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SUMMARY In this paper, the performance of the incompressible SPH (ISPH) method and an improved weakly compressible SPH (IWCSPH) method for free surface incompressible flows are compared and analyzed. In both methods, the Navier-Stokes equations are solved, and no artificial viscosity is used. The ISPH algorithm in this paper is based on the classical SPH projection method with common treatments on solid boundaries and free surfaces. The IWCSPH model includes some advanced corrective algorithms in density approximation and solid boundary treatment (SBT). In density approximation, the moving least squares (MLS) approach is applied to re-initialize density every several steps to obtain smoother and more stable pressure fields. An improved coupled dynamic SBT algorithm is implemented to obtain stable pressure values near solid wall areas and, thus, to minimize possible numerical oscillations brought in by the solid boundaries. Three representative numerical examples, including a benchmark test for hydrostatic pressure, a dam breaking problem and a liquid sloshing problem, are comparatively analyzed with ISPH and IWCSPH. It is demonstrated that the present IWCSPH is more attractive than ISPH in modeling free surface incompressible flows as it is more accurate and more stable with comparable or even less computational efforts. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a collocation spectral numerical algorithm is presented for solving nonlinear systems of fractional partial differential equations subject to different types of conditions. A proposed error analysis investigates the convergence of the mentioned algorithm. Some numerical examples confirm the efficiency and accuracy of the method. [ABSTRACT FROM AUTHOR]

In this paper we introduce fast numerical algorithms for the solution of the model. For each variable, background illumination, amplitude modulation and phase map, we develop a fixed point method. Then, we write all three algorithms in the same framework and analyze their convergence rates, local smoothing factors by means of local Fourier analysis and present experimental evidence of their performance on synthetic and real world problems. [ABSTRACT FROM AUTHOR]

In this paper a new TLM model for the two-dimensional wave equation is introduced. It is synthesized directly from a FD algorithm. The FD algorithm is second-order-accurate in both space and time, and is explicitly time-stepped. The spatial derivatives in the FD algorithm are approximated by the weighted combination of two standard central difference stencils, one oriented as usual, the other rotated by 45° with its arms extended by a factor of (2)½. The TLM model is realized as the weighted connection of two original models (with the same geometrical configuration as the PD algorithm). The weighting in the TLM model is accomplished by using a variable intrinsic impedance for specific elemental transmission lines. The FD and TLM methods possess identical dispersion relations if the former is operated at its upper limit of stability. Therefore, under these conditions both represent identical models for the simulation of wave propagation. The propagation characteristics of the new model are investigated and the conditions for approximate numerical isotropy are provided. The numerical implementation (scattering matrix and transfer event) is described. To validate the new model, the calculation of cutoff frequencies of various modes in rectangular waveguide is performed. Comparison with analytical results (for an unfilled waveguide) and other numerical results (for a waveguide partially filled with a dielectric) validate the implementation of the model. [ABSTRACT FROM AUTHOR]

Abstract: Based on Adomian decomposition method, a new algorithm for solving boundary value problem (BVP) of nonlinear partial differential equations on the rectangular area is proposed. The solutions obtained by the method precisely satisfy all boundary conditions, except the small pieces near the four corners of the rectangular area. A theorem on the boundary error is given. Hence, the Adomian decomposition method is more efficiently applied to BVPs for partial differential equations. Segmented and weighted analytical solutions with a high accuracy for the BVP of nonlinear groundwater equations on a rectangular area are obtained by the present algorithm. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

To overcome the staircase effects and simultaneously avoid edge blurring, here the authors proposed a novel algorithm of adaptive fourth-order partial differential equation (PDE) for image smoothing. The proposed method is based on You and Kaveh (YK) model that first performs a pre-processing using coherence-enhancing diffusion filter before the smoothing. Then, the authors select an adaptive edge-stopping function founded on the Tansig function to effectively remove the noise and insignificant details without destruction of important edges and textures in the image. MATLAB simulation results validate the efficiency of the proposed model quantitatively and visually, when compared with other state-of-the-art methods. [ABSTRACT FROM AUTHOR]

