Showing papers in "Applications of Mathematics in 2017"
TL;DR: This article designs a simple modification of the classic differential evolution algorithm that limits the possible premature convergence to local minima and ensures the asymptotic global convergence.
Abstract: Differential evolution algorithms represent an up to date and efficient way of solving complicated optimization tasks. In this article we concentrate on the ability of the differential evolution algorithms to attain the global minimum of the cost function. We demonstrate that although often declared as a global optimizer the classic differential evolution algorithm does not in general guarantee the convergence to the global minimum. To improve this weakness we design a simple modification of the classic differential evolution algorithm. This modification limits the possible premature convergence to local minima and ensures the asymptotic global convergence. We also introduce concepts that are necessary for the subsequent proof of the asymptotic global convergence of the modified algorithm. We test the classic and modified algorithm by numerical experiments and compare the efficiency of finding the global minimum for both algorithms. The tests confirm that the modified algorithm is significantly more efficient with respect to the global convergence than the classic algorithm.
25 citations
TL;DR: In this paper, a general form of PDE for pricing of Asian option contracts on two assets is presented, based on an ideal pure diffusion process for two risky asset prices with an additional pathdependent variable for continuous arithmetic average.
Abstract: Option pricing models are an important part of financial markets worldwide. The PDE formulation of these models leads to analytical solutions only under very strong simplifications. For more general models the option price needs to be evaluated by numerical techniques. First, based on an ideal pure diffusion process for two risky asset prices with an additional path-dependent variable for continuous arithmetic average, we present a general form of PDE for pricing of Asian option contracts on two assets. Further, we focus only on one subclass—Asian options with floating strike—and introduce the concept of the dimensionality reduction with respect to the payoff leading to PDE with two spatial variables. Then the numerical option pricing scheme arising from the discontinuous Galerkin method is developed and some theoretical results are also mentioned. Finally, the aforementioned model is supplemented with numerical results on real market data.
17 citations
TL;DR: A residual-based a posteriori error estimator for a discontinuous Galerkin approximation of the Steklov eigenvalue problem is derived and the reliability and efficiency of this estimator are proved.
Abstract: We derive a residual-based a posteriori error estimator for a discontinuous Galerkin approximation of the Steklov eigenvalue problem. Moreover, we prove the reliability and efficiency of the error estimator. Numerical results are provided to verify our theoretical findings.
14 citations
TL;DR: In this article, the control variational problem of a nonlinear beam model is solved by a simple linear state problem and solved by the conditioned gradient method, where the normal compliance condition is employed.
Abstract: This paper deals with a nonlinear beam model which was published by D.Y.Gao in 1996. It is considered either pure bending or a unilateral contact with elastic foundation, where the normal compliance condition is employed. Under additional assumptions on data, higher regularity of solution is proved. It enables us to transform the problem into a control variational problem. For basic types of boundary conditions, suitable transformations of the problem are derived. The control variational problem contains a simple linear state problem and it is solved by the conditioned gradient method. Illustrative numerical examples are introduced in order to compare the Gao beam with the classical Euler-Bernoulli beam.
13 citations
TL;DR: A new weighted version of the Gompertz distribution is introduced in this paper, where the model represents a mixture of classical GOMpertz and second upper record value of Gomertz densities, and using a certain transformation it gives a new version of two-parameter Lindley distribution.
Abstract: A new weighted version of the Gompertz distribution is introduced It is noted that the model represents a mixture of classical Gompertz and second upper record value of Gompertz densities, and using a certain transformation it gives a new version of the two-parameter Lindley distribution The model can be also regarded as a dual member of the log-Lindley-X family Various properties of the model are obtained, including hazard rate function, moments, moment generating function, quantile function, skewness, kurtosis, conditional moments, mean deviations, some types of entropy, mean residual lifetime and stochastic orderings Estimation of the model parameters is justified by the method of maximum likelihood Two real data sets are used to assess the performance of the model among some classical and recent distributions based on some evaluation goodness-of-fit statistics As a result, the variance-covariance matrix and the confidence interval of the parameters, and some theoretical measures have been calculated for such data for the proposed model with discussions
12 citations
TL;DR: A complex algorithm is proposed which controls discretization and algebraic errors and drives the adaptation of the mesh in the close to optimal manner with respect to the given quantity of interest.
