**Applied Numerical Mathematics**[8 followers] Follow

**Hybrid journal**(

*It can contain Open Access articles*)

ISSN (Print) 0168-9274 - ISSN (Online) 0168-9274

Published by

**Elsevier**[2556 journals] [SJR: 1.208] [H-I: 44]

**A modified quasi-boundary value method for an inverse source problem of**

the time-fractional diffusion equation**Abstract:**Publication date: Available online 31 December 2013

Source:Applied Numerical Mathematics

Author(s): Ting Wei , Jungang Wang

In this paper, we consider an inverse source problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine a space-dependent source term in the time-fractional diffusion equation from a noisy final data. Based on a series expression of the solution, we can transform the original inverse problem into a first kind integral equation. The uniqueness and a conditional stability for the space-dependent source term can be obtained. Further, we propose a modified quasi-boundary value regularization method to deal with the inverse source problem and obtain two kinds of convergence rates by using an a priori and an a posteriori regularization parameter choice rule, respectively. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.

**PubDate:**2014-01-03T17:17:03Z

**Strong approximation of solutions of stochastic differential equations**

with time–irregular coefficients via randomized Euler algorithm**Abstract:**Publication date: Available online 31 December 2013

Source:Applied Numerical Mathematics

Author(s): Paweł Przybyłowicz , Paweł Morkisz

We investigate pointwise approximation of the solution of a scalar stochastic differential equation in case when drift coefficient is a Carathéodory mapping and diffusion coefficient is only piecewise Hölder continuous with Hölder exponent ϱ ∈ ( 0 , 1 ] . Since under imposed assumptions drift is only measurable with respect to the time variable, the classical Euler algorithm does not converge in general to the solution of such equation. We give a construction of the randomized Euler scheme and prove that it has the error O ( n − min { ϱ , 1 / 2 } ) , where n is the number of discretization points. We also investigate the optimality of the defined algorithm.

**PubDate:**2014-01-03T17:17:03Z

**Editorial Board****Abstract:**Publication date: March 2014

Source:Applied Numerical Mathematics, Volume 77

**PubDate:**2013-12-22T06:57:35Z

**A unified approach to identifying an unknown spacewise dependent source in**

a variable coefficient parabolic equation from final and integral

overdeterminations**Abstract:**Publication date: Available online 14 December 2013

Source:Applied Numerical Mathematics

Author(s): Alemdar Hasanov , Burhan Pektaş

An adjoint problem approach with subsequent conjugate gradient algorithm (CGA) for a class of problems of identification of an unknown spacewise dependent source in a variable coefficient parabolic equation u t = ( k ( x ) u x ) x + F ( x ) H ( t ) , ( x , t ) ∈ ( 0 , l ) × ( 0 , T ] is proposed. The cases of final time and time-average, i.e. integral type, temperature observations are considered. We use well-known Tikhonov regularization method and show that the adjoint problems, corresponding to inverse problems ISPF1 and ISPF2 can uniquely be derived by the Lagrange multiplier method. This result allows us to obtain representation formula for the unique solutions of each regularized inverse problem. Using standard Fourier analysis, we show that series solutions for the case in which the governing parabolic equation has constant coefficient, coincide with the Picardʼs singular value decomposition. It is shown that use of these series solutions in CGA as an initial guess substantially reduces the number of iterations. A comparative numerical analysis between the proposed version of CGA and the Fourier method is performed using typical classes of sources, including oscillating and discontinuous functions. Numerical experiments for variable coefficient parabolic equation with different smoothness properties show the effectiveness of the proposed version of CGA.

**PubDate:**2013-12-15T03:09:00Z

**A Dirichlet–Neumann reduced basis method for homogeneous domain**

decomposition problems**Abstract:**Publication date: Available online 12 December 2013

Source:Applied Numerical Mathematics

Author(s): I. Maier , B. Haasdonk

Reduced basis methods allow efficient model reduction of parametrized partial differential equations. In the current paper, we consider a reduced basis method based on an iterative Dirichlet-Neumann coupling for homogeneous domain decomposition of elliptic PDEʼs. We gain very small basis sizes by an efficient treatment of problems with a-priori known geometry. Moreover iterative schemes may offer advantages over other approaches in the context of parallelization. We prove convergence of the iterative reduced scheme, derive rigorous a-posteriori error bounds and provide a full offline/online decomposition. Different methods for basis generation are investigated, in particular a variant of the POD-Greedy procedure. Experiments confirm the rigor of the error estimators and identify beneficial basis construction procedures.

**PubDate:**2013-12-15T03:09:00Z

**A discontinuous Galerkin method for a model arising from stratigraphy****Abstract:**Publication date: Available online 13 December 2013

Source:Applied Numerical Mathematics

Author(s): Roland Becker , Guy Vallet , Abdelaziz Taakili

We investigate a mathematical problem arising from the modeling of maximal erosion rates in geological stratigraphy. A global constraint on ∂ t u , the time-derivative of the solution, is the main feature of this model. This leads to a non-linear pseudoparabolic equation with a diffusion coefficient which is a nonlinear function of ∂ t u . Moreover, the problem degenerates in order to take implicitly into account the constraint. In this paper, we develop a numerical scheme based on the discontinuous Galerkin finite element method (DgFem) for its numerical approximation. With a particular choice of the flux at the interface, we prove that the constraint is implicitly satisfied by using piecewise constant approximation. This is confirmed by some numerical experiments.