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The paper presents an extension of the composite programmable graph grammar (CP-graph grammar) suitable for modeling the parallel direct solver algorithm utilized by the hp finite element method (hp-FEM). In the proposed graph grammar model, the computational mesh is represented by a CP-graph. The presented graph grammar models the solver algorithm by a set of graph grammar productions. The graph grammar model makes it possible to examine the concurrency of the algorithm by analyzing the interdependence between the atomic tasks, tasks and super-tasks. The atomic tasks correspond to the graph grammar productions, representing basic undividable parts of the algorithms. The level of atomic tasks models the concurrency for the shared memory architectures. On the other hand, the tasks correspond to the groups of atomic tasks with predefined inter-task communication channels. They constitute the grain for the decomposition of the parallel algorithm for the distributed memory architecture. Finally, the super-tasks correspond to a group of tasks resulting from the execution of load balancing algorithm. The solver algorithm is tested on distributed memory linux cluster for up to 192 processors. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

With the ubiquity of large-scale computing resources has come significant attention to practical details of fast algorithms for the numerical solution of partial differential equations. Included in this group are the class of multigrid and algebraic multigrid algorithms that are effective solvers for many of the large matrix problems arising from the discretization of elliptic operators. Algebraic multigrid (AMG) is especially effective for many problems with discontinuous coefficients, discretized on unstructured grids, or over complex geometries. While much effort has been invested in improving the practical performance of AMG, little theoretical understanding of this performance has emerged. This paper presents a two-level convergence theory for a reduction-based variant of AMG, called AMGr, which is particularly appropriate for linear systems that have M-matrix-like properties. For situations where less is known about the problem matrix, an adaptive version of AMGr that automatically determines the form of the reduction needed by the AMGr process is proposed. The adaptive cycle is shown, in both theory and practice, to yield an effective AMGr algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study five different types of problems in potential and viscous flows past intersecting circles or spheres in two and three dimensions, respectively, with different angles of intersection. We observe a striking resemblance in the form of the solutions in all these cases by introducing certain operators L and M which generate a group G. By introducing a procedure called ‘closure’ which determines the order of the group G, we give a general method to discuss the problem of flow past two intersecting circles or spheres in potential and Stokes flows with different angles of intersection. [ABSTRACT FROM AUTHOR]

This article develops a new algorithm named TTRISK to solve high‐dimensional risk‐averse optimization problems governed by differential equations (ODEs and/or partial differential equations [PDEs]) under uncertainty. As an example, we focus on the so‐called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid nonsmoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the Karush–Kuhn–Tucker (KKT) system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by large‐scale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID‐19. The results indicate that the risk‐averse framework is feasible with the tensor approximations under tens of random variables. [ABSTRACT FROM AUTHOR]

This is a new international journal "Numerical Linear Algebra With Applications," which publishes refereed papers describing significant developments in numerical linear algebra and the application of such method; to the solution of practical problems, such as occur in engineering, for instance. Numerical linear algebra algorithms form the core of almost any solution algorithm found in mathematical modeling. The models originate in physics, chemistry, engineering, ecology, economy and social sciences. Frequently they involve solutions of systems of algebraic equations and eigenvalue problems derived by some discretization method used for partial differential equations.

In this paper a method for fast computations with the inverse to weakly singular, hypersingular and double layer potential boundary integral operators associated with the Laplacian on Lipschitz domains is proposed and analyzed. It is based on the representation formulae suggested for above-mentioned boundary operators in terms of the Poincaré­Stcklov interface mappings generated by the special decompositions of the interior and exterior domains. Computations with the discrete counterparts of these formulae can he efficiently performed by iterative substructuring algorithms provided some asymptotically optimal techniques for treatment of interface operators on subdomain boundaries. For both two- and three-dimensional cases the computation cost and memory needs arc of the order O(N IogP N) and O(N log2 N), respectively, with 1 ≤ p ≤ 3, where N is the number of degrees of freedom on the boundary under consideration (some kinds of polygons and polyhedra). The proposed algorithms are well suited for serial and parallel computations. [ABSTRACT FROM AUTHOR]

This paper presents a general method to associate the operator splitting for the Peaceman-Rachford procedure with a decomposition of the domain in problems arising from finite element discretization of partial differential equations. The algorithm is provably convergent without any symmetry requirement. Moreover, this method possesses the significant advantage of making the linear systems of the Peaceman-Rachford iteration block diagonal and therefore perfectly appropriate for parallel processing. Not only is sparsity not affected but a reduction of the bandwidth occurs. In fact, for appropriate choices of nonconforming finite element spaces, this method makes directly possible elementwise processing. This option remains available in general for higher-dimensional problems by applying the splitting algorithm recursively. Practical implementation requires nothing more than the standard finite element assembly procedure and some bookkeeping to relate a few different orderings of the nodes. In addition to all these attractive features, the method is rapidly convergent and remains highly competitive even when used on a serial machine. [ABSTRACT FROM AUTHOR]

