experimental results show that the proposed method has better clustering results and less computing time. [ABSTRACT FROM AUTHOR]

In this paper, we consider the one-dimensional initial–boundary problem for a pseudo-parabolic equation with a time delay in the second spatial derivative. To solve this problem numerically, we construct a higher-order difference method and obtain the error estimate for its solution. Based on the method of energy estimates the fully discrete scheme is shown to be convergent of order four in space and of order two in time. Given numerical results illustrate the convergence and effectiveness of the numerical method. [ABSTRACT FROM AUTHOR]

The rational fraction system is a special nonlinear system, the existence of the denominator polynomial leads to the difficulty of identifying rational fraction models. Inspired by the gradient search and the Newton method, the gradient-based iterative algorithm and Newton iterative algorithm are presented to estimate the parameters of rational fraction system models. Furthermore, in order to avoid a large amount of calculation and complex equations encountered in the process of solving partial derivatives, the model transformation-based gradient iterative algorithm and the model transformation-based Newton iterative algorithm are proposed for parameter identification. Two examples are carried out to show the effectiveness of the proposed algorithms. This paper focuses on solving the identification problem of rational fraction systems. [ABSTRACT FROM AUTHOR]

In this paper, we propose the tensor Noda iteration (NI) and its inexact version for solving the eigenvalue problem of a particular class of tensor pairs called generalized M -tensor pairs. A generalized M -tensor pair consists of a weakly irreducible nonnegative tensor and a nonsingular M -tensor within a linear combination. It is shown that any generalized M -tensor pair admits a unique positive generalized eigenvalue with a positive eigenvector. A modified tensor Noda iteration (MTNI) is developed for extending the Noda iteration for nonnegative matrix eigenproblems. In addition, the inexact generalized tensor Noda iteration method (IGTNI) and the generalized Newton-Noda iteration method (GNNI) are also introduced for more efficient implementations and faster convergence. Under a mild assumption on the initial values, the convergence of these algorithms is guaranteed. The efficiency of these algorithms is illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

The flexibility introduced by evolutionary algorithms (EAs) has allowed the use of virtually arbitrary objective functions and constraints—even when evaluations require, as for real-world problems, running complex mathematical and/or procedural simulations of the systems under analysis. Even so, EAs are not a panacea. Traditionally, the solution search process has been totally oblivious of the specific problem being solved, and optimization processes have been applied regardless of the size, complexity, and domain of the problem. In this paper, we justify our claim that far-reaching benefits may be obtained from more directly influencing how searches are performed. We propose using data mining techniques as a step for dynamically generating knowledge that can be used to improve the efficiency of solution search processes. In this paper, we use Kohonen SOMs and show an application for a well-known benchmark problem in the water distribution system design literature. The result crystallizes the conceptual rules for the EA to apply at certain stages of the evolution, which reduces the search space and accelerates convergence. [ABSTRACT FROM AUTHOR]

In this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the interpolating element-free Galerkin approach to approximate the spatial derivatives. The element-free Galerkin method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the element-free Galerkin method test and trial functions are moving least squares approximation shape functions. Also, in the element-free Galerkin method, we do not use any triangular, quadrangular or other type of meshes. It is a global method while finite element method is a local one. The element free Galerkin method is not a truly meshless method and for integration employs a background mesh. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. We show that convergence order of the time discrete scheme is O ( τ ) . Since the shape functions of moving least squares approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, the improved moving least squares shape functions that have the mentioned property are employed. An error estimate for the method proposed in the current paper is obtained. Also, the two-dimensional version of both equations on different complex geometries is solved. The aim of this paper is to show that the meshless method based on the weak form is also suitable for the treatment of the nonlinear partial differential equations and to obtain an error bound for the new method. Numerical examples confirm the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