**PubDate:**2013-12-15T03:09:00Z

**Pricing of early-exercise Asian options under Lévy processes based on**

Fourier cosine expansions**Abstract:**Publication date: Available online 6 December 2013

Source:Applied Numerical Mathematics

Author(s): B. Zhang , C.W. Oosterlee

In this article, we propose a pricing method for Asian options with early–exercise features. It is based on a two–dimensional integration and a backward recursion of the Fourier coefficients, in which several numerical techniques, like Fourier cosine expansions, Clenshaw–Curtis quadrature and the Fast Fourier transform (FFT) are employed. Rapid convergence of the pricing method is illustrated by an error analysis. Its performance is further demonstrated by various numerical examples, where we also show the power of an implementation on Graphics Processing Units (GPUs).

**PubDate:**2013-12-06T23:48:09Z

**Some techniques for improving the resolution of finite difference**

component-wise WENO schemes for polydisperse sedimentation models**Abstract:**Publication date: Available online 4 December 2013

Source:Applied Numerical Mathematics

Author(s): M.C. Martí , P. Mulet

Polydisperse sedimentation models can be described by a system of conservation laws for the concentration of each species of solids. Some of these models, as the Masliyah-Locket-Bassoon model, can be proven to be hyperbolic, but its full characteristic structure cannot be computed in closed form. Component-wise finite difference WENO schemes may be used in these cases, but these schemes suffer from an excessive diffusion and may present spurious oscillations near shocks. In this work we propose to use a flux-splitting that prescribes less numerical viscosity for component-wise finite difference WENO schemes. We compare this technique with others to alleviate the diffusion and oscillatory behavior of the solutions obtained with component-wise finite difference WENO methods.

**PubDate:**2013-12-06T23:48:09Z

**Editorial Board****Abstract:**Publication date: February 2014

Source:Applied Numerical Mathematics, Volume 76

**PubDate:**2013-11-24T14:41:45Z

**Superconvergence using pointwise interpolation in**

convection–diffusion problems**Abstract:**Publication date: February 2014

Source:Applied Numerical Mathematics, Volume 76

Author(s): Sebastian Franz

Considering a singularly perturbed convection–diffusion problem, we present an analysis for a superconvergence result using pointwise interpolation of Gauß–Lobatto type for higher-order streamline diffusion FEM. We show a useful connection between two different types of interpolation, namely a vertex–edge–cell interpolant and a pointwise interpolant. Moreover, different postprocessing operators are analysed and applied to model problems.

**PubDate:**2013-11-24T14:41:45Z

**A Riemann problem solution methodology for a class of evolutionary mixture**

equations with an arbitrary number of components**Abstract:**Publication date: February 2014

Source:Applied Numerical Mathematics, Volume 76

Author(s): M.W. Crochet , K.A. Gonthier

The solution of the two-phase Riemann problem is a critical component of upwind finite-volume numerical schemes used to solve systems of evolutionary equations, which are routinely used to model compaction and combustion phenomena in gas–granular explosive mixtures. Extensions of a common two-phase model are currently being used to analyze the thermomechanics and combustion of explosive mixtures consisting of N components. Although a solution to the two-phase Riemann problem has been formulated, there is currently no available analogue for the N-phase system in the literature, due to the inherent difficulty of determining the correct wave ordering within the Riemann solver. The development of a solution for these systems is therefore an important step in the formulation of numerical schemes applied to N-phase mixtures. Here, an extension of the exact two-phase solution methodology is proposed for the N-phase case, which may be utilized in the construction of finite-volume schemes for multiphase systems, and can be used with general, convex equations of state. Finally, example problems for three-phase mixtures are considered to illustrate the accuracy of the solution compared to the results of a centered numerical scheme. These solutions also demonstrate the complexity of the possible wave configurations that arise when multiple solid phases are present, as well as the algorithmic challenges which must be addressed to provide a robust implementation.

**PubDate:**2013-11-24T14:41:45Z

**Asymptotically exact discontinuous Galerkin error estimates for linear**

symmetric hyperbolic systems**Abstract:**Publication date: February 2014

Source:Applied Numerical Mathematics, Volume 76

Author(s): Slimane Adjerid , Thomas Weinhart

We present an a posteriori error analysis for the discontinuous Galerkin discretization error of first-order linear symmetric hyperbolic systems of partial differential equations with smooth solutions. We perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of Legendre polynomials of degree p and p + 1 . We apply these asymptotic results to observe that projections of the error are pointwise O ( h p + 2 ) -superconvergent in some cases. Then we solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error. We present computational results for several linear hyperbolic systems in acoustics and electromagnetism.

**PubDate:**2013-11-24T14:41:45Z

**Model reduction of linear time-varying systems over finite horizons****Abstract:**Publication date: Available online 18 November 2013

Source:Applied Numerical Mathematics

Author(s): Samuel A. Melchior , Paul Van Dooren , Kyle A. Gallivan

We consider the problem of approximating a linear time-varying p × m discrete-time state space model S of high dimension by another linear time-varying p × m discrete-time state space model S ˆ of much smaller dimension, using an error criterion defined over a finite time interval. We derive the gradients of the norm of the approximation error and show how this can be solved via a fixed point iteration. We compare this to the classical H 2 norm approximation problem for the infinite horizon time-invariant case and show that our solution extends this to the time-varying and finite horizon case.