In this paper, we present and analyse some local and parallel covolume algorithms for second-order elliptic problems. We first prove some local a priori error estimates for covolume method. With these results, we shall apply the two-grid strategy to covolume approximation to give two novel covolume algorithms, and show that the convergence rates of these algorithms are optimal in piecewise H1-norm. Numerical results confirming the theory are also provided. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The redistribution and embedding are two main adaptive-grid methods. With use of initial structured grid, it is possible to combine these adaptive-grid techniques. It means that before the grid becomes unstructured, we employ redistribution technique and then use embedding method. It is used to solve three model problems, transonic and supersonic inviscid flows in channels with circular arc bump and with convergent part. The Euler equations are integrated to steady state by an explicit, finite volume, Ni's Lax–Wendroff type. The combination of adaptive-grid redistribution and embedding is complex, but it is shown in this paper that it is very efficient and worth to use it in large CPU time consuming problems. The results also show that the combination requires less grid points and therefore less memory space for prescribed accuracy. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The authors present a technique of deriving basic hypergeometric identities from specializations using fewer parameters, by using the classical Cauchy identity on the expansion of the power of $x$ in terms of the $q$-binomial coefficients. This method is referred to as ‘Cauchy augmentation’. Despite its simple appearance, the Cauchy identity plays a key role in parameter augmentation. For example, one can reach the $q$-Gauss summation formula from the Euler identity by using the Cauchy augmentation twice. This idea also applies to Jackson's $_2\phi_1$ to $_3\phi_1$ transformation formula. Moreover, a transformation formula analogous to Jackson's formula is obtained. [ABSTRACT FROM AUTHOR]

In this paper numerical methods for solving first-order hyperbolic partial differential equations are developed. These methods are developed by approximating the first-order spatial derivative by third-order finite-difference approximations and a matrix exponential function by a third-order rational approximation having distinct real poles. Then parallel algorithms are developed and tested on a sequential computer for an advection equation with constant coefficient and a non-linear problem. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Performing data assimilation (DA) at low cost is of prime concern in Earth system modeling, particularly in the era of Big Data, where huge quantities of observations are available. Capitalizing on the ability of neural network techniques to approximate the solution of partial differential equations (PDEs), we incorporate deep learning (DL) methods into a DA framework. More precisely, we exploit the latent structure provided by autoencoders (AEs) to design an ensemble transform Kalman filter with model error (ETKF‐Q) in the latent space. Model dynamics are also propagated within the latent space via a surrogate neural network. This novel ETKF‐Q‐Latent (ETKF‐Q‐L) algorithm is tested on a tailored instructional version of Lorenz 96 equations, named the augmented Lorenz 96 system, which possesses a latent structure that accurately represents the observed dynamics. Numerical experiments based on this particular system evidence that the ETKF‐Q‐L approach both reduces the computational cost and provides better accuracy than state‐of‐the‐art algorithms such as the ETKF‐Q. [ABSTRACT FROM AUTHOR]

We prove extension theorems in the norms described by Stokes and Lamé operators for the three-dimensional case with periodic boundary conditions. For the Lamé equations, we show that the extension theorem holds for nearly incompressible media, but may fail in the opposite limit, i.e. for case of absolutely compressible media. We study carefully the latter case and associate it with the Cosserat problem. Extension theorems serve as an important tool in many applications, e.g. in domain decomposition and fictitious domain methods, and in analysis of finite element methods. We consider an application of established extension theorems to an efficient iterative solution technique for the isotropic linear elasticity equations for nearly incompressible media and for the Stokes equations with highly discontinuous coefficients. The iterative method involves a special choice for an initial guess and a preconditioner based on solving a constant coefficient problem. Such preconditioner allows the use of well-known fast algorithms for preconditioning. Under some natural assumptions on smoothness and topological properties of subdomains with small coefficients, we prove convergence of the simplest Richardson method uniform in the jump of coefficients. For the Lamé equations, the convergence is also uniform in the incompressible limit. Our preliminary numerical results for two-dimensional diffusion problems show fast convergence uniform in the jump and in the mesh size parameter. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents an integrated approach and a unified algorithm that combine the benefits of PDE surfaces and powerful physics-based modeling techniques within one single modeling framework, in order to realize the full potential of PDE surfaces. We have developed a novel system that allows direct manipulation and interactive sculpting of PDE surfaces at arbitrary location, hence supporting various interactive techniques beyond the conventional boundary control. Our prototype software affords users to interactively modify point, normal, curvature, and arbitrary region of PDE surfaces in a predictable way. We employ several simple, yet effective numerical techniques including the finite-difference discretization of the PDE surface, the multigrid-like subdivision on the PDE surface, the mass-spring approximation of the elastic PDE surface, etc. to achieve real-time performance. In addition, our dynamic PDE surfaces can also be approximated using standard bivariate B-spline finite elements, which can subsequently be sculpted and deformed directly in real-time subject to intrinsic PDE constraints. Our experiments demonstrate many attractive advantages of our dynamic PDE formulation such as intuitive control, real-time feedback, and usability to the general public. [ABSTRACT FROM AUTHOR]