Abstract: We deal with a posteriori error control of discontinuous Galerkin approximations for linear boundary value problems. The computational error is estimated in the framework of the Dual Weighted Residual method (DWR) for goal-oriented error estimation which requires to solve an additional (adjoint) problem. We focus on the control of the algebraic errors arising from iterative solutions of algebraic systems corresponding to both the primal and adjoint problems. Moreover, we present two different reconstruction techniques allowing an efficient evaluation of the error estimators. Finally, we propose a complex algorithm which controls discretization and algebraic errors and drives the adaptation of the mesh in the close to optimal manner with respect to the given quantity of interest.
12 citations
TL;DR: In particular, optimal interpolation properties of linear simplicial elements in ℝd that degenerate in some way are proved.
Abstract: The maximum angle condition of J. L. Synge was originally introduced in interpolation theory and further used in finite element analysis and applications for triangular and later also for tetrahedral finite element meshes. In this paper we present some of its generalizations to higher-dimensional simplicial elements. In particular, we prove optimal interpolation properties of linear simplicial elements in ℝd that degenerate in some way.
11 citations
TL;DR: In this article, the inverse scattering problem of determining the shape and location of a crack surrounded by a known inhomogeneous media is considered and a reciprocity relationship between the background Green function and the solution of an auxiliary scattering problem is proved.
Abstract: We consider the inverse scattering problem of determining the shape and location of a crack surrounded by a known inhomogeneous media. Both the Dirichlet boundary condition and a mixed type boundary conditions are considered. In order to avoid using the background Green function in the inversion process, a reciprocity relationship between the Green function and the solution of an auxiliary scattering problem is proved. Then we focus on extending the factorization method to our inverse shape reconstruction problems by using far field measurements at fixed wave number. We remark that this is done in a non intuitive space for the mixed type boundary condition as we indicate in the sequel.
11 citations
TL;DR: In this article, the Sakurai-Sugiura method with Rayleigh-Ritz projection (SS-RR) was used to solve polynomial eigenvalue problems. But, the SS-RR method suffers from backward instability when the norms of the coefficient matrices of the projected PEP vary widely.
Abstract: One of the most efficient methods for solving the polynomial eigenvalue problem (PEP) is the Sakurai-Sugiura method with Rayleigh-Ritz projection (SS-RR), which finds the eigenvalues contained in a certain domain using the contour integral. The SS-RR method converts the original PEP to a small projected PEP using the Rayleigh-Ritz projection. However, the SS-RR method suffers from backward instability when the norms of the coefficient matrices of the projected PEP vary widely. To improve the backward stability of the SS-RR method, we combine it with a balancing technique for solving a small projected PEP. We then analyze the backward stability of the SS-RR method. Several numerical examples demonstrate that the SS-RR method with the balancing technique reduces the backward error of eigenpairs of PEP.
8 citations
TL;DR: In this paper, the second order Navier-Stokes viscosity was used to model the viscous and dissipative effects of hydrodynamic flows in astrophysical disks.
Abstract: We calculate self-consistent time-dependent models of astrophysical processes. We have developed two types of our own (magneto) hydrodynamic codes, either the operator-split, finite volume Eulerian code on a staggered grid for smooth hydrodynamic flows, or the finite volume unsplit code based on the Roe's method for explosive events with extremely large discontinuities and highly supersonic outbursts. Both the types of the codes use the second order Navier-Stokes viscosity to realistically model the viscous and dissipative effects. They are transformed to all basic orthogonal curvilinear coordinate systems as well as to a special non-orthogonal geometric system that fits to modeling of astrophysical disks. We describe mathematical background of our codes and their implementation for astrophysical simulations, including choice of initial and boundary conditions. We demonstrate some calculated models and compare the practical usage of numerically different types of codes.
7 citations
TL;DR: In this paper, the authors focus on the controllability of complex systems, defined as the minimum set of controls that are needed in order to steer the whole system toward any desired state, and focus the study on the obtention of the set of all B making the system (A, B) exact controllable.