**PubDate:**2013-11-20T23:57:49Z

**FGMRES for linear discrete ill-posed problems****Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): Keiichi Morikuni , Lothar Reichel , Ken Hayami

GMRES is one of the most popular iterative methods for the solution of large linear systems of equations. However, GMRES does not always perform well when applied to the solution of linear systems of equations that arise from the discretization of linear ill-posed problems with error-contaminated data represented by the right-hand side. Such linear systems are commonly referred to as linear discrete ill-posed problems. The FGMRES method, proposed by Saad, is a generalization of GMRES that allows larger flexibility in the choice of solution subspace than GMRES. This paper explores application of FGMRES to the solution of linear discrete ill-posed problems. Numerical examples illustrate that FGMRES with a suitably chosen solution subspace may determine approximate solutions of higher quality than commonly applied iterative methods.

**PubDate:**2013-11-12T14:08:14Z

**A note on superoptimal generalized circulant preconditioners****Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): Silvia Noschese , Lothar Reichel

Circulant matrices can be effective preconditioners for linear systems of equations with a Toeplitz matrix. Several approaches to construct such preconditioners have been described in the literature. This paper focuses on the superoptimal circulant preconditioners proposed by Tyrtyshnikov, and investigates a generalization obtained by allowing generalized circulant matrices. Numerical examples illustrate that the new preconditioners so obtained can give faster convergence than available preconditioners based on circulant and generalized circulant matrices.

**PubDate:**2013-11-12T14:08:14Z

**Fast computation of convolution operations via low-rank approximation****Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): Tristan A. Hearn , Lothar Reichel

Methods for the approximation of 2D discrete convolution operations are derived for the case when a low-rank approximation of one of the input matrices is available. Algorithms based on explicit computation of discrete convolution and on the Fast Fourier Transform are both described. Applications of the described methods to the computation of cross-correlation and autocorrelation are discussed and illustrated by examples. Both theory and numerical experiments show that the use of low-rank approximations makes it possible to determine accurate approximations of convolution, cross-correlation, and autocorrelation operations at competitive speeds.

**PubDate:**2013-11-12T14:08:14Z

**A dynamic contact problem in thermoviscoelasticity with two temperatures****Abstract:**Publication date: Available online 8 November 2013

Source:Applied Numerical Mathematics

Author(s): M.I.M. Copetti , J.R. Fernández

This work is concerned with the study of a one dimensional dynamic contact problem arising in thermoviscoelasticity with two temperatures. The existence and uniqueness of a solution to the continuous problem is established using the Faedo–Galerkin method. A finite element approximation is proposed, a convergence result given and some numerical simulations described.

**PubDate:**2013-11-12T14:08:14Z

**A New Jacobi Rational-Gauss Collocation Method for Numerical Solution of**

Generalized Pantograph Equations**Abstract:**Publication date: Available online 11 November 2013

Source:Applied Numerical Mathematics

Author(s): E.H. Doha , A.H. Bhrawy , D. Baleanu , R.M. Hafez

This manuscript is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this article, a new spectral collocation method is applied to solve the generalized pantograph equation with variable coefficients on a semi-infinite domain. This method is based on Jacobi rational functions and Gauss quadrature integration. The Jacobi rational-Gauss method reduces solving the generalized pantograph equation to a system of algebraic equations. Reasonable numerical results are obtained by selecting few Jacobi rational-Gauss collocation points. The proposed Jacobi rational-Gauss method is favorably compared with other methods. Numerical results demonstrate its accuracy, efficiency, and versatility on the half-line.

**PubDate:**2013-11-12T14:08:14Z

**Editorial Board****Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

**PubDate:**2013-11-12T14:08:14Z

**Non-uniform multiresolution analysis for surfaces and applications****Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): M.A. Fortes , M.L. Rodríguez

Haar wavelet can exactly represent any piecewise constant function. Beam and Warming proved later that the supercompact wavelets can exactly represent any piecewise polynomial function in one variable, attaining higher level of accuracy by increasing the polynomial order of the supercompact wavelets. The approach of Beam and Warming, which is based on multiwavelets (family of wavelets) constructed in a one dimensional context, uses orthogonal basis defined over sequences of uniform partitions of [ 0 , 1 ] . The work of Beam and Warming has been recently extended by Fortes and Moncayo to the case of surfaces by using orthogonal basis defined over sequences of uniform triangulations of [ 0 , 1 ] 2 . In that work the authors propose applications to data compression and to discontinuities detection, but both applications have the constraint that it is necessary to know information (at least) at the vertex of the triangulation, and so the data must be uniformly distributed. In the present work we overcome this constraint by considering a multiresolution scheme based on non-uniform triangulations. We develop the multiresolution algorithms and present two examples of the application of the algorithms to compress data and to detect discontinuities of data sets which need not to be uniformly distributed.