This paper describes an analytical model for a beam system, based on a modified Timoshenko theory, where the beam is pinned to a hub driven by an actuator at one end and is subject to a heavy load at the other end. A new efficient computational algorithm is then proposed for solving the higher-order non-canonical partial differential equation model, which is developed based on the generalized difference method. This allows a suitable selection of different trial and test spaces, so as to improve the computational efficiency while preserving the high convergence rate of the standard finite element method. With the trial space of cubic Hermite finite elements and the test space of piecewise linear functions, the computational scheme reduces to a semi-discretized or even fully discretized computational algorithm. A numerical simulation result is included to visualize the theoretical modelling and computational results. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Based on the concept of the Laplacian matrix of a graph, this paper presents the SGPD (spectral graph pseudoperipheral and pseudodiameter) algorithm for finding a pseudoperipheral vertex or the end-points of a pseudodiameter in a graph. This algorithm is compared with the ones by Grimes eta!. (1990), George and Liu (1979), and Gibbs et a!. (1976). Numerical results from a collection of benchmark test problems show the effectiveness of the proposed algorithm. Moreover, it is shown that this algorithm can be efficiently used in conjunction with heuristic algorithms for ordering sparse matrix equations. Such heuristic algorithms, of course, must be the ones which use the pseudoperipheral vertex or pseudodiameter concepts [ABSTRACT FROM AUTHOR]

Distributed systems such as networks of workstations are becoming an increasingly viable alternative to traditional supercomputer systems for running complex scientific applications. A large number of these applications require solving sets of partial differential equations (PDEs). In this paper, we describe the implementation and performance of SPEED (Scalable Partial differential Equation Environment on Distributed systems), a parallel platform which provides an efficient solution for time-dependent PDEs. SPEED allows the inclusion of a wide range of parameters and programming aids. PVM is employed as the underlying message-passing system. The parallel implementation has been performed using two algorithms. The first algorithm is a two-phase scheme which uses the conventional technique of alternating phases of computation and communication. The second algorithm employs a pre-computation technique that allows overlapping of computation and communication. Both methods yield significant speedups. The pre-computation technique reduces the communication time between the workstations but incurs additional overhead in buffer management. Hence, if the saving in communication time is larger than the overhead, the pre-computation technique outperforms the two-phase algorithm. SPEED also provides a performance prediction methodology that can accurately predict the performance of a given application on the system before running the application. This methodology allows the user to tune various parameters in order to identify system bottlenecks and maximize the performance. [ABSTRACT FROM AUTHOR]

The fast adaptive composite grid method (FAC) is an algorithm that uses uniform grids, both global and local, to solve partial differential equations. This method is known to be highly efficient on scalar or singe-processor vector computers, due to its effective use of uniform grids and multiple levels of resolution of the solution. On distributed memory multiprocesssors, such methods benefit from their tendency to create multiple isolated refinement regions, which may be effectively treated in parallel. However, they suffer from the way in which the levels of refinement are treated sequentially in each region. Specifically, the finer levels must wait m be processed until the coarse-level approximations have been computed and passed to them; conversely, the coarser levels must wait until the finer level approximations have been computed and used to correct their equations. The asynchronous fast adaptive composite method (AFAC) eliminates this bottleneck of parallelism. Through a simple mechanism used to reduce interlevel dependence, individual refinements levels can be processed by AFAC in parallel, The result is that the convergence rate of AFAC is the square root of that for FAC. Therefore, since both AFAc and FAC have roughly the same number of floating-point operations, AFAC requires twice the serial computational time as FAC, but AFAC, may be much more efficiently parallelized. This paper presents a comparison of FAC and AFAC under a variety of conditions, including vectorization and parallelization. Results are presented for distributed memory multiprocessor architectures (SUPRENUM, Intel iPSC/2 and iPSC/860). The most crucial factor for the evaluation of both algorithms is the different computation/communication cost ratios of these architectures. It is shown that, under many circumstances, AFAC is superior to FAC in a parallel environment. [ABSTRACT FROM AUTHOR]

