Applied Numerical Mathematics [SJR: 1.163] [H-I: 49] [7 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 0168-9274 - ISSN (Online) 0168-9274 Published by Elsevier [2812 journals] |
- A predictor–corrector approach for pricing American options under
the finite moment log-stable model- Abstract: Publication date: November 2015
Source:Applied Numerical Mathematics, Volume 97
Author(s): Wenting Chen, Xiang Xu, Song-Ping Zhu
This paper investigates the pricing of American options under the finite moment log-stable (FMLS) model. Under the FMLS model, the price of American-style options is governed by a highly nonlinear fractional partial differential equation (FPDE) system, which is much more complicated to solve than the corresponding Black–Scholes (B–S) system, with difficulties arising from the semi-globalness of the fractional operator, in conjunction with the nonlinearity associated with the early exercise nature of American-style options. Albeit difficult, in this paper, we propose a new predictor–corrector scheme based on the spectral-collocation method to solve for the prices of American options under the FMLS model. In the current approach, the nonlinearity of the pricing system is successfully dealt with using the predictor–corrector framework, whereas the non-localness of the fractional operator is elegantly handled. We have also provided an elegant error analysis for the current approach. Various numerical experiments suggest that the current method is fast and efficient, and can be easily extended to price American-style options under other fractional diffusion models. Based on the numerical results, we have also examined quantitatively the influence of the tail index on American put options.
PubDate: 2015-07-28T09:04:27Z
- Abstract: Publication date: November 2015
- Fourier collocation algorithm for identifying the spacewise-dependent
source in the advection–diffusion equation from boundary data
measurements- Abstract: Publication date: November 2015
Source:Applied Numerical Mathematics, Volume 97
Author(s): Alemdar Hasanov, Balgaisha Mukanova
In this study, we investigate the inverse problem of identifying an unknown spacewise-dependent source F ( x ) in the one-dimensional advection–diffusion equation u t = Du xx − vu x + F ( x ) H ( t ) , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T ] , based on boundary concentration measurements g ( t ) : = u ( ℓ , t ) . Most studies have attempted to reconstruct an unknown spacewise-dependent source F ( x ) from the final observation u T ( x ) : = u ( x , T ) , but from an engineering viewpoint, the above boundary data measurements are feasible. Thus, we propose a new algorithm for reconstructing the spacewise-dependent source F ( x ) . This algorithm is based on Fourier expansion of the direct problem solution followed by minimization of the cost functional by taking a partial K-sum of the Fourier expansion. Tikhonov regularization is then applied to the ill-posed problem that is obtained. The proposed approach also allows us to estimate the degree of ill-posedness for the inverse problem considered in this study. We then establish the relationship between the noise level γ > 0 , the parameter of regularization α > 0 , and the truncation (or cut-off) parameter K. A new numerical filtering algorithm is proposed for smoothing the noisy output data. Our numerical results demonstrated that the results obtained for random noisy data up to noise levels of 7% had sufficiently high accuracy for all reconstructions.
PubDate: 2015-07-28T09:04:27Z
- Abstract: Publication date: November 2015
- Editorial Board
- Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
PubDate: 2015-07-28T09:04:27Z
- Abstract: Publication date: October 2015
- On a fictitious domain method with distributed Lagrange multiplier for
interface problems- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Ferdinando Auricchio , Daniele Boffi , Lucia Gastaldi , Adrien Lefieux , Alessandro Reali
In this paper we propose a new variational formulation for an elliptic interface problem and discuss its finite element approximation. Our formulation fits within the framework of fictitious domain methods with distributed Lagrange multipliers. For the underlying mixed scheme we prove stability and convergence. Some preliminary numerical tests confirm the theoretical investigations.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Efficient and accurate implementation of hp-BEM for the Laplace operator
in 2D- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Markus Bantle , Stefan Funken
We discuss the accurate and efficient implementation of hp-BEM for the Laplace operator in two dimensions. Using Legendre polynomials and their antiderivatives as local bases for the discrete ansatz spaces, we are able to reduce both the evaluation of potentials and the computation of Galerkin entries to the evaluation of basic integrals. For the computation of these integrals we derive recurrence relations and discuss their accurate evaluation. Our implementation of p- and hp-BEM produces accurate results even for large polynomial degrees ( p > 1000 ) while still being efficient. While this work only treats Symm's integral equation for the Laplace operator in 2D, our approach can be used to solve Symm's, hypersingular and mixed integral equations for Laplace, Lamé and Stokes problems in two dimensions.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Finite element potentials
- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Ana Alonso Rodríguez , Alberto Valli
We present an explicit and efficient way for constructing finite elements with assigned gradient, or curl, or divergence. Some simple notions of homology theory and graph theory applied to the finite element mesh are basic tools for devising the solution algorithms.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Energy norm based error estimators for adaptive BEM for hypersingular
integral equations- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Markus Aurada , Michael Feischl , Thomas Führer , Michael Karkulik , Dirk Praetorius