**PubDate:**2013-11-12T14:08:14Z

**Quadratic choreographies****Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): P. Ryckelynck , L. Smoch

This paper addresses the classical and discrete Euler–Lagrange equations for systems of n particles interacting quadratically in R d . By highlighting the role played by the center of mass of the particles, we solve the previous systems via the classical quadratic eigenvalue problem (QEP) and its discrete transcendental generalization. Next, we state a conditional convergence result, in the Hausdorff sense, for the roots of the discrete QEP to the roots of the classical one. At last, we focus especially on periodic and choreographic solutions and we provide some numerical experiments which confirm the convergence.

**PubDate:**2013-11-12T14:08:14Z

**Necessary and sufficient conditions for GMRES complete and partial**

stagnation**Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): Gérard Meurant

In this paper we give necessary and sufficient conditions for the complete or partial stagnation of the GMRES iterative method for solving real linear systems. Our results rely on a paper by Arioli, Pták and Strakoš (1998), characterizing the matrices having a prescribed convergence curve for the residual norms. We show that we have complete stagnation if and only if the matrix A is orthonormally similar to an upper or lower Hessenberg matrix having a particular first row or column or a particular last row or column. Partial stagnation is characterized by a particular pattern of the matrix Q in the QR factorization of the upper Hessenberg matrix generated by the Arnoldi process.

**PubDate:**2013-11-12T14:08:14Z

**Perron vector optimization applied to search engines****Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): Olivier Fercoq

In the last years, Googleʼs PageRank optimization problems have been extensively studied. In that case, the ranking is given by the invariant measure of a stochastic matrix. In this paper, we consider the more general situation in which the ranking is determined by the Perron eigenvector of a nonnegative, but not necessarily stochastic, matrix, in order to cover Kleinbergʼs HITS algorithm. We also give some results for Tomlinʼs HOTS algorithm. The problem consists then in finding an optimal outlink strategy subject to design constraints and for a given search engine. We study the relaxed versions of these problems, which means that we should accept weighted hyperlinks. We provide an efficient algorithm for the computation of the matrix of partial derivatives of the criterion, that uses the low rank property of this matrix. We give a scalable algorithm that couples gradient and power iterations and gives a local minimum of the Perron vector optimization problem. We prove convergence by considering it as an approximate gradient method. We then show that optimal linkage strategies of HITS and HOTS optimization problems satisfy a threshold property. We report numerical results on fragments of the real web graph for these search engine optimization problems.

**PubDate:**2013-11-12T14:08:14Z

**A mathematical analysis of optimized waveform relaxation for a small RC**

circuit**Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): Mohammad Al-Khaleel , Martin J. Gander , Albert E. Ruehli

Waveform relaxation techniques are an important tool for the simulation of very large scale circuits. They are based on a partition of the circuit into sub-circuits, and then use an iteration between sub-circuits to converge to the solution of the entire circuit. Their importance has increased with the wide availability of parallel computers with a large number of processors. Unfortunately classical waveform relaxation is hampered by slow convergence, but this can be addressed by better transmission conditions, which led to the new class of optimized waveform relaxation methods. In these methods, both voltage and current information is exchanged in a combination which can be optimized for the performance of the method. We prove in this paper a conjecture for the optimal combination for the particular case of a small RC circuit, and also present and analyze a transmission condition which includes a time derivative.

**PubDate:**2013-11-12T14:08:14Z

**Numerical simulations for a seawater intrusion problem in a free aquifer****Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): P. Marion , K. Najib , C. Rosier

We simulate a sharp interface model issuing from a seawater intrusion problem in a free aquifer. We model the evolution of the sea front and of the upper free surface of the aquifer. We use a P1 finite element method for the space discretization combined with a semi-implicit in time scheme.

**PubDate:**2013-11-12T14:08:14Z

**Approximation of the matrix exponential operator by a structure-preserving**

block Arnoldi-type method**Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): Atika Archid , Abdeselem Hafid Bentbib

The approximation of exp ( A ) V , where A is a real matrix and V a rectangular matrix, is the key ingredient of many exponential integrators for solving systems of ordinary differential equations. The use of Krylov subspace techniques in this context has been actively investigated; see Calledoni and Moret (1997) [10], Hochbruck and Lubich (1997) [17], Saad (1992) [20]. An appropriate structure preserving block method for approximating exp ( A ) V , where A is a large square real matrix and V a rectangular matrix, is given in Lopez and Simoncini (2006) [18]. A symplectic Krylov method to approximate exp ( A ) V was also proposed in Agoujil et al. (2012) [2] with V ∈ R 2 n × 2 . The purpose of this work is to describe a structure preserving block Krylov method for approximating exp ( A ) V when A is a Hamiltonian or skew-Hamiltonian 2n-by-2n real matrix and V is a 2n-by-2s matrix ( s ≪ n ). Our approach is based on block Krylov subspace methods that preserve Hamiltonian and skew-Hamiltonian structures.