The fast adaptive composite grid method (FAC) is a multi-level adaptive method for the discretization and solution of partial differential equations. This paper compares the speed of solution of the resulting discrete equations by the FAC algorithm and a preconditioned conjugate gradient scheme. Comparisons and information concerning the solutions are given for several model problems. [ABSTRACT FROM AUTHOR]

Boundary element formulations employing time-independent fundamental solutions are becoming popular for treating transient problems described by parabolic or hyperbolic partial differential equations. This paper describes the extension of one such formation, named the dual reciprocity boundary element method, to non-linear transient heat conduction problems with temperature-dependent material parameters and boundary conditions of the radiative type. The original non-linear diffusion equation is solved in a transform space, where it appears in a pseudo-linear form, since it contains a modified time variable which is itself a function of position. The problem is solved by using an iterative algorithm of the Newton- Raphson type. [ABSTRACT FROM AUTHOR]

In contrast to most triangulation algorithms which implicitly assume that triangulation point locations are fixed. 'Laplacian' smoothing focuses on moving point locations to improve triangulation. Laplacian smoothing is attractive for its simplicity but it does require an existing triangulation. In this paper the effect of Laplacian smoothing on Delaunay triangulations is explored. It will become clear that constraining Laplacian smoothing to maintain a Delaunay triangulation measurably improves Laplacian smoothing. [ABSTRACT FROM AUTHOR]

This paper proposes a quasi-competitive network in which the mass reaction intensity of the elements in the intermediate layer is kept constant as a model that approximates the continuous nonlinear function. As the loss function, the local loss function is considered. It is composed of the error and the local model loss term reflecting the local model size. The learning algorithm of the quasi-competitive network including the change of the number of elements is derived from the local loss function. It is shown by numerical experiment that the quasi-competitive network can form quickly the structure reflecting the target function and has a high generalization power. It is shown also that the high generalization power is due to the constant mass reaction intensity of the elements in the intermediate layer of the quasi-competitive network. [ABSTRACT FROM AUTHOR]

Often, partial differential equations for the case of numerical computation are solved. However, to solve partial differential equations whose order of dimensions are more than 2, main memory is not large enough even in the main frame or super computers, not to mention workstations or personal computers. Thus, we have to use external memory to solve large partial differential equations numerically. This paper proposes an algorithm to solve numerically by explicit methods, large partial differential equations which are time-dependent and have fixed boundary conditions. In this algorithm, neighbor mesh points are placed together in main memory. Also, even partially, they are computed toward the time axis as far as possible. In other words, this method computes mesh points toward the time axis with priority, reusing computed data in main memory. The proposed algorithm is unique in this point. It is faster than conventional methods in proportion to D&redic;P where P is the bulk of the main memory and D is the order of dimensions. [ABSTRACT FROM AUTHOR]

We have developed a new numerical method based on Haar wavelet (HW) in this article for the numerical solution (NS) of one‐ and two‐dimensional hyperbolic Telegraph equations (HTEs). The proposed technique is utilized for one‐ and two‐dimensional linear and nonlinear problems, which shows its advantage over other existing numerical methods. In this technique, we approximated both space and temporal derivatives by the truncated Haar series. The algorithm of the method is simple and we can implement easily in any other programming language. The technique is tested on some linear and nonlinear examples from literature. The maximum absolute errors (MAEs), root mean square errors (RMSEs), and computational convergence rate are calculated for different number of collocation points (CPs) and also some 3D graphs are also drawn. The results show that the proposed technique is simply applicable and accurate. [ABSTRACT FROM AUTHOR]

Editorial. Examines the development of algebraic based versions used in solving linear and nonlinear partial differential equations. Elements of the algebraic multilevel iteration methods; Computational effort required in each iteration step; Effect of the emergence of parallel computers on numerical algorithms.