For hypersingular integral equations in 2D and 3D, we analyze easy-to-implement error estimators like ( h − h / 2 ) -based estimators, two-level estimators, and averaging on large patches and prove their equivalence. Moreover, we introduce some ZZ-type error estimators. All of these a posteriori error estimators are analyzed within the framework of localization techniques for the energy norm.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Parallel multilevel solvers for the cardiac electro-mechanical coupling
- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): P. Colli Franzone , L.F. Pavarino , S. Scacchi
We develop a parallel solver for the cardiac electro-mechanical coupling. The electric model consists of two non-linear parabolic partial differential equations (PDEs), the so-called Bidomain model, which describes the spread of the electric impulse in the heart muscle. The two PDEs are coupled with a non-linear elastic model, where the myocardium is considered as a nearly-incompressible transversely isotropic hyperelastic material. The discretization of the whole electro-mechanical model is performed by Q1 finite elements in space and a semi-implicit finite difference scheme in time. This approximation strategy yields at each time step the solution of a large scale ill-conditioned linear system deriving from the discretization of the Bidomain model and a non-linear system deriving from the discretization of the finite elasticity model. The parallel solver developed consists of solving the linear system with the Conjugate Gradient method, preconditioned by a Multilevel Schwarz preconditioner, and the non-linear system with a Newton–Krylov-Algebraic Multigrid solver. Three-dimensional parallel numerical tests on a Linux cluster show that the parallel solver proposed is scalable and robust with respect to the domain deformations induced by the cardiac contraction.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Editorial Board
- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Special Issue: Fourth Chilean Workshop on Numerical Analysis of Partial
Differential Equations (WONAPDE 2013), Universidad de Concepción,
Chile- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Raimund Bürger , Gabriel N. Gatica , Norbert Heuer , Rodolfo Rodríguez , Mauricio Sepúlveda
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Application of optimal control to the cardiac defibrillation problem using
a physiological model of cellular dynamics- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Nagaiah Chamakuri , Karl Kunisch , Gernot Plank
Optimal control techniques are investigated with the goal of terminating reentry waves in cardiac tissue models. In this computational study the Luo–Rudy phase-I ventricular action potential model is adopted which accounts for more biophysical details of cellular dynamics as compared to previously used phenomenological models. The parabolic and ordinary differential equations are solved as a coupled system and an AMG preconditioner is used to solve the discretized elliptic equation. The numerical results demonstrate that defibrillation is possible by delivering a single strong shock. The optimal control approach also leads to successful defibrillation and demands less total current. The present study motivates us to further investigate optimal control techniques on realistic geometries by incorporating the structural heterogeneity in the cardiac tissue.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Modified fully discretized projection method for the incompressible
Navier–Stokes equations- Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Daniel X. Guo
The stability and convergence of a second-order fully discretized projection method for the incompressible Navier–Stokes equations is studied. In order to update the pressure field faster, modified fully discretized projection methods are proposed. It results in a nearly second-order method. This method sacrifices a little of accuracy, but it requires much less computations at each time step. It is very appropriate for actual computations. The comparison with other methods for the driven-cavity problem is presented.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: October 2015
- A note on the stability of cut cells and cell merging
- Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Marsha Berger
Embedded boundary meshes may have cut cells of arbitrarily small volume which can lead to stability problems in finite volume computations with explicit time stepping. We show that time step constraints are not as strict as often believed. We prove this in one dimension for linear advection and the first order upwind scheme. Numerical examples in two dimensions demonstrate that this carries over to more complicated situations. This analysis sheds light on the choice of time step when using cell merging to stabilize the arbitrarily small cells that arise in embedded boundary schemes.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: October 2015
- Mechanical quadrature methods and their extrapolations for solving the
first kind boundary integral equations of Stokes equation- Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Chong Chen , Xiaoming He , Jin Huang
In this article the mechanical quadrature methods (MQMs) and their extrapolations are proposed and analyzed for solving the first kind boundary integral equations of Stokes equation with closed smooth boundary or closed piecewise curved boundary. It is straightforward and cost efficient to obtain the entries in the linear system arising from the MQMs. The condition numbers of the discrete matrices are of only O ( h − 1 ) and the MQMs achieve higher accuracy than the collocation and Galerkin methods. The analysis of MQMs is more challenging than that of the collocation and Galerkin methods since its theory is no longer within the framework of the projection theory. In this article the convergence of the MQM solutions and the asymptotic expansions of the MQM solution errors are proved for both of the two types of boundary. In order to further improve the accuracy, a Richardson extrapolation is constructed for the mechanical quadrature solution on the smooth boundary and a splitting extrapolation is constructed for the mechanical quadrature solution on the piecewise curved boundary based on the asymptotic expansions of the errors. Numerical examples are provided to illustrate the features of the proposed numerical methods.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: October 2015