**PubDate:**2013-11-12T14:08:14Z

**Preface****Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): Khalide Jbilou , Lothar Reichel , Hassane Sadok

**PubDate:**2013-11-12T14:08:14Z

**Carathéodory–Toeplitz based mathematical methods and their**

algorithmic applications in biometric image processing**Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): Khalid Saeed

In this paper the application of bounded series theory due to Carathéodory and Toeplitz is explored to study Brune positive real rational function (PRF). The main goal is to find the necessary and sufficient conditions for PRF coefficients. The introduced algorithms and assertions present an appropriate mathematical model derived from the developed analytical functions. The suggested solution is based on the results of Carathéodory, Toeplitz, Schur and their achievements at the beginning of the twentieth century. Toeplitz matrix lowest eigenvalues are constructed by the coefficients of the bounded power series representing Carathéodory function to establish a new simple and general algorithm for testing the nonnegativeness of real rational functions. The achieved results have shown engineering interests in two different areas of research: the electrical and mechanical circuit theory from one side and the image analysis and processing from the other side. The involvement in these methods has recently drawn the attention of researchers due to the increasing demand for simple methods of electrical and mechanical network synthesis. The author has proved the reasonability of Carathéodory–Toeplitz theory and modified it for using in other new areas of research. The most important achievements that describe relevant applications in such fields as digital filter design, speech signal and object image processing are discussed in the paper. Examples are introduced to illustrate these applications with emphasis on biometrics.

**PubDate:**2013-11-12T14:08:14Z

**Convergence of a semi-discrete scheme for an abstract nonlinear second**

order evolution equation**Abstract:**Publication date: January 2014

Source:Applied Numerical Mathematics, Volume 75

Author(s): J. Rogava , M. Tsiklauri

In the paper there is considered the Cauchy problem for an abstract nonlinear second order evolution equation in the Hilbert space. This equation represents a generalization of a nonlinear Kirchhoff-type beam equation. For approximate solution of this problem, we introduce a three-layer semi-discrete scheme, where the value of the gradient in the nonlinear term is taken at the middle point. This makes possible to reduce the finding of the approximate solution on each time step to solution of the linear problem. It is proved that the solution of the nonlinear discrete problem, as well as its corresponding difference analog of the first order derivative, is uniformly bounded. For the corresponding linear discrete problem, the high order a priori estimates are obtained using classic Chebyshev polynomials. Based on these facts, for nonlinear discrete problem, the a priori estimates are proved, whence the stability and error estimates of the approximate solution follow. Using the constructed scheme, numerical calculations for various test problems are carried out.

**PubDate:**2013-11-12T14:08:14Z

**Analysis of a multi-frequency electromagnetic imaging functional for thin,**

crack-like electromagnetic inclusions**Abstract:**Publication date: Available online 7 November 2013

Source:Applied Numerical Mathematics

Author(s): Won-Kwang Park

Recently, a non-iterative multi-frequency subspace migration imaging algorithm was developed based on an asymptotic expansion formula for thin, curve-like electromagnetic inclusions and the structure of singular vectors in the Multi-Static Response (MSR) matrix. The present study examines the structure of subspace migration imaging functional and proposes an improved imaging functional weighted by the frequency. We identify the relationship between the imaging functional and Bessel functions of integer order of the first kind. Numerical examples for single and multiple inclusions show that the presented algorithm not only retains the advantages of the traditional imaging functional but also improves the imaging performance.

**PubDate:**2013-11-08T10:59:40Z

**A very fast and accurate boundary element method for options with moving**

barrier and time-dependent rebate**Abstract:**Publication date: Available online 5 November 2013

Source:Applied Numerical Mathematics

Author(s): Luca Vincenzo Ballestra , Graziella Pacelli

A numerical method to price options with moving barrier and time-dependent rebate is proposed. In particular, using the so-called Boundary Element Method, an integral representation of the barrier option price is derived in which one of the integrand functions is not given explicitly but must be obtained solving a Volterra integral equation of the first kind. This equation is affected by several kinds of singularities, some of which are removed using a suitable change of variables. Then the transformed equation is solved using a low-order finite element method based on product integration. Numerical experiments are carried out showing that the method proposed is extraordinarily fast and accurate. In particular a high level of accuracy is achieved also when the initial price of the underlying asset is close to the barrier, when the barrier and the rebate are not differentiable functions, or when the optionʼs maturity is particularly long.

**PubDate:**2013-11-08T10:59:40Z

**A fourth-order approximate projection method for the incompressible**

Navier–Stokes equations on locally-refined periodic domains**Abstract:**Publication date: Available online 7 November 2013

Source:Applied Numerical Mathematics

Author(s): Qinghai Zhang

In this follow-up of our previous work [Zhang et. al., A fourth-order accurate finite-volume method with structured adaptive mesh refinement for solving the advection–diffusion equation, SIAM J. Sci. Comput. 34 (2012) B179-B201], the author proposes a high-order semi-implicit method for numerically solving the incompressible Navier–Stokes equations on locally-refined periodic domains. Fourth-order finite-volume stencils are employed for spatially discretizing various operators in the context of structured adaptive mesh refinement (AMR). Time integration adopts a fourth-order, semi-implicit, additive Runge–Kutta method to treat the non-stiff convection term explicitly and the stiff diffusion term implicitly. The divergence-free condition is fulfilled by an approximate projection operator. Altogether, these components yield a simple algorithm for simulating incompressible viscous flows on periodic domains with fourth-order accuracies both in time and in space. Results of numerical tests show that the proposed method is superior to previous second-order methods in terms of accuracy and efficiency. A major contribution of this work is the analysis of a fourth-order approximate projection operator.