Multigrid methods are among the most efficient algorithms for solving discretized partial differential equations. Typically, a multigrid system offers various configuration options to tune performance for different applications and hardware platforms. However, knowing the best performing configuration in advance is difficult, because measuring all multigrid system variants is costly. Instead of direct measurements, we use machine learning to predict the performance of the variants. Selecting a representative set of configurations for learning is nontrivial, although, but key to prediction accuracy. We investigate different sampling strategies to determine the tradeoff between accuracy and measurement effort. In a nutshell, we learn a performance-influence model that captures the influences of configuration options and their interactions on the time to perform a multigrid iteration and relate this to existing domain knowledge. In an experiment on a multigrid system working on triangular grids, we found that combining pair-wise sampling with the D-Optimal experimental design for selecting a learning set yields the most accurate predictions. After measuring less than 1 % of all variants, we were able to predict the performance of all variants with an accuracy of 95.9 %. Furthermore, we were able to verify almost all knowledge on the performance behavior of multigrid methods provided by 2 experts. [ABSTRACT FROM AUTHOR]

The multisymplectic schemes have been used in numerical simulations for the RLW-type equation successfully. They well preserve the local geometric property, but not other local conservation laws. In this article, we propose three novel efficient local structure-preserving schemes for the RLW-type equation, which preserve the local energy exactly on any time-space region and can produce richer information of the original problem. The schemes will be mass- and energy-preserving as the equation is imposed on appropriate boundary conditions. Numerical experiments are presented to verify the efficiency and invariant-preserving property of the schemes. Comparisons with the existing nonconservative schemes are made to show the behavior of the energy affects the behavior of the solution.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1678-1691, 2017 [ABSTRACT FROM AUTHOR]

The algorithm for the solution of the first biharmonic problem by the Trefftz method is presented. The solution is purely mathematical and follows Mikhlin [29] approach to the solution of the harmonic problem. As such it fills in the gap in the approximate methods of Mathematical Physics. Validity of the algorithm is demonstrated on two examples. The boundary values of the sought function and its normal derivative must be given explicitly to achieve correct results by the algorithm. For that reason, the presented algorithm is, however, not adequate for the plane problems of the theory of elasticity or for plates with free or simply supported edges. [ABSTRACT FROM AUTHOR]

SUMMARY In many large-scale computations, systems of equations arise in the form Au = b, where A is a linear operation to be performed on the unknown data u, producing the known right-hand side, b, which represents some constraint of known or assumed behavior of the system being modeled. Because such systems can be very large, solving them directly can be too slow. In contrast, a multigrid method removes different components of the error at different resolutions using smoothers that reduce high-frequency components of the error more readily than low. Here, we present an open-source multigrid solver written only in Python. OpenMG is a pure Python experimentation environment for testing multigrid concepts, not a production solver. The particular restriction method implemented is for 'standard' multigrid. By making the code simple and modular, we make the algorithmic details clear. The resulting solver is tested on an implicit pressure reservoir simulation problem with satisfactory results.Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In many applications of multiple objective network programming (MONP) problems, only integer solutions are acceptable as the final optimal solution. Representative efficient solutions are usually obtained by sampling the efficient set through the solution of augmented weighted Tchebycheff network programs. Because such efficient solutions are usually not integer solutions, a branch-and-bound (BB) algorithm is developed to find integer efficient solutions. The purpose of the BB algorithm is to support interactive procedures by generating representative integer efficient solutions. To be computationally efficient, the algorithm takes advantage of the network structure as much as possible. An algorithm, used in the BB algorithm and performed on the key tree, is developed to construct feasible solutions from infeasible solutions and basic solutions from nonbasic solutions when bounds on branching variables change. The BB algorithm finds basic and nonbasic or supported and unsupported integer efficient solutions as long as they are optimal. Details of the algorithm are presented, an example is provided and computational results are reported. Computational results show that the BB algorithm performs well. Although the BB algorithm is developed for the purpose of generating integer efficient solutions for MONP problems, it can also solve more general integer network flow problems with linear side constraints. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013 [ABSTRACT FROM AUTHOR]