Inexact Restoration methods have been proved to be effective to solve constrained optimization problems in which some structure of the feasible set induces a natural way of recovering feasibility from arbitrary infeasible points. Sometimes natural ways of dealing with minimization over tangent approximations of the feasible set are also employed. A recent paper [Banihashemi and Kaya (2013)] suggests that the Inexact Restoration approach can be competitive with well-established nonlinear programming solvers when applied to certain control problems without any problem-oriented procedure for restoring feasibility. This result motivated us to revisit the idea of designing general-purpose Inexact Restoration methods, especially for large-scale problems. In this paper we introduce affordable algorithms of Inexact Restoration type for solving arbitrary nonlinear programming problems and we perform the first experiments that aim to assess their reliability. Initially, we define a purely local Inexact Restoration algorithm with quadratic convergence. Then, we modify the local algorithm in order to increase the chances of success of both the restoration and the optimization phase. This hybrid algorithm is intermediate between the local algorithm and a globally convergent one for which, under suitable assumptions, convergence to KKT points can be proved. [ABSTRACT FROM AUTHOR]

In this paper, we consider a backward space-fractional diffusion problem. We propose an a posteriori parameter choice rule for the regularization method given in Zheng and Wei (2010), where the authors proposed a regularization method called convolution regularization method, and gave an a priori parameter choice strategy. In this paper, we study the same problem but give a new a posteriori parameter choice based on a modified version of the discrepancy principle, and obtain a log -type error estimate under an additional source condition. Numerical results show that our method is feasible. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of finding poles of rational functions from function values on open curves in the complex plane. For this problem, Nara and Ando recently proposed an algorithm that reduces the problem to a system of linear equations through contour integration. The main aim of this paper is to analyze and improve this algorithm by giving a new interpretation to the algorithm in terms of orthogonal polynomials. It is demonstrated that the system of linear equations is not always uniquely solvable and that this difficulty can be remedied by doubling the number of the linear equations. Moreover, to cope with discretization errors caused by numerical integration, we introduce new polynomials similar, in spirit, to discrete orthogonal polynomials, which yield an algorithm free from discretization errors. [ABSTRACT FROM AUTHOR]

In this paper, using the properties of graph convergence of H(.,.)-accretive operators, we construct a perturbed algorithm for solving systems of ill-posed variational inclusions involving H(.,.)-accretive operators. Under some suitable conditions, we prove that the sequence generated by the proposed method is strongly convergent to the unique solution of the problem. The main results presented in this paper generalize and improve some recent results in the literature. [ABSTRACT FROM AUTHOR]

This paper is concerned with the solution of large-scale linear discrete ill-posed problems. The determination of a meaningful approximate solution of these problems requires regularization. We discuss regularization by the Tikhonov method and by truncated iteration. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. The present paper describes the construction of square regularization matrices from finite difference equations with a focus on the boundary conditions. The regularization matrices considered have a structure that makes them easy to apply in iterative methods, including methods based on the Arnoldi process. Numerical examples illustrate the properties and effectiveness of the regularization matrices described. [ABSTRACT FROM AUTHOR]

In this paper, the improved modulus-based matrix splitting iteration methods are proposed to solve the quasi-complementarity problems. It is proved that the proposed iteration methods are convergent under certain conditions by using some new arguments. Numerical experiments are given to illustrate the high efficiency of the new iteration methods. Numerical results show that the proposed methods have better performance than the existing methods. [ABSTRACT FROM AUTHOR]

The stochastic Broyden–Fletcher–Goldfarb–Shanno (BFGS) method has effectively solved strongly convex optimization problems. However, this method frequently encounters the near-singularity problem of the Hessian. Additionally, obtaining the optimal solution necessitates a long convergence time. In this paper, we present a regularized stochastic Nesterov's accelerated quasi-Newton method that combines Nesterov acceleration with a novel momentum coefficient to effectively accelerate convergence speed and avoid the near-singularity problem of the Hessian update in the stochastic BFGS method. Moreover, we show the almost sure convergence of the generated subsequence of iterates to an optimal solution of the strongly convex optimization problems. We examined our approach to real-world datasets. The experiment results confirmed the effectiveness and superiority of the proposed method compared with other methods in solving classification problems. [Display omitted] • A novel quasi-Newton algorithm for solving strongly convex optimization problems. • A novel momentum coefficient is introduced to accelerate the convergence speed • Classification problems can be solved using the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a class of nonlinear semi-infinite optimization problems. These problems involve continuous inequality constraints that need to be satisfied at every point in an infinite index set, as well as conventional equality and inequality constraints. By introducing a novel penalty function to penalize constraint violations, we form an approximate optimization problem in which the penalty function is minimized subject to only bound constraints. We then show that this penalty function is exact—that is, when the penalty parameter is sufficiently large, any local solution of the approximate problem can be used to generate a corresponding local solution of the original problem. On this basis, the original problem can be solved as a sequence of approximate nonlinear programming problems. We conclude the paper with some numerical results demonstrating the applicability of our approach to PID control and filter design. [Copyright &y& Elsevier]