**PubDate:**2013-11-08T10:59:40Z

**Editorial Board****Abstract:**Publication date: December 2013

Source:Applied Numerical Mathematics, Volume 74

**PubDate:**2013-11-04T09:19:43Z

**A Trust Region Method for Constructing Triangle-mesh Approximations of**

Parametric Minimal Surfaces**Abstract:**Publication date: Available online 31 October 2013

Source:Applied Numerical Mathematics

Author(s): Robert J. Renka

Given a function f 0 defined on the unit square Ω with values in R 3 , we construct a piecewise linear function f on a triangulation of Ω such that f agrees with f 0 on the boundary nodes, and the image of f has minimal surface area. The problem is formulated as that of minimizing a discretization of a least squares functional whose critical points are uniformly parameterized minimal surfaces. The nonlinear least squares problem is treated by a trust region method in which the trust region radius is defined by a stepwise-variable Sobolev metric. Test results demonstrate the effectiveness of the method.

**PubDate:**2013-11-04T09:19:43Z

**A posteriori error estimates of hp-adaptive IPDG-FEM for elliptic obstacle**

problems**Abstract:**Publication date: Available online 25 October 2013

Source:Applied Numerical Mathematics

Author(s): Lothar Banz , Ernst P. Stephan

A variational inequality (VI) and a mixed formulation for an elliptic obstacle problem are considered. Both formulations are discretized by a hp-FE interior penalty discontinuous Galerkin (IPDG) method. In the case of the mixed method, the discrete Lagrange multiplier is a linear combination of biorthogonal basis functions. In particular, also the discrete problems are equivalent. For these formulations a residual based a posteriori error estimate and a hierarchical a posteriori error estimate are derived. For the mixed method the residual based estimate is constructed explicitly, for which the approximation error is split into a discretization error of a linear variational equality problem and additional consistency and obstacle condition terms. For the VI-method a hierarchical estimate based on an overkill solution is derived explicitly. This includes the h − h / 2 and p − ( p + 1 ) -estimates. The numerical experiments show exponential convergence up to the desired tolerance.

**PubDate:**2013-10-27T18:54:43Z

**On the construction of second derivative diagonally implicit multistage**

integration methods for ODEs**Abstract:**Publication date: Available online 25 October 2013

Source:Applied Numerical Mathematics

Author(s): A. Abdi , M. Braś , G. Hojjati

Second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods (SGLMs) have been divided into four types, depending on the nature of the differential system to be solved and the computer architecture that is used to implement these methods. In this paper, we describe the construction of SDIMSIMs for all types with Runge–Kutta stability property. Examples of ( p , q , r , s ) SDIMSIMs are given with p = q = r = s ⩽ 4 , where p is the order, q is the stage order, r is the number of external stages, and s is the number of internal stages of the method. Efficiency of the constructed methods is shown by numerical experiments.

**PubDate:**2013-10-27T18:54:43Z

**Split-step θ-method for stochastic delay differential equations****Abstract:**Publication date: Available online 22 October 2013

Source:Applied Numerical Mathematics

Author(s): Wanrong Cao , Peng Hao , Zhongqiang Zhang

In this paper we study the mean-square stability and convergence of the split-step θ-method for stochastic differential equations with fixed time delay. Under mild assumptions, the split-step θ-method is proved to be exponentially mean-square stable and converge with strong order 1/2. Numerical examples show how mean-square stability of the split-step θ-method depends on the parameter θ and the step size h for both linear and nonlinear models.

**PubDate:**2013-10-23T10:52:01Z

**Efficient and stable perfectly matched layer for CEM****Abstract:**Publication date: Available online 22 October 2013

Source:Applied Numerical Mathematics

Author(s): Kenneth Duru , Gunilla Kreiss

An efficient unsplit perfectly matched layer for numerical simulation of electromagnetic waves in unbounded domains is derived via a complex change of variables. In order to surround a Cartesian grid with the PML, the time–dependent PML requires only one (scalar) auxiliary variable in two space dimensions and six (scalar) auxiliary variables in three space dimensions. It is therefore cheap and straightforward to implement. We use Fourier and energy methods to prove the stability of the PML. We extend the stability result to a semi-discrete PML approximated by central finite differences of arbitrary order of accuracy and to a fully discrete problem for the ‘Leap-Frog’ schemes. This makes precise the usefulness of the derived PML model for longtime simulations. Numerical experiments are presented, illustrating the accuracy and stability of the PML.

**PubDate:**2013-10-23T10:52:01Z

**A local projection stabilization of fictitious domain method for elliptic**

boundary value problems**Abstract:**Publication date: Available online 18 October 2013

Source:Applied Numerical Mathematics

Author(s): S. Amdouni , M. Moakher , Yves Renard

In this paper, a new consistent method based on local projections for the stabilization of a Dirichlet condition is presented in the framework of finite element method with a fictitious domain approach. The presentation is made on the Poisson problem but the theoretical and numerical results can be straightforwardly extended to any elliptic boundary value problem. A numerical comparison is performed with the Barbosa-Hughes stabilization technique. The advantage of the new stabilization technique is to affect only the equation on multipliers and thus to be equation independent.

**PubDate:**2013-10-19T19:07:52Z

**Higher order uniformly convergent continuous/discontinuous Galerkin**

methods for singularly perturbed problems of convection-diffusion type**Abstract:**Publication date: Available online 15 October 2013

Source:Applied Numerical Mathematics

Author(s): Peng Zhu , Shenglan Xie

In this paper, we propose and analyze a higher order continuous/discontinuous Galerkin methods for solving singularly perturbed convection-diffusion problems. Based on piecewise polynomial approximations of degree k ⩾ 1 , a uniform convergence rate O ( N − k ln k N ) in associated norm is established on Shishkin mesh, where N is the number of elements. Numerical experiments complement the theoretical results.