- Timestepping schemes for the 3d Navier–Stokes equations
- Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Youngjoon Hong , Djoko Wirosoetisno
It is well known that the (exact) solutions of the 3d Navier–Stokes equations remain bounded for all time if the initial data and the forcing are sufficiently small relative to the viscosity. They also remain bounded for a finite time for arbitrary initial data in L 2 . In this article, we consider two temporal discretisations (semi-implicit and fully implicit) of the 3d Navier–Stokes equations in a periodic domain and prove that their solutions remain uniformly bounded in H 1 subject to essentially the same respective smallness conditions as the continuous system (on initial data and forcing or on the time of existence) provided the time step is small.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: October 2015
- The study of a fourth-order multistep ADI method applied to nonlinear
delay reaction–diffusion equations- Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Dingwen Deng
In this paper, a high-order compact alternating direction implicit (HOC ADI) method, which combines fourth-order compact difference approximation to spatial derivatives and second order backward differentiation formula (BDF2) for temporal integration, is derived for nonlinear delay reaction–diffusion equations. By the discrete energy method, its optimal error estimates in L 2 - and H 1 -norms are constructively obtained. Then, a class of Richardson extrapolation algorithms (REAs) are established to improve computational efficiency. Besides, a modified HOC ADI solver is devised to reduce time cost as delay is very small. Numerical results confirm the theoretical results and performance of our algorithms.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: October 2015
- Penalization of Robin boundary conditions
- Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Bouchra Bensiali , Guillaume Chiavassa , Jacques Liandrat
This paper is devoted to the mathematical analysis of a method based on fictitious domain approach. Boundary conditions of Robin type (also known as Fourier boundary conditions) are enforced using a penalization method. A complete description of the method and a full analysis are provided for univariate elliptic and parabolic problems using finite difference approximation. Numerical evidence of the predicted estimations is provided as well as numerical results for a nonlinear problem and a first extension of the method in the bivariate situation is proposed.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: October 2015
- Uniformly convergent difference schemes for a singularly perturbed third
order boundary value problem- Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Hans-Goerg Roos , Ljiljana Teofanov , Zorica Uzelac
In this paper we consider a numerical approximation of a third order singularly perturbed boundary value problem by an upwind finite difference scheme on a Shishkin mesh. The behavior of the solution, and the stability of the continuous problem are discussed. The proof of the uniform convergence of the proposed numerical method is based on the strongly uniform stability and a weak consistency property of the discrete problem. Numerical experiments verify our theoretical results.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: October 2015
- Comparison results for the Stokes equations
- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): C. Carstensen , K. Köhler , D. Peterseim , M. Schedensack
This paper enfolds a medius analysis for the Stokes equations and compares different finite element methods (FEMs). A first result is a best approximation result for a P 1 non-conforming FEM. The main comparison result is that the error of the P 2 P 0 -FEM is a lower bound to the error of the Bernardi–Raugel (or reduced P 2 P 0 ) FEM, which is a lower bound to the error of the P 1 non-conforming FEM, and this is a lower bound to the error of the MINI-FEM. The paper discusses the converse direction, as well as other methods such as the discontinuous Galerkin and pseudostress FEMs. Furthermore this paper provides counterexamples for equivalent convergence when different pressure approximations are considered. The mathematical arguments are various conforming companions as well as the discrete inf-sup condition.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Post-processing procedures for an elliptic distributed optimal control
problem with pointwise state constraints- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Susanne C. Brenner , Li-Yeng Sung , Yi Zhang
We consider an elliptic distributed optimal control problem with state constraints and compare three post-processing procedures that compute approximations of the optimal control from the approximation of the optimal state obtained by a quadratic C 0 interior penalty method.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Integral equations requiring small numbers of Krylov-subspace iterations
for two-dimensional smooth penetrable scattering problems- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Yassine Boubendir , Oscar Bruno , David Levadoux , Catalin Turc
This paper presents a class of boundary integral equations for the solution of problems of electromagnetic and acoustic scattering by two-dimensional homogeneous penetrable scatterers with smooth boundaries. The new integral equations, which, as is established in this paper, are uniquely solvable Fredholm equations of the second kind, result from representations of fields as combinations of single and double layer potentials acting on appropriately chosen regularizing operators. As demonstrated in this text by means of a variety of numerical examples (that resulted from a high-order Nyström computational implementation of the new equations), these “regularized combined equations” can give rise to important reductions in computational costs, for a given accuracy, over those resulting from previous iterative boundary integral equation solvers for transmission problems.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- A numerical study of divergence-free kernel approximations
- Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Arthur A. Mitrano , Rodrigo B. Platte
Approximation properties of divergence-free vector fields by global and local solenoidal bases are studied. A comparison between interpolants generated with radial kernels and multivariate polynomials is presented. Numerical results show higher rates of convergence for derivatives of the vector field being approximated in directions enforced by the divergence operator when a rectangular grid is used. We also compute the growth of Lebesgue constants for uniform and clustered nodes and study the flat limit of divergence-free interpolants based on radial kernels. Numerical results are presented for two- and three-dimensional vector fields.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: October 2015
- Pure Lagrangian and semi-Lagrangian finite element methods for the
numerical solution of Navier–Stokes equations- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): M. Benítez , A. Bermúdez
In this paper we propose a unified formulation to introduce Lagrangian and semi-Lagrangian velocity and displacement methods for solving the Navier–Stokes equations. This formulation allows us to state classical and new numerical methods. Several examples are given. We combine them with finite element methods for spatial discretization. In particular, we propose two new second-order characteristics methods in terms of the displacement, one semi-Lagrangian and the other one pure Lagrangian. The pure Lagrangian displacement methods are useful for solving free surface problems and fluid-structure interaction problems because the computational domain is independent of the time and fluid–solid coupling at the interphase is straightforward. However, for moderate to high-Reynolds number flows, they can lead to high distortion in the mesh elements. When this happens it is necessary to remesh and reinitialize the transformation to the identity. In order to assess the performance of the obtained numerical methods, we solve different problems in two space dimensions. In particular, numerical results for a sloshing problem in a rectangular tank and the flow in a driven cavity are presented.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- A priori hp-estimates for discontinuous Galerkin approximations to linear
hyperbolic integro-differential equations- Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Samir Karaa , Amiya K. Pani , Sangita Yadav
An hp-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz–Volterra projection, a priori hp-error estimates in L ∞ ( L 2 ) -norm of the velocity as well as of the displacement, which are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p are derived. For optimal estimates of the displacement in L ∞ ( L 2 ) -norm with reduced regularity on the exact solution, a variant of Baker's nonstandard energy formulation is developed and analyzed. Results on order of convergence which are similar in spirit to linear elliptic and parabolic problems are established for the semidiscrete case after suitably modifying the numerical fluxes. For the completely discrete scheme, an implicit-in-time procedure is formulated, stability results are derived and a priori error estimates are discussed. Finally, numerical experiments on two dimensional domains are conducted which confirm the theoretical results.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: October 2015
- Two essential properties of (q,h)-Bernstein–Bézier curves
- Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Ron Goldman , Plamen Simeonov
The ( q , h ) -Bernstein–Bézier curves are generalizations of both the h-Bernstein–Bézier curves and the q-Bernstein–Bézier curves. We investigate two essential features of ( q , h ) -Bernstein bases and ( q , h ) -Bézier curves: the variation diminishing property and the degree elevation algorithm. We show that the ( q , h ) -Bernstein bases for a non-empty interval [ a , b ] satisfy Descartes' law of signs on [ a , b ] when q > − 1 , q ≠ 0 , and h ≤ min { ( 1 − q ) a , ( 1 − q ) b } . We conclude that the corresponding ( q , h ) -Bézier curves are variation diminishing. We also derive a degree elevation formula for ( q , h ) -Bernstein bases and ( q , h ) -Bézier curves over arbitrary intervals [ a , b ] . We show that these degree elevation formulas depend only on the parameter q and are independent of both the parameter h and the interval [ a , b ] . We investigate the convergence of the control polygons generated by repeated degree elevation. We show that unlike classical Bézier curves, the control polygons generated by repeated degree elevation for ( q , h ) -Bézier curves with 0 < q < 1 do not converge to the original ( q , h ) -Bézier curve, but rather to a piecewise linear curve with vertices that depend only on q and the monomial coefficients of the q-Bézier curve having the same control points as the original ( q , h ) -Bézier curve. A similar result holds when q > 1 . Here the control polygons generated by repeated degree elevation converge to a piecewise linear curve that depends only on q and the monomial coefficients of the 1 / q -Bézier curve with the control points of the original ( q , h ) -Bézier curve in reverse order.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: October 2015
- On the numerical solution of a boundary integral equation for the exterior
Neumann problem on domains with corners- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): L. Fermo , C. Laurita
The authors propose a “modified” Nyström method to approximate the solution of a boundary integral equation connected with the exterior Neumann problem for Laplace's equation on planar domains with corners. They prove the convergence and the stability of the method and show some numerical tests.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: August 2015
- Convergence analysis of the summation of the factorially divergent Euler
series by Padé approximants and the delta transformation- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Riccardo Borghi , Ernst Joachim Weniger
Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far from satisfactory. The Euler series E ( z ) ∼ ∑ n = 0 ∞ ( − 1 ) n n ! z n is a very important model for the ubiquitous factorially divergent perturbation expansions in theoretical physics and for the divergent asymptotic expansions for special functions. In this article, we analyze the summation of the Euler series by Padé approximants and by the delta transformation, which is a powerful nonlinear Levin-type transformation that works very well in the case of strictly alternating convergent or divergent series. Our analysis is based on a very recent factorial series representation of the truncation error of the Euler series. We derive explicit expressions for the transformation errors of Padé approximants and of the delta transformation. A subsequent asymptotic analysis proves rigorously the convergence of both Padé and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Padé. This is in agreement with previous numerical results.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: August 2015
- Method of lines for physiologically structured models with diffusion
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Agnieszka Bartłomiejczyk , Henryk Leszczyński
We deal with a size-structured model with diffusion. Partial differential equations are approximated by a large system of ordinary differential equations. Due to a maximum principle for this approximation method its solutions preserve positivity and boundedness. We formulate theorems on stability of the method of lines and provide suitable numerical experiments.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: August 2015
- Bounding matrix functionals via partial global block Lanczos decomposition
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): M. Bellalij , L. Reichel , G. Rodriguez , H. Sadok
Approximations of expressions of the form I f : = trace ( W T f ( A ) W ) , where A ∈ R m × m is a large symmetric matrix, W ∈ R m × k with k ≪ m , and f is a function, can be computed without evaluating f ( A ) by applying a few steps of the global block Lanczos method to A with initial block-vector W. This yields a partial global Lanczos decomposition of A. We show that for suitable functions f upper and lower bounds for I f can be determined by exploiting the connection between the global block Lanczos method and Gauss-type quadrature rules. Our approach generalizes techniques advocated by Golub and Meurant for the standard Lanczos method (with block size one) to the global block Lanczos method. We describe applications to the computation of upper and lower bounds of the trace of f ( A ) and consider, in particular, the computation of upper and lower bounds for the Estrada index, which arises in network analysis. We also discuss an application to machine learning.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: August 2015
- A new boundary element integration strategy for retarded potential
boundary integral equations- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): S. Falletta , L. Scuderi
We consider the retarded potential boundary integral equation, arising from the 3D Dirichlet exterior wave equation problem. For its numerical solution we use compactly supported temporal basis functions in time and a standard collocation method in space. Since the accurate computation of the integrals involved in the numerical scheme is a key issue for the numerical stability, we propose a new efficient and competitive quadrature strategy. We compare this approach with the one that uses the Lubich time convolution quadrature, and show pros and cons of both methods.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: August 2015
- Generalized convolution quadrature with variable time stepping. Part II:
Algorithm and numerical results- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Maria Lopez-Fernandez , Stefan Sauter
In this paper we address the implementation of the Generalized Convolution Quadrature (gCQ) presented and analyzed by the authors in a previous paper for solving linear parabolic and hyperbolic convolution equations. Our main goal is to overcome the current restriction to uniform time steps of Lubich's Convolution Quadrature (CQ). A major challenge for the efficient realization of the new method is the evaluation of high-order divided differences for the transfer function in a fast and stable way. Our algorithm is based on contour integral representation of the numerical solution and quadrature in the complex plane. As the main application we consider the wave equation in exterior domains, which is formulated as a retarded boundary integral equation. We provide numerical experiments to illustrate the theoretical results.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: August 2015
- An analysis of the Prothero–Robinson example for constructing new
adaptive ESDIRK methods of order 3 and 4- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Joachim Rang
Explicit singly-diagonally-implicit (ESDIRK) Runge–Kutta methods have usually order reduction if they are applied on stiff ODEs, such as the example of Prothero and Robinson. It can be observed that the numerical order of convergence decreases to the stage order, which is limited to two. In this paper we analyse the Prothero–Robinson example and derive new order conditions to avoid order reduction. New third and fourth order ESDIRK methods are created, which are applied to the Prothero–Robinson example and to an index-2 DAE. Numerical examples show that the new methods have better convergence properties than usual ESDIRK methods.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: August 2015
- Carleman estimates for the regularization of ill-posed Cauchy problems
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Michael V. Klibanov
This work is a survey of results for ill-posed Cauchy problems for PDEs of the author with co-authors starting from 1991. A universal method of the regularization of these problems is presented here. Even though the idea of this method was previously discussed for specific problems, a universal approach of this paper was not discussed, at least in detail. This approach consists in constructing of such Tikhonov functionals which are generated by unbounded linear operators of those PDEs. The approach is quite general one, since it is applicable to all PDE operators for which Carleman estimates are valid. Three main types of operators of the second order are among them: elliptic, parabolic and hyperbolic ones. The key idea is that convergence rates of minimizers are established using Carleman estimates. Generalizations to nonlinear inverse problems, such as problems of reconstructions of obstacles and coefficient inverse problems are also feasible.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: August 2015