Abstract: In recent years there has been growing interest in the descriptive analysis of complex systems, permeating many aspects of daily life, obtaining considerable advances in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the controllability of the dynamics taking place on them. Concretely, for complex systems it is of interest to study the exact controllability; this measure is defined as the minimum set of controls that are needed in order to steer the whole system toward any desired state. In this paper, we focus the study on the obtention of the set of all B making the system (A, B) exact controllable.
TL;DR: In this paper, the authors used Almansi-type decomposition of bi-harmonic functions and proved that an approximate solution exists uniquely and that the approximation error decays exponentially with respect to the number of singular points.
Abstract: The aim of this paper is to analyze mathematically the method of fundamental solutions applied to the biharmonic problem. The key idea is to use Almansi-type decomposition of biharmonic functions, which enables us to represent the biharmonic function in terms of two harmonic functions. Based on this decomposition, we prove that an approximate solution exists uniquely and that the approximation error decays exponentially with respect to the number of the singular points. We finally present results of numerical experiments, which verify the sharpness of our error estimate.
TL;DR: In this paper, the numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain, where the discrete fractional Laplacian is approximated with a matrix power.
Abstract: Numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain. An appropriate extension is applied to have a well-posed problem on R2 and the solution on the square is regarded as a localization. For the numerical approximation a finite difference method is applied combined with the matrix transformation method. Here the discrete fractional Laplacian is approximated with a matrix power instead of computing the complicated approximations of fractional order derivatives. The spatial convergence of this method is proved and demonstrated by some numerical experiments.
TL;DR: A new Broyden method is proposed for solving systems of nonlinear equations, which uses the first derivatives, but is more efficient than the Newton method (measured by the computational time) for larger dense systems.
Abstract: We propose a new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR or LU decompositions of nonsymmetric approximations of the Jacobian matrix, so it requires O(n
2) arithmetic operations per iteration in contrast with the Newton method, which requires O(n
3) operations per iteration. Computational experiments confirm the high efficiency of the new method.
TL;DR: In this article, Nanda et al. generalize their results and characterize some distributions through functions used by them and Glaser's function, and obtain bounds for the expected values of selected functions in reliability theory.
Abstract: Nanda (2010) and Bhattacharjee et al. (2013) characterized a few distributions with help of the failure rate, mean residual, log-odds rate and aging intensity functions. In this paper, we generalize their results and characterize some distributions through functions used by them and Glaser’s function. Kundu and Ghosh (2016) obtained similar results using reversed hazard rate, expected inactivity time and reversed aging intensity functions. We also, via w(·)-function defined by Cacoullos and Papathanasiou (1989), characterize exponential and logistic distributions, as well as Type 3 extreme value distribution and obtain bounds for the expected values of selected functions in reliability theory. Moreover, a bound for the varentropy of random variable X is provided.
TL;DR: In this paper, the authors extended the results of numerical pricing of path-dependent multi-asset options to the case of Asian option contracts with fixed strike and applied the discontinuous Galerkin (DG) method to solve the problem.
Abstract: The evaluation of option premium is a very delicate issue arising from the assumptions made under a financial market model, and pricing of a wide range of options is generally feasible only when numerical methods are involved. This paper is based on our recent research on numerical pricing of path-dependent multi-asset options and extends these results also to the case of Asian options with fixed strike. First, we recall the three-dimensional backward parabolic PDE describing the evolution of European-style Asian option contracts on two assets, whose payoff depends on the difference of the strike price and the average value of the basket of two underlying assets during the life of the option. Further, a suitable transformation of variables respecting this complex form of a payoff function reduces the problem to a two-dimensional equation belonging to the class of convection-diffusion problems and the discontinuous Galerkin (DG) method is applied to it in order to utilize its solving potentials. The whole procedure is accompanied with theoretical results and differences to the floating strike case are discussed. Finally, reference numerical experiments on real market data illustrate comprehensive empirical findings on Asian options.
TL;DR: For simplices of dimension n ≥ 3, acuteness is defined by demanding that all dihedral angles between (n−1)-dimensional faces are smaller than π/2 as mentioned in this paper.