**PubDate:**2013-10-15T23:02:49Z

**A note on iterative refinement for seminormal equations****Abstract:**Publication date: Available online 16 September 2013

Source:Applied Numerical Mathematics

Author(s): Miroslav Rozložník , Alicja Smoktunowicz , Jiří Kopal

We present a roundoff error analysis of the method for solving the linear least squares problem min x ‖ b − A x ‖ 2 with full column rank matrix A, using only factors Σ and V from the SVD decomposition of A = U Σ V T . This method (called S N E S V D here) is an analogue of the method of seminormal equations ( S N E Q R ), where the solution is computed from R T R x = A T b using only the factor R from the QR factorization of A. Such methods have practical applications when A is large and sparse and if one needs to solve least squares problems with the same matrix A and multiple right-hand sides. However, in general both S N E Q R and S N E S V D are not forward stable. We analyze one step of fixed precision iterative refinement to improve the accuracy of the S N E S V D method. We show that, under the condition O ( u ) κ 2 ( A ) < 1 , this method (called C S N E S V D ) produces a forward stable solution, where κ ( A ) denotes the condition number of the matrix A and u is the unit roundoff. However, for problems with only O ( u ) κ ( A ) < 1 it is generally not forward stable, and has similar numerical properties to the corresponding C S N E Q R method. Our forward error bounds for the C S N E S V D are slightly better than for the C S N E Q R since the terms O ( u 2 ) κ 3 ( A ) are not present. We illustrate our analysis by numerical experiments.

**PubDate:**2013-09-19T18:24:56Z

**Stopping rules for iterative methods in nonnegatively constrained**

deconvolution**Abstract:**Publication date: Available online 17 September 2013

Source:Applied Numerical Mathematics

Author(s): P. Favati , G. Lotti , O. Menchi , F. Romani

We consider the two-dimensional discrete nonnegatively constrained deconvolution problem, whose goal is to reconstruct an object x ⁎ from its image b obtained through an optical system and affected by noise. When the large size of the problem prevents regularization through a direct method, iterative methods enjoying the semi-convergence property, coupled with suitable strategies for enforcing nonnegativity, are suggested. For these methods an accurate detection of the stopping index is essential. In this paper we analyze various stopping rules and, with the aid of a large experimentation, we test their effect on three different widely used iterative regularizing methods.

**PubDate:**2013-09-19T18:24:56Z

**The Krein method and the globally convergent method for experimental data****Abstract:**Publication date: Available online 18 September 2013

Source:Applied Numerical Mathematics

Author(s): Andrey L. Karchevsky , Michael V. Klibanov , Lam Nguyen , Natee Pantong , Anders Sullivan

Comparison of numerical performances of two methods for coefficient inverse problems is described. The first one is the classical Krein integral equation method, and the second one is the recently developed approximately globally convergent numerical method. This comparison is performed for both computationally simulated and experimental data.

**PubDate:**2013-09-19T18:24:56Z

**A meshless method for the numerical computation of the solution of steady**

Burgers-type equations**Abstract:**Publication date: Available online 13 September 2013

Source:Applied Numerical Mathematics

Author(s): A. Bouhamidi , M. Hached , K. Jbilou

In this paper, we discuss a meshless method for solving steady Burgers-type equations with Dirichlet boundary conditions. The numerical approximation of the solution in the given domain is obtained by using thin plate spline approximation, leading to a large-scale nonlinear matrix equation. The main difficulty of the proposed method is the numerical computation of a solution of the derived nonlinear matrix equation. We will show how to combine Newtonʼs method with some matrix Krylov subspace techniques such as the global GMRES to solve these nonlinear problems. Numerical examples are given to illustrate the proposed method.

**PubDate:**2013-09-15T05:09:27Z

**Stability of ADI schemes for multidimensional diffusion equations with**

mixed derivative terms**Abstract:**Publication date: Available online 6 September 2013

Source:Applied Numerical Mathematics

Author(s): K.J. in ʼt Hout , C. Mishra

In this paper the unconditional stability of four well-known ADI schemes is analyzed in the application to time-dependent multidimensional diffusion equations with mixed derivative terms. Necessary and sufficient conditions on the parameter θ of each scheme are obtained that take into account the actual size of the mixed derivative coefficients. Our results generalize results obtained previously by Craig & Sneyd (1988) and In ʼt Hout & Welfert (2009). Numerical experiments are presented illustrating our main theorems.