- Stiff convergence of force-gradient operator splitting methods
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Emil Kieri
We consider force-gradient, also called modified potential, operator splitting methods for problems with unbounded operators. We prove that force-gradient operator splitting schemes retain their classical orders of accuracy for linear time-dependent partial differential equations of parabolic and Schrödinger types, provided that the solution is sufficiently regular.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: August 2015
- Second-order differential equations in the Laguerre–Hahn class
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): A. Branquinho , A. Foulquié Moreno , A. Paiva , M.N. Rebocho
Laguerre–Hahn families on the real line are characterized in terms of second-order differential equations with matrix coefficients for vectors involving the orthogonal polynomials and their associated polynomials, as well as in terms of second-order differential equation for the functions of the second kind. Some characterizations of the classical families are derived.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: August 2015
- Guaranteed a posteriori error estimates for nonconforming finite element
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
Author(s): Bei Zhang , Shaochun Chen , Jikun Zhao
We derive guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem. First, an abstract a posteriori error bound is derived under a special equilibration condition. Based on conservative flux reconstruction, two error estimators are proposed and provide actual upper error bounds in the usual energy norm without unknown constants, one of which can be directly constructed without solving local Neumann problems and provide practical computable error bounds. The error estimators also provide local lower bounds but with the multiplicative constants dependent on the diffusion coefficient and mesh size, where the constants can be bounded for enough small mesh size comparable with the square root of the diffusion coefficient. By adding edge jumps with weights to the energy norm, two modified error estimators with additional edge tangential jumps are shown to be robust with respect to the diffusion coefficient and provide guaranteed upper bounds on the error in the modified norm. Finally, the performance of the estimators are illustrated by the numerical results.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: August 2015
- Editorial Board
- Abstract: Publication date: August 2015
Source:Applied Numerical Mathematics, Volume 94
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: August 2015
- A hybrid discontinuous Galerkin method for
advection–diffusion–reaction problems- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Dong-wook Shin , Youngmok Jeon , Eun-Jae Park
A hybrid discontinuous Galerkin (HDG) method for the Poisson problem introduced by Jeon and Park can be viewed as a hybridizable discontinuous Galerkin method using a Baumann–Oden type local solver. In this work, an upwind HDG method with super-penalty is proposed to solve advection–diffusion–reaction problems. A super-penalty formulation facilitates an optimal order convergence in the L 2 norm as well as the energy norm. Several numerical examples are presented to show the performance of the method.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Numerical solution of a multidimensional sedimentation problem using
finite volume-element methods- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Ricardo Ruiz-Baier , Héctor Torres
We are interested in the reliable simulation of the sedimentation of monodisperse suspensions under the influence of body forces. At the macroscopic level, the complex interaction between the immiscible fluid and the sedimentation of a compressible phase may be governed by the Navier–Stokes equations coupled to a nonlinear advection–diffusion–reaction equation for the local solids concentration. A versatile and effective finite volume element (FVE) scheme is proposed, whose formulation relies on a stabilized finite element (FE) method with continuous piecewise linear approximation for velocity, pressure and concentration. Some numerical simulations in two and three spatial dimensions illustrate the features of the present FVE method, suggesting their applicability in a wide range of problems.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- A relaxation Riemann solver for compressible two-phase flow with phase
transition and surface tension- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Christian Rohde , Christoph Zeiler
The dynamics of two-phase flows depend crucially on interfacial effects like surface tension and phase transition. A numerical method for compressible inviscid flows is proposed that accounts in particular for these two effects. The approach relies on the solution of Riemann-like problems across the interface that separates the liquid and the vapour phase. Since the analytical solutions of the Riemann problems are only known in particular cases an approximative Riemann solver for arbitrary settings is constructed. The approximative solutions rely on the relaxation technique. The local well-posedness of the approximative solver is proven. Finally we present numerical experiments for radially symmetric configurations that underline the reliability and efficiency of the numerical scheme.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Nonlinear PDE based numerical methods for cell tracking in zebrafish
embryogenesis- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Karol Mikula , Róbert Špir , Michal Smíšek , Emmanuel Faure , Nadine Peyriéras