Abstract: In this paper, an effective method based upon the piecewise homotopy perturbation method (PHPM) is proposed for finding the numerical solutions of Fredholm integro-differential equations with weakly singular kernels. For these equations, the traditional homotopy perturbation method is divergent. So we present a modified homotopy perturbation method for solving this kind of equation. The advantages of the approach of this paper lie in the feature that this technique not only achieves an approximate solution with high accuracy, but also improves the calculation speed. Two examples are given to show the effectiveness and convenience of this method. [Copyright &y& Elsevier]

Recently, a direct method of the time–frequency approach, called the signal separation operator (SSO), which is based on sinusoidal signal approximation, was introduced to solving the inverse problem of multicomponent signal separation. In a very recent paper " Direct signal separation via extraction of local frequencies with adaptive time-varying parameters ", the authors obtained a more accurate component recovery formula derived from the linear chirp (also called linear frequency modulation signal) approximation at any local time. However the theoretical analysis of the recovery formula derived from linear chirp local approximation has not been studied there. In this paper, we carry out the analysis of SSO based on the adaptive short-time Fourier transform (STFT). We study both the sinusoidal signal-based model and the linear chirp-based model, and obtain the error bounds for the instantaneous frequency estimation and component recovery. The error bounds are derived by studying the approximation to the STFT of each component and by the assumption of the decrease of the Fourier transform of the window function for STFT. These results provide a mathematical guarantee to the proposed adaptive STFT-based non-stationary multicomponent signal separation method. In addition, experiments are provided to illustrate the general theory. [ABSTRACT FROM AUTHOR]

Abstract: In this paper numerical methods for solving stochastic differential equations with Markovian switching (SDEwMSs) are developed by pathwise approximation. The proposed family of strong predictor–corrector Euler–Maruyama methods is designed to overcome the propagation of errors during the simulation of an approximate path. This paper not only shows the strong convergence of the numerical solution to the exact solution but also reveals the order of the error under some conditions on the coefficient functions. A natural analogue of the -stability criterion is studied. Numerical examples are given to illustrate the computational efficiency of the new predictor–corrector Euler–Maruyama approximation. [Copyright &y& Elsevier]

Abstract: This paper presents an optimization technique for solving a maximum flow problem arising in widespread applications in a variety of settings. On the basis of the Karush–Kuhn–Tucker (KKT) optimality conditions, a neural network model is constructed. The equilibrium point of the proposed neural network is then proved to be equivalent to the optimal solution of the original problem. It is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the maximum flow problem. Several illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper. [Copyright &y& Elsevier]

Abstract: The linear complementarity problem is to find a vector in satisfying , ,, where and are given. In this paper, we use the fact that solving is equivalent to solving the nonlinear equation where is a function from into itself defined by . We build a sequence of smooth functions which is uniformly convergent to the function . We show that, an approximation of the solution of the (when it exists) is obtained by solving for a parameter large enough. Then we give a globally convergent hybrid algorithm which is based on vector divisions and the secant method for solving . We close our paper with some numerical simulations to illustrate our theoretical results, and to show that this method can solve efficiently large-scale linear complementarity problems. [Copyright &y& Elsevier]