**PubDate:**2013-09-11T12:35:18Z

**High resolution methods for scalar transport problems in compliant systems**

of arteries**Abstract:**Publication date: Available online 26 August 2013

Source:Applied Numerical Mathematics

Author(s): M. Tavelli , M. Dumbser , V. Casulli

In this paper a new semi-implicit high resolution scheme for the simulation of advection-diffusion problems in compliant arterial systems is proposed. Such transport problems are not only of great importance for the modelling of drug delivery processes, but also for the simulation of continuous processes occurring in the human body such as the exchange of oxygen, carbon dioxide, nutrients and toxics. Assuming cylindrical geometry and axially symmetric blood flow, a finite volume scheme for scalar transport on unstructured staggered grids is derived. It is shown how both mass conservation and maximum principle can be assured by the present method. Since the discrete maximum principle imposes a CFL type restriction on the time step, the scalar transport equation is solved using a consistent local time stepping approach in order to not affect the unconditional stability of the underlying semi-implicit scheme for the hydrodynamics. It is a key feature of the present approach that the radial profiles of axial velocity and scalar concentration are computed directly from first principles and that no heuristic model for the velocity profile is needed as in classical one-dimensional approaches, which are still frequently used for the simulation of artery trees. The knowledge of radial velocity and concentration gradients is fundamental for the exchange processes happening across the vessel walls. The accuracy of the proposed approach is validated on one and two-dimensional test problems with exact solution. An example for scalar transport in a model artery tree with 55 branches rounds off the numerical test problems discussed in this paper.

**PubDate:**2013-08-30T14:07:27Z

**Error bounds for explicit ERKN integrators for systems of multi-frequency**

oscillatory second-order differential equations**Abstract:**Publication date: Available online 26 August 2013

Source:Applied Numerical Mathematics

Author(s): Bin Wang , Xinyuan Wu , Jianlin Xia

A substantial issue of numerical analysis is concerned with the investigation and estimation of the errors. In this paper, we pay attention to the error analysis for the extended Runge–Kutta–Nyström (ERKN) integrators proposed by Wu et al. [Comput. Phys. Comm. 181 (2010)] for systems of multi-frequency oscillatory second-order differential equations q ″ ( t ) + M q ( t ) = f ( q ( t ) ) . The ERKN integrators are important generalizations of the classical Runge–Kutta–Nyström methods in the sense that both the updates and internal stages have been reformed so that the quantitative behavior of ERKN integrators is adapted to the oscillatory properties of the true solution. By the expansions for the errors of explicit ERKN integrators, we derive stiff order conditions up to order three and present the error bounds. We show that the explicit ERKN integrator fulfilling stiff order p converges with order p, and for an important particular case where M is a symmetric and positive semi-definite matrix, the error bound of ‖ q n − q ( t n ) ‖ is independent of ‖ M ‖ ( ‖ ⋅ ‖ denotes the Euclidean norm). The stiff order conditions provided in the error analysis allow us to design new and efficient explicit ERKN integrators for multi-frequency oscillatory systems. We propose a novel explicit third order multi-frequency and multidimensional ERKN integrator with minimal dispersion error and dissipation error. Numerical experiments carried out show that our new explicit multi-frequency and multidimensional ERKN integrator is more efficient than various other existing effective methods in the scientific literature. We use the first problem to show that the methods perform well with nonsymmetric matrices. In particular, for the well-known Fermi–Pasta–Ulam problem, the numerical behavior of our new explicit ERKN integrator supports our theoretical analysis.

**PubDate:**2013-08-30T14:07:27Z

**Preconditioning of complex symmetric linear systems with applications in**

optical tomography**Abstract:**Publication date: Available online 19 August 2013

Source:Applied Numerical Mathematics

Author(s): S. Arridge , H. Egger , M. Schlottbom

We consider the numerical solution of linear systems of the form ( A + i κ B ) x = y , which arise in many applications, e.g., in time-harmonic acoustics, electromagnetics, or radiative transfer. We propose and analyze a class of preconditioners leading to complex symmetric iteration operators and investigate convergence of corresponding preconditioned iterative methods. Under mild assumptions on the operators A and B, we establish parameter and dimension independent convergence. The proposed methods are then applied to the solution of even-parity formulations of time-harmonic radiative transfer. For this application, we verify all assumptions required for our convergence analysis. The performance of the preconditioned iterations is then demonstrated by numerical tests supporting the theoretical results.

**PubDate:**2013-08-22T22:09:25Z

**Algebraic cubature by linear blending of elliptical arcs****Abstract:**Publication date: Available online 21 August 2013

Source:Applied Numerical Mathematics

Author(s): Gaspare Da Fies , Alvise Sommariva , Marco Vianello

We construct a cubature formula of algebraic degree of exactness n with n 2 / 2 + O ( n ) nodes, on the bidimensional domains generated by linear blending of two arcs of ellipses corresponding to the same angular interval. The construction is based on recent results on “subperiodic” trigonometric quadrature. Our formula generalizes several recent cubature formulas on standard circular sections. Among its numerous possible applications, we quote for example integration of functions with singularities, and integration on nonstandard circular sections arising in optical design or in meshfree methods with compactly supported radial bases.

**PubDate:**2013-08-22T22:09:25Z

**An equi-directional generalization of adaptive cross approximation for**

higher-order tensors**Abstract:**Publication date: Available online 21 August 2013

Source:Applied Numerical Mathematics

Author(s): M. Bebendorf , A. Kühnemund , S. Rjasanow

A scheme for the equi-directional approximation of third-order tensors by sums of products of vectors is presented and analyzed. The new method is a generalization of the adaptive cross approximation (ACA) method initially designed for the low-rank approximation of matrices using only few of its original entries. Moreover, a recursive formula is presented which generalizes the three-way ACA to tensors of arbitrary order.

**PubDate:**2013-08-22T22:09:25Z