The paper presents numerical algorithms leading to an automated cell tracking and reconstruction of the cell lineage tree during the first hours of animal embryogenesis. We present results obtained for large-scale 3D+time two-photon laser scanning microscopy images of early stages of zebrafish (Danio rerio) embryo development. Our approach consists of three basic steps – the image filtering, the cell centers detection and the cell trajectories extraction yielding the lineage tree reconstruction. In all three steps we use nonlinear partial differential equations. For the filtering the geodesic mean curvature flow in level set formulation is used, for the cell center detection the motion of level sets by a constant speed regularized by mean curvature flow is used and the solution of the eikonal equation is essential for the cell trajectories extraction. The core of our new tracking method is an original approach to cell trajectories extraction based on finding a continuous centered paths inside the spatio-temporal tree structures representing cell movement and divisions. Such paths are found by using a suitably designed distance function from cell centers detected in all time steps of the 3D+time image sequence and by a backtracking in the steepest descent direction of a potential field based on this distance function. We also present efficient and naturally parallelizable discretizations of the aforementioned nonlinear PDEs and discuss properties and results of our new tracking method on artificial and real 4D data.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Analysis and discretization of the volume penalized Laplace operator with
Neumann boundary conditions- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Dmitry Kolomenskiy , Romain Nguyen van yen , Kai Schneider
We study the properties of an approximation of the Laplace operator with Neumann boundary conditions using volume penalization. For the one-dimensional Poisson equation we compute explicitly the exact solution of the penalized equation and quantify the penalization error. Numerical simulations using finite differences allow then to assess the discretization and penalization errors. The eigenvalue problem of the penalized Laplace operator with Neumann boundary conditions is also studied. As examples in two space dimensions, we consider a Poisson equation with Neumann boundary conditions in rectangular and circular domains.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Deterministic particle method approximation of a contact inhibition
cross-diffusion problem- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Gonzalo Galiano , Virginia Selgas
We use a deterministic particle method to produce numerical approximations to the solutions of an evolution cross-diffusion problem for two populations. According to the values of the diffusion parameters related to the intra- and inter-population repulsion intensities, the system may be classified in terms of an associated matrix. When the matrix is definite positive, the problem is well posed and the finite element approximation produces convergent approximations to the exact solution. A particularly important case arises when the matrix is only positive semi-definite and the initial data are segregated: the contact inhibition problem. In this case, the solutions may be discontinuous and hence the (conforming) finite element approximation may exhibit instabilities in the neighborhood of the discontinuity. In this article we deduce the particle method approximation to the general cross-diffusion problem and apply it to the contact inhibition problem. We then provide some numerical experiments comparing the results produced by the finite element and the particle method discretizations.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Semi-implicit finite volume level set method for advective motion of
interfaces in normal direction- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Peter Frolkovič , Karol Mikula , Jozef Urbán
In this paper a semi-implicit finite volume method is proposed to solve the applications with moving interfaces using the approach of level set methods. The level set advection equation with a given speed in normal direction is solved by this method. Moreover, the scheme is used for the numerical solution of eikonal equation to compute the signed distance function and for the linear advection equation to compute the so-called extension speed [1]. In both equations an extrapolation near the interface is used in our method to treat Dirichlet boundary conditions on implicitly given interfaces. No restrictive CFL stability condition is required by the semi-implicit method that is very convenient especially when using the extrapolation approach. In summary, we can apply the method for the numerical solution of level set advection equation with the initial condition given by the signed distance function and with the advection velocity in normal direction given by the extension speed. Several advantages of the proposed approach can be shown for chosen examples and application. The advected numerical level set function approximates well the property of remaining the signed distance function during whole simulation time. Sufficiently accurate numerical results can be obtained even with the time steps violating the CFL stability condition.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- An adaptive multiresolution method for ideal magnetohydrodynamics using
divergence cleaning with parabolic–hyperbolic correction- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Anna Karina Fontes Gomes , Margarete Oliveira Domingues , Kai Schneider , Odim Mendes , Ralf Deiterding
We present an adaptive multiresolution method for the numerical simulation of ideal magnetohydrodynamics in two space dimensions. The discretization uses a finite volume scheme based on a Cartesian mesh and an explicit compact Runge–Kutta scheme for time integration. Harten's cell average multiresolution allows to introduce a locally refined spatial mesh while controlling the error. The incompressibility of the magnetic field is controlled by using a Generalized Lagrangian Multiplier (GLM) approach with a mixed hyperbolic–parabolic correction. Different applications to two-dimensional problems illustrate the properties of the method. For each application CPU time and memory savings are reported and numerical aspects of the method are discussed. The accuracy of the adaptive computations is assessed by comparison with reference solutions computed on a regular fine mesh.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- A simple weighted essentially non-oscillatory limiter for the correction