The wave digital concept for numerical integration of partial differential equations leads to algorithms with highly advantageous features as robustness, full localness and massive parallelism. However, the required synthesis of an internally multidimensionally passive reference circuit, from which the algorithm is derived, usually demands an in-depth knowledge of circuit theory and a high level of intuition. In this practical guide, a step-by-step approach for the synthesis of such reference circuits is introduced to relax these requirements, using the nonlinear fluid dynamic equations as a nontrivial example. General implementation issues for the wave digital algorithm are discussed as well as applying arbitrary passive linear multistep methods in place of the commonly used trapezoidal rule. As an example, we take the well-known numerically critical shock tube problem, the solution of which is problematic when the trapezoidal rule is used as unwanted oscillations occur. These oscillations are suppressed when using the second-order accurate Gear method instead. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We construct a sequence of functions that uniformly converge (on compact sets) to the price of an Asian option, which is written on a stock whose dynamics follow a jump diffusion. The convergence is exponentially fast. We show that each element in this sequence is the unique classical solution of a parabolic partial differential equation (not an integro-differential equation). As a result we obtain a fast numerical approximation scheme whose accuracy versus speed characteristics can be controlled. We analyze the performance of our numerical algorithm on several examples. [ABSTRACT FROM AUTHOR]

Image restoration has been an important field of image processing for decades. Various methods of restoration have been studied. Meanwhile, regularization presents a very general methodology for image restoration. However, more regularization will remove more noise but will also lose more image detail, and vice versa. In this article, a novel image restoration algorithm is presented, which is derived from regularization by minimizing error energy of noise and ring artifact. Thus, this proposed algorithm is not only immune to noise but also ringing artifact, where a designed high-pass filter is used in the algorithm proposed here such as the difference of a delta function and a blurring function or an edge operator like a Laplacian operator. This proposed algorithm can achieve high-resolution images by being implemented with a 1D noiseless signal to reveal its ability of generating high frequencies beyond the diffraction limit and with a 2D computer-generated noisy image to show the ability of being immune to noise as well as with a 94-GHz millimeter-wave image to validate its reliability. © 2010 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 20, 308-315, 2010 [ABSTRACT FROM AUTHOR]

Lax-Wendroff and Nyström methods are numerical algorithms of temporal approximations for solving differential equations. These methods provide efficient algorithms for high-accuracy seismic modeling. In the context of spatial pseudospectral discretizations, I explore these two kinds of methods in a comparative way. Their stability and dispersion relation are discussed in detail. Comparison between the fourth-order Lax-Wendroff method and a fourth-order Nyström method shows that the Nyström method has smaller stability limit but has a better dispersion relation, which is closer to the sixth-order Lax-Wendroff method. The structure-preserving property of these methods is also revealed. The Lax-Wendroff methods are a second-order symplectic algorithm, which is independent of the order of the methods. This result is useful for understanding the error growth of Lax-Wendroff methods. Numerical experiments based on the scalar wave equation are performed to test the presented schemes and demonstrate the advantages of the symplectic methods over the nonsymplectic ones. [ABSTRACT FROM AUTHOR]

The article discusses the use of a modeling and optimization system that can integrate computational fluid dynamics and polymer electrolyte membrane fuel cells (PEMFC). A multidimensional and multiphysical PEMFC is used to calculate gas channel and membrane electrode assembly transport processes. The result, a nonlinear partial differential-algebraic equation, is discretized with a finite volume method. After this, it is attached to an interior point optimization algorithm. This method can be used to solve parameter estimation issues. It is also useful in parametric studies for distribution characteristics and water transport mechanisms.

In the context of the univariate Gaussian mixture with grouped data, it is shown that the global maximum of the likelihood may correspond to a situation where a Dirac lies in any non-empty interval. Existence of a domain of attraction near such a maximizer is discussed and we establish that the expectation-maximization (EM) iterates move extremely slowly inside this domain. These theoretical results are illustrated both by some Monte-Carlo experiments and by a real data set. To help practitioners identify and discard these potentially dangerous degenerate maximizers, a specific stopping rule for EM is proposed. [ABSTRACT FROM AUTHOR]

Efficient and unconditionally stable perfectly matched layer (PML) formulations are presented for truncating the scalar wave-equation finite-difference time-domain (WE-FDTD) grids. The proposed formulations are based on incorporating the alternating-direction-implicit FDTD (ADI-FDTD) scheme into the scalar wave-equation derived in the PML region at the domain boundaries. A numerical example carried out in the 2D domain is included and it is observed that the time step of the proposed formulations is not bounded by the CFL stability limit of the conventional WE-FDTD algorithm, but by the required accuracy level. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 1029–1032, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21591 [ABSTRACT FROM AUTHOR]