Abstract In this paper, a numerical method is applied for solving delay fractional optimal control problems (DFOCPs). The fractional derivative is described in the Caputo sense. Since the fractional derivative of Müntz polynomials can be expressed in terms of the same polynomials, those polynomials can accurately represent properties of fractional calculus. In some situations such as in the frequency response based analysis of control systems containing a time-delay, it is necessary to substitute exponential function with an approximation in the form of a rational function. The most common approximation is the Padé approximation. At the first step, using Padé approximation, the delay problem is transformed to a non-delay problem. Next, using the operational matrix of the fractional derivative of Müntz polynomials and pseudospectral method, fractional optimal control problem (FOCP) is reduced to a nonlinear programming problem. Some numerical examples are given to illustrate the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

Abstract Although frames, which are a generalization of bases, are important tools used in signal processing, their potential in other fields of engineering and applied mathematics (e.g. acoustics) has not been fully explored yet. Gabor frames, that are specific type of frames, are very well adapted to oscillating functions, and therefore have a great potential to efficiently represent functions in connection with the Helmholtz operator. In this paper representations of the solution of a scattering problem in 2D using Gabor frames based on B-splines as building blocks are investigated. Practical issues concerning the implementation of frames like the restriction of the frame elements to a finite interval and methods to determine the unknown coefficients for the representation, i.e. by using dual frames or by solving a least squares problem, are discussed. Numerical experiments are made comparing the different ways to determine the unknown expansion coefficients and the representations are compared in terms of their efficiency, i.e. the number of used frame coefficients versus the accuracy of the approximation. In all cases the coefficients calculated with a slightly modified orthogonal matching pursuit algorithm provide the best accuracy versus sparsity ratio. [ABSTRACT FROM AUTHOR]

Abstract Bayesian hypothesis testing procedures are constructed by means of test statistics which are functions of the posterior distribution. Usually, the whole sample vector is selected to form the sufficient empirical part of the posterior distribution. But, in certain problems, one may prefer to use well-established one-dimensional sufficient statistics in place of the sample vector. This paper introduces a Bayesian Monte Carlo procedure specially designed for such cases. It is shown that the performance of this new approach is arbitrarily close to the exact Bayesian test. In addition, for arbitrary desired precisions, we develop a theoretical rule of thumb for choosing the minimum number m 0 of Monte Carlo simulations. Surprisingly, m 0 does not depend on the shape of loss/cost functions when those are used to compound the test statistic. The method is illustrated for testing mean vectors in high-dimension and for detecting spatial clusters of diseases in aggregated maps. [ABSTRACT FROM AUTHOR]

Abstract In this paper, an efficient numerical method is proposed for the 1D elliptic type interface problems. We first construct a broken reproducing kernel space and then apply the simplified reproducing kernel method to the interface problems. The convergence of the proposed method is discussed. Finally, some representative numerical examples are presented to illustrate the accuracy and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract Many inverse problems can be described by a PDE model with unknown parameters that need to be calibrated based on measurements related to its solution. This can be seen as a constrained minimization problem where one wishes to minimize the mismatch between the observed data and the model predictions, including an extra regularization term, and use the PDE as a constraint. Often, a suitable regularization parameter is determined by solving the problem for a whole range of parameters –e.g. using the L-curve –which is computationally very expensive. In this paper we derive two methods that simultaneously solve the inverse problem and determine a suitable value for the regularization parameter. The first one is a direct generalization of the Generalized Arnoldi Tikhonov method for linear inverse problems. The second method is a novel method based on similar ideas, but with a number of advantages for nonlinear problems. [ABSTRACT FROM AUTHOR]

Abstract This paper studies an optimal robust investment and reinsurance problem for an insurer under a surplus model with ambiguity that the insurer does not have perfect information in drift terms of the risky asset and insurance risk. The ambiguity is described by a set of equivalent measures to a benchmark. The insurer aims to maximize a robust value involving the probability of reaching a high goal before ruin and a penalization of model ambiguity. We characterize the robust value function as the unique classical solution to the Hamilton–Jacobi–Bellman (HJB) equation. We derive the value function in explicit form by solving the HJB equation, and find the optimal investment-reinsurance strategy and drift distortions of the optimal measure in feedback forms. We observe that the optimal strategy under the scenario of ambiguity penalization coincides with the optimal strategy under maximization of the goal-reaching probability in the benchmark case without ambiguity. We further demonstrate the impact of ambiguity-aversion on the insurer's evaluation and the selection of optimal drift distortions. [ABSTRACT FROM AUTHOR]