procedure via reconstruction (CPR) framework- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): Jie Du , Chi-Wang Shu , Mengping Zhang
In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes [45], to the correction procedure via reconstruction (CPR) framework for solving conservation laws. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper is particularly simple to implement and will not harm the conservativeness of the CPR framework. Also, it uses information only from the target cell and its immediate neighbors, and thus can maintain the compactness of the CPR framework. Since the CPR framework with the WENO limiter does not in general preserve positivity of the solution, we also extend the positivity-preserving limiters in [43,44,42] to the CPR framework. Numerical results in one and two dimensions are provided to illustrate the good behavior of this procedure.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Numerical identification of constitutive functions in scalar nonlinear
convection–diffusion equations with application to batch
sedimentation- Abstract: Publication date: September 2015
Source:Applied Numerical Mathematics, Volume 95
Author(s): S. Diehl
A fast and simple method for the identification of nonlinear constitutive functions in scalar convection–diffusion equations is presented. No a priori information is needed on the form of the constitutive functions, which are obtained as continuous piecewise affine functions. Accurate and frequent measurements in space and time are required. Synthetic data of batch sedimentation of particles in a liquid and traffic flow are chosen as examples where a convective flux function and a function modelling compression are identified. Real data should first undergo a denoising procedure, which is also presented. It consists of a sequence of convex optimization problems, whose constraints originate from fundamental physical properties. The methodology is applied on data from a batch sedimentation experiment of activated sludge in wastewater treatment.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: September 2015
- Stability analysis and classification of Runge–Kutta methods for
index 1 stochastic differential-algebraic equations with scalar noise- Abstract: Publication date: October 2015
Source:Applied Numerical Mathematics, Volume 96
Author(s): Dominique Küpper , Anne Kværnø , Andreas Rößler
The problem of solving stochastic differential-algebraic equations (SDAEs) of index 1 with a scalar driving Wiener process is considered. Recently, the authors have proposed a class of stiffly accurate stochastic Runge–Kutta (SRK) methods that do not involve any pseudo-inverses or projectors for the numerical solution of the problem. Based on this class of approximation methods, classifications for the coefficients of stiffly accurate SRK methods attaining strong order 0.5 as well as strong order 1.0 are calculated. Further, the mean-square stability of the considered class of SRK methods is analyzed. As the main result, families of A-stable efficient order 0.5 and 1.0 stiffly accurate SRK methods with a minimal number of stages for SDEs as well as for SDAEs are presented.
PubDate: 2015-07-06T14:29:41Z
- Abstract: Publication date: October 2015
- Simultaneous optical flow and source estimation: space-time discretization
and preconditioning- Abstract: Publication date: Available online 14 May 2015
Source:Applied Numerical Mathematics
Author(s): R. Andreev , O. Scherzer , W. Zulehner
We consider the simultaneous estimation of an optical flow field and an illumination source term in a movie sequence. The particular optical flow equation is obtained by assuming that the image intensity is a conserved quantity up to possible sources and sinks which represent varying illumination. We formulate this problem as an energy minimization problem and propose a space-time simultaneous discretization for the optimality system in saddle-point form. We investigate a preconditioning strategy that renders the discrete system well-conditioned uniformly in the discretization resolution. Numerical experiments complement the theory.
PubDate: 2015-05-20T13:22:30Z
- Abstract: Publication date: Available online 14 May 2015
- Projected finite elements for reaction-diffusion systems on stationary
closed surfaces- Abstract: Publication date: Available online 29 April 2015
Source:Applied Numerical Mathematics
Author(s): N. Tuncer , A. Madzvamuse , A.J. Meir
In this paper we present a robust, efficient and accurate finite element method for solving reaction-diffusion systems on stationary spheroidal surfaces (these are surfaces which are deformations of the sphere such as ellipsoids, dumbbells, and heart-shaped surfaces) with potential applications in the areas of developmental biology, cancer research, wound healing, tissue regeneration, and cell motility among many others where such models are routinely used. Our method is inspired by the radially projected finite element method introduced in [39]. (Hence the name “projected” finite element method). The advantages of the projected finite element method are that it is easy to implement and that it provides a conforming finite element discretization which is “logically” rectangular. To demonstrate the robustness, applicability and generality of this numerical method, we present solutions of reaction-diffusion systems on various spheroidal surfaces. We show that the method presented in this paper preserves positivity of the solutions of reaction-diffusion equations which is not generally true for Galerkin type methods. We conclude that surface geometry plays a pivotal role in pattern formation. For a fixed set of model parameter values, different surfaces give rise to different pattern generation sequences of either spots or stripes or a combination (observed as circular spot-stripe patterns). These results clearly demonstrate the need for detailed theoretical analytical studies to elucidate how surface geometry and curvature influence pattern formation on complex surfaces.
PubDate: 2015-05-13T12:42:30Z
- Abstract: Publication date: Available online 29 April 2015