A new two-level black-box preconditioner based on the hybrid domain decomposition technique is proposed and studied. The preconditioner is a combination of an additive Schwarz preconditioner and a special smoother. The smoother removes dependence of the condition number on the number of subdomains and variations of the diffusion coefficient and leaves minor sensitivity to the problem size. The algorithm is parallel and pure algebraic which makes it a convenient framework for the construction parallel black-box preconditioners on unstructured meshes. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We present a new domain decomposition algorithm for the parallel finite element solution of elliptic partial differential equations. As with most parallel domain decomposition methods each processor is assigned one or more subdomains and an iteration is devised which allows the processors to solve their own subproblem(s) concurrently. The novel feature of this algorithm however is that each of these subproblems is defined over the entire domain—although the vast majority of the degrees of freedom for each subproblem are associated with a single subdomain (owned by the corresponding processor). This ensures that a global mechanism is contained within each of the subproblems tackled and so no separate coarse grid solve is required in order to achieve rapid convergence of the overall iteration. Furthermore, by following the paradigm introduced in 15, it is demonstrated that this domain decomposition solver may be coupled easily with a conventional mesh refinement code, thus allowing the accuracy, reliability and efficiency of mesh adaptivity to be utilized in a well load-balanced manner. Finally, numerical evidence is presented which suggests that this technique has significant potential, both in terms of the rapid convergence properties and the efficiency of the parallel implementation. Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We investigate multigrid algorithms on locally refined quadrilateral meshes. In contrast to a standard multigrid algorithm, where the hierarchy of meshes is generated by global refinement, we suppose that the finest mesh results from an adaptive refinement algorithm using bisection and ‘hanging nodes’. We discuss the additional difficulties introduced by these meshes and investigate two different algorithms. The first algorithm uses merely the local refinement regions per level, leading to optimal solver complexity even on strongly locally refined meshes, whereas the second one constructs the lower level meshes by agglomeration of cells. In this note, we are mainly interested in implementation details and practical performance of the two multigrid schemes. Copyright © 2000 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We implement the Du Fort–Frankel modified explicit finite-difference beam propagation method (MEFD) to model three-dimensional optical devices using parallel computers. Accuracy comparisons with other parallel FD–BPMs are made, and we observe that the MEFD is very accurate and efficient. The parallel implementation of MEFD shows a large run-time computer savings compared to other parallel FD–BPM algorithms. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 24: 179–182, 2000. [ABSTRACT FROM AUTHOR]

We show how Van Loan's method for annulling the (2,1) block of skew-Hamiltonian matrices by symplectic-orthogonal similarity transformation generalizes to general matrices and provides a numerical algorithm for solving the general quadratic matrix equation: $$X \cdot A + B \cdot X + Q - X \cdot P \cdot X = 0$$ For skew-Hamiltonian matrices we find their canonical form under a similarity transformation and find the class of all symplectic-orthogonal similarity transformations for annulling the (2,1) block and simultaneously bringing the (1,1) block to Hessenberg form. We present a structure-preserving algorithm for the solution of continuous-time algebraic Riccati equation. Unlike other methods in the literature, the final transformed Hamiltonian matrix is not in Hamiltonian–Schur form. Three applications are presented: (a) for a special system of partial differential equations of second order for a single unknown function, we obtain the matrix of partial derivatives of second order of the unknown function by only algebraic operations and differentiation of functions; (b) for a similar transformation of a complex matrix into a symmetric (and three-diagonal) one by applying only finite algebraic transformations; and (c) for finite-step reduction of the eigenvalues–eigenvectors problem of a Hermitian matrix to the eigenvalues– eigenvectors problem of a real symmetric matrix of the same dimension. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A propagator integral interpretation for general TLM processes is presented and applied to discretized Maxwell's equations. The approach provides exact algebraic solutions of finite difference equations along a quite universal scheme. Specifically, Johns's process based on the symmetrical condensed node is reconstructed and generalized as a clear-cut FD algorithm. The fields computed are exact time-domain solutions of the FD equations, provided that total voltages and currents are evaluated and stub quantities are not externally excited. Losses are very naturally incorporated into the TLM algorithm. The structural similarity of the TLM process (properly operated) with the propagator integral appears just as fundamental as general and should have further applications. [ABSTRACT FROM AUTHOR]