Abstract: Acute triangles are defined by having all angles less than π/2, and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension n ≥ 3, acuteness is defined by demanding that all dihedral angles between (n−1)-dimensional faces are smaller than π/2. However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property of acuteness is logically independent of the property of containing the circumcenter when the dimension is greater than two. In this article, we show that the latter property is also quite rare in higher dimensions. In a natural probability measure on the set of n-dimensional simplices, we show that the probability that a uniformly random n-simplex contains its circumcenter is 1/2
n
.
TL;DR: The estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian, useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities.
Abstract: Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian. The estimates depend on the decomposition and discretization parameters and gluing conditions. We also show how to plug the results into the analysis of H-TFETI methods and compare the estimates with numerical experiments. The results are useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities.
TL;DR: A low-rank tensor structured representation of Slatertype and Hydrogen-like orbital basis functions that can be used in electronic structure calculations using the tensor representation of basis functions and can take advantage of parallel computing.
Abstract: The paper focuses on a low-rank tensor structured representation of Slatertype and Hydrogen-like orbital basis functions that can be used in electronic structure calculations. Standard packages use the Gaussian-type basis functions which allow us to analytically evaluate the necessary integrals. Slater-type and Hydrogen-like orbital functions are physically more appropriate, but they are not analytically integrable. A numerical integration is too expensive when using the standard discretization techniques due the dimensionality of the problem. However, it can be effectively performed using the tensor representation of basis functions. Furthermore, this approach can take advantage of parallel computing.
TL;DR: In this paper, the authors provided a correction to the mean convergence and complete convergence theorems for sequences of m-linearly negative quadrant dependent random variables, which were incorrect.
Abstract: The authors provide a correction to “Some mean convergence and complete convergence theorems for sequences of m-linearly negative quadrant dependent random variables”.
TL;DR: In this paper, the authors extend the analysis of the recently proposed nonlinear EIS scheme applied to the partial eigenvalue problem to the case where the Rayleigh quotient iteration is used as the smoother on the fine-level.
Abstract: We extend the analysis of the recently proposed nonlinear EIS scheme applied to the partial eigenvalue problem. We address the case where the Rayleigh quotient iteration is used as the smoother on the fine-level. Unlike in our previous theoretical results, where the smoother given by the linear inverse power method is assumed, we prove nonlinear speed-up when the approximation becomes close to the exact solution. The speed-up is cubic. Unlike existent convergence estimates for the Rayleigh quotient iteration, our estimates take advantage of the powerful effect of the coarse-space.
TL;DR: A natural connection between Galerkin time discretizations and Runge-Kutta methods together with order reduction phenomenon is discussed and the theory of stability for stiff ordinary differential equations is explained.
Abstract: The aim of this work is to give an introductory survey on time discretizations for liner parabolic problems. The theory of stability for stiff ordinary differential equations is explained on this problem and applied to Runge-Kutta and multi-step discretizations. Moreover, a natural connection between Galerkin time discretizations and Runge-Kutta methods together with order reduction phenomenon is discussed.
TL;DR: In this article, the T-TLS method is applied to the problem AX ≈ B, where B = [b1,..., bd] is a matrix. And the corresponding filter factors are explicitly derived.
Abstract: The total least squares (TLS) and truncated TLS (T-TLS) methods are widely known linear data fitting approaches, often used also in the context of very ill-conditioned, rank-deficient, or ill-posed problems. Regularization properties of T-TLS applied to linear approximation problems Ax ≈ b were analyzed by Fierro, Golub, Hansen, and O’Leary (1997) through the so-called filter factors allowing to represent the solution in terms of a filtered pseudoinverse of A applied to b. This paper focuses on the situation when multiple observations b1,..., bd are available, i.e., the T-TLS method is applied to the problem AX ≈ B, where B = [b1,..., bd] is a matrix. It is proved that the filtering representation of the T-TLS solution can be generalized to this case. The corresponding filter factors are explicitly derived.