Abstract Quantifying and modeling shape variation within a population, identifying morphological contrasts across groups, and categorizing individuals or objects according to morphological similarity are central problems in numerous domains of science and applications. In this paper, we present an approach to optimal shape categorization through a new family of metrics for shapes presented as a finite collection of labeled landmark points. We develop a technique to learn metrics that optimally differentiate and categorize shapes using Monte Carlo optimization methods. We discuss the theory and the practice of the methods and apply them to the analysis of synthetic data and the classification of multiple species of fruit flies based on the shape of their wings. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers a one-dimensional cutting stock and assortment problem. One of the main difficulties in formulating and solving these kinds of problems is the use of the set of cutting patterns as a parameter set in the mathematical model. Since the total number of cutting patterns to be generated may be very huge, both the generation and the use of such a set lead to computational difficulties in solution process. The purpose of this paper is therefore to develop a mathematical model without the use of cutting patterns as model parameters. We propose a new, two-objective linear integer programming model in the form of simultaneous minimization of two contradicting objectives related to the total trim loss amount and the total number of different lengths of stock rolls to be maintained as inventory, in order to fulfill a given set of cutting orders. The model does not require pre-specification of cutting patterns. We suggest a special heuristic algorithm for solving the presented model. The superiority of both the mathematical model and the solution approach is demonstrated on test problems. [Copyright &y& Elsevier]

Abstract: In this paper, we study the following minimization problem where is a bounded linear operator, is some constant, is a potential function for , is the set of fixed points of nonexpansive mapping and is the solution set of an equilibrium problem. This paper introduces two new algorithms (one implicit and one explicit) that can be used to find the solution of the above minimization problem. [Copyright &y& Elsevier]

Abstract: In this paper, we present a mathematical study of the method of fundamental solutions (MFS) applied to reduced wave problems with Dirichlet boundary conditions in the exterior domain of a disk. A theorem in this paper shows that the MFS with source points in equi-distantly equally phased arrangement with assignment parameter , which characterizes the position of the source points and the collocation points, gives an approximate solution with error of if the Fourier coefficients of the boundary data decay exponentially. This error estimate is an extension of the results of the previous studies. Numerical examples make good agreements with the results of the theoretical study. [Copyright &y& Elsevier]

Image deblurring is a fundamental problem in imaging, usually solved with computationally intensive optimization procedures. The goal of this paper is to provide new efficient strategies to reduce computing times for simple deblurring models regularized using orthogonal wavelet transforms. We show that the minimization can be significantly accelerated by leveraging the fact that images and blur operators are compressible in the same orthogonal wavelet basis. The proposed methodology consists of three ingredients: (i) a sparse approximation of the blur operator in wavelet bases, (ii) a diagonal preconditioner and (iii) an implementation on massively parallel architectures. Combining the three ingredients leads to acceleration factors ranging from 4 to 250 on a typical workstation. For instance, a 1024 × 1024 image can be deblurred in 0.15 s. [ABSTRACT FROM AUTHOR]

In this paper, using the modified Adomian decomposition method, a novel fast scheme, has been presented, being interactive, to be used for solving a class of hypersingular integral equations of the first kind. Compared to the standard Adomian decomposition method, this method expedites the quick accumulation of a series of solutions. Different problems, for exemplification, have been solved in order to display the accuracy and validity of the aforementioned method. According to the results, it gets known that the novel fast modified Adomian decomposition method is so efficient and simple which gives the exact solution, and also shuns the complicated function of long theoretical calculations of collocation methods based on polynomial producing the exact solution. [ABSTRACT FROM AUTHOR]

Recently, Goh et al. (2012) [ 28 ] proposed a numerical technique based on quartic B-spline collocation for solving a class of singular boundary value problems (SBVP) with Neumann and Dirichlet boundary conditions (BC). This method is only fourth-order accurate. In this paper, we propose an optimal numerical technique for solving a more general class of nonlinear SBVP subject to Neumann and Robin BC. The method is based on high order perturbation of the problem under consideration. The convergence of the proposed method is analyzed. To demonstrate the applicability and efficiency of the method, we consider four numerical examples, three of which arise in various physical models in applied science and engineering. A comparison with other available numerical solutions has been carried out to justify the advantage of the proposed technique. Numerical result reveals that the proposed method is sixth order convergent, which in turn is two orders of magnitude larger than in Goh et al. (2012) [ 28 ]. [ABSTRACT FROM AUTHOR]

In this paper, we develop an online basis enrichment method with the mortar mixed finite element method, using the oversampling technique, to solve for flow problems in highly heterogeneous media. We first compute a coarse grid solution with a certain number of offline basis functions per edge, which are chosen as standard polynomials basis functions. We then iteratively enrich the multiscale solution space with online multiscale basis functions computed by using residuals. The iterative solution converges to the fine scale solution rapidly. We also propose an oversampling online method to achieve faster convergence speed. The oversampling refers to using larger local regions in computing the online multiscale basis functions. We present extensive numerical experiments(including both 2D and 3D) to demonstrate the performance of our methods for both steady state flow, and two-phase flow and transport problems. In particular, for the time dependent two-phase flow and transport problems, we apply the online method to the initial model, without updating basis along the time evolution. Our numerical results demonstrate that by using a few number of online basis functions, one can achieve a fast convergence. [ABSTRACT FROM AUTHOR]

The standard application of the Lehmann–Goerisch method for lower bounds on eigenvalues of symmetric elliptic second-order partial differential operators relies on determination of fluxes σ ̃ i that approximate co-gradients of exact eigenfunctions scaled by corresponding eigenvalues. Fluxes σ ̃ i are usually computed by solving a global saddle point problem with mixed finite element methods. In this paper we propose a simpler global problem that yields fluxes σ ̃ i of the same quality. The simplified problem is smaller, it is positive definite, and any H ( div , Ω ) conforming finite elements, such as Raviart–Thomas elements, can be used for its solution. In addition, these global problems can be split into a number of independent local problems on patches, which allows for trivial parallelization. The computational performance of these approaches is illustrated by numerical examples for Laplace and Steklov type eigenvalue problems. These examples also show that local flux reconstructions enable computation of lower bounds on eigenvalues on considerably finer meshes than the traditional global reconstructions. [ABSTRACT FROM AUTHOR]

The problem of accurate and efficient numerical integration of ordinary differential equations (ODEs) is a key issue of computation science, which has a valued practical impact in various topics of research. In this respect, effective global error estimation and control strategies are of particular theoretical and practical significance. They allow many applied dynamic systems to be treated numerically for user-supplied accuracy conditions in automatic mode. The property of double quasi-consistency plays an outstanding role in the mentioned topic because the global (or true) error of a doubly quasi-consistent method is asymptotically equal to its local error. Therefore, this entails quite a nontrivial implication that the true error of numerical integration can be cheaply evaluated and regulated within a single solution run. The latter was considered to be impossible for decades. The property of double quasi-consistency goes back to 2009, when Kulikov studied it in the first time but failed to find doubly quasi-consistent numerical methods among Nordsieck formulas. Later on, Kulikov and Weiner proved the existence of doubly quasi-consistent numerical schemes among superconvergent explicit two-step parallel peer methods and examined their practical efficiency in comparison to explicit Matlab ODE solvers in 2010. The focus of the present research is on accurate numerical integration formulas for treating stiff ODEs, which often arise in practice and for which explicit methods are shown to be ineffective. Here, we make the first step towards an accurate and efficient numerical solution of stiff ODEs and prove existence of implicit doubly quasi-consistent formulas. We fulfill our investigation of double quasi-consistency within the family of fixed-stepsize implicit two-step peer schemes and construct two methods of convergence orders 3 and 4, which possess excellent stability properties. Our theoretical results are confirmed with numerical integration of both nonstiff and stiff test problems in this paper. [ABSTRACT FROM AUTHOR]