TL;DR: This work considers the finite element method for the time-dependent Stokes problem with the slip boundary condition in a smooth domain and proposes reduced and non-reduced integration schemes for the penalty method, and obtains an error estimate for velocity and pressure.
Abstract: We consider the finite element method for the time-dependent Stokes problem with the slip boundary condition in a smooth domain. To avoid a variational crime of numerical computation, a penalty method is introduced, which also facilitates the numerical implementation. For the continuous problem, the convergence of the penalty method is investigated. Then we study the fully discretized finite element approximations for the penalty method with the P1/P1-stabilization or P1b/P1 element. For the discretization of the penalty term, we propose reduced and non-reduced integration schemes, and obtain an error estimate for velocity and pressure. The theoretical results are verified by numerical experiments.
TL;DR: The paper presents shorter proofs, extended and new results compared to earlier publications on two solution methods involving very efficient preconditioned matrices based on a Schur complement reduction and a transformation matrix with a perturbation of one of the given matrix blocks.
Abstract: Two-by-two block matrices of special form with square matrix blocks arise in important applications, such as in optimal control of partial differential equations and in high order time integration methods. Two solution methods involving very efficient preconditioned matrices, one based on a Schur complement reduction of the given system and one based on a transformation matrix with a perturbation of one of the given matrix blocks are presented. The first method involves an additional inner solution with the pivot matrix block but gives a very tight condition number bound when applied for a time integration method. The second method does not involve this matrix block but only inner solutions with a linear combination of the pivot block and the off-diagonal matrix blocks. Both the methods give small condition number bounds that hold uniformly in all parameters involved in the problem, i.e. are fully robust. The paper presents shorter proofs, extended and new results compared to earlier publications.
TL;DR: This work proposes a new block conjugate gradient type method based on the Schur complement of a certain 2-by-2 real block form that consists of building blocks that involve only real arithmetic with real symmetric matrices of the original size.
Abstract: We consider solving complex symmetric linear systems with multiple right-hand sides. We assume that the coefficient matrix has indefinite real part and positive definite imaginary part. We propose a new block conjugate gradient type method based on the Schur complement of a certain 2-by-2 real block form. The algorithm of the proposed method consists of building blocks that involve only real arithmetic with real symmetric matrices of the original size. We also present the convergence property of the proposed method and an efficient algorithmic implementation. In numerical experiments, we compare our method to a complex-valued direct solver, and a preconditioned and nonpreconditioned block Krylov method that uses complex arithmetic.
TL;DR: In this paper, the meshless element-free Galerkin method is developed for numerical analysis of hyperbolic initial-boundary value problems, where only scattered nodes are required in the domain.
Abstract: The meshless element-free Galerkin method is developed for numerical analysis of hyperbolic initial-boundary value problems. In this method, only scattered nodes are required in the domain. Computational formulae of the method are analyzed in detail. Error estimates and convergence are also derived theoretically and verified numerically. Numerical examples validate the performance and efficiency of the method.
TL;DR: It is demonstrated on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error, that the Galerkin projection for second-order elliptic problems shows a dependence on the maximum angle of all triangles in the triangulation.
Abstract: Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H1 norm best approximation error estimates for H2 functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.
TL;DR: The main goal is in proposing block diagonal preconditionings to systems arising from the discretization of the Biot-Barenblatt model by a mixed finite element method in space and implicit Euler method in time and estimating the condition number for such preconditionsing.
Abstract: Poroelastic systems describe fluid flow through porous medium coupled with deformation of the porous matrix. In this paper, the deformation is described by linear elasticity, the fluid flow is modelled as Darcy flow. The main focus is on the Biot-Barenblatt model with double porosity/double permeability flow, which distinguishes flow in two regions considered as continua. The main goal is in proposing block diagonal preconditionings to systems arising from the discretization of the Biot-Barenblatt model by a mixed finite element method in space and implicit Euler method in time and estimating the condition number for such preconditioning. The investigation of preconditioning includes its dependence on material coefficients and parameters of discretization.
TL;DR: A full multigrid finite element method for semilinear elliptic equations that only needs the Lipschitz continuation in some sense of the nonlinear term for solving linear boundary value problems.
Abstract: A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.