Finite Difference Matrices Research Articles

An adaptive wavelet optimized finite difference B-spline polynomial chaos method for random partial differential equations

In this paper, wavelet optimized finite difference B-spline polynomial chaos method is proposed for solving stochastic partial differential equations. The generalized polynomial chaos is applied by considering linear B-spline wavelet basis. Then, stochastic Galerkin projection is executed for evaluating the deterministic coefficients of the generalized polynomial chaos. In the next step, the system of equations is discretized by using Crank-–Nicolson scheme for time integration and for approximating the differential operators, central finite difference matrices are considered. An adaptive grid is generated using the linear B-spline generalized polynomial chaos for optimizing the numerical solution. The method is then tested on three problems namely heat equation with uncertain initial conditions, Burger’s equation with random initial conditions and Burger’s equation with random viscosity as well as uncertain initial conditions. For the three test problems, grid modifications are displayed by taking periodic boundary conditions into consideration. Mean and standard deviation are plotted for each test problem. Moreover, for the third test problem, computational time comparison is performed by computing CPU time taken by the proposed method and the CPU time taken by the finite difference method on a uniform grid.

On the ℓ∞-norms of the singular vectors of arbitrary powers of a difference matrix with applications to sigma-delta quantization

Let ‖A‖max:=maxi,j⁡|Ai,j| denote the maximum magnitude of entries of a given matrix A. In this paper we show thatmax⁡{‖Ur‖max,‖Vr‖max}≤(Cr)6rN, where Ur and Vr are the matrices whose columns are, respectively, the left and right singular vectors of the r-th order finite difference matrix Dr with r≥2, and where D is the N×N finite difference matrix with 1 on the diagonal, −1 on the sub-diagonal, and 0 elsewhere. Here C is a universal constant that is independent of both N and r. Among other things, this establishes that both the right and left singular vectors of such finite difference matrices are Bounded Orthonormal Systems (BOSs) with known upper bounds on their BOS constants, objects of general interest in classical compressive sensing theory. Such finite difference matrices are also fundamental to standard rth order Sigma-Delta quantization schemes more specifically, and as a result the new bounds provided herein on the maximum ℓ∞-norms of their ℓ2-normalized singular vectors allow for several previous Sigma-Delta quantization results to be generalized and improved.

Alternating direction ghost-fluid methods for solving the heat equation with interfaces

This work presents two new alternating direction implicit (ADI) schemes for solving parabolic interface problems in two and three dimensions. First, the ghost fluid method (GFM) is adopted for the first time in the literature to treat interface jumps in the ADI framework, which results in symmetric and tridiagonal finite difference matrices in each ADI step. The proposed GFM-ADI scheme achieves a first order of accuracy in both space and time, as confirmed by numerical experiments involving complex interface shapes and spatial–temporal dependent jumps. The GFM-ADI scheme also maintains the ADI efficiency — the computational complexity for each time step scales as O(N) for a total degree of freedom N in higher dimensions. Second, a new matched interface and boundary (MIB) scheme is constructed, which downgrades the quadratic polynomial bases in the existing second order MIB to linear ones. Interestingly, the resulting MIB-ADI or mADI scheme produces the same finite difference matrices as the GFM-ADI scheme in all dimensions. Hence, the mADI scheme can be regarded as an improvement of the GFM, because it calculates tangential jumps which are omitted in the GFM. Consequently, the present mADI scheme is constantly more accurate than the GFM-ADI in all numerical examples, while keeping the same computational efficiency. Nevertheless, the mADI scheme is semi-implicit due to tangential jump approximations, while the GFM-ADI scheme is fully implicit without tangential corrections. Thus, the GFM-ADI scheme could be more stable than the mADI scheme when a huge contrast is presented in diffusion coefficients.

Fast and Accurate Computation of Vertical Modes

AbstractThe vertical modes of linearized equations of motion are widely used by the oceanographic community in numerous theoretical and observational contexts. However, the standard approach for solving the generalized eigenvalue problem using second‐order finite difference matrices produces errors for all but the few lowest modes, and increasing resolution quickly becomes too slow as the computational complexity of eigenvalue algorithms increases as . Existing methods are therefore inadequate for computing a full spectrum of internal waves, such as needed for initializing a numerical model with a full internal wave spectrum. Here we show that rewriting the eigenvalue problem in stretched coordinates and projecting onto Chebyshev polynomials results in substantially more accurate modes than finite differencing at a fraction of the computational cost. We also compute the surface quasigeostrophic modes using the same methods. All spectral and finite difference algorithms are made available in a suite of Matlab classes that have been validated against known analytical solutions in constant and exponential stratification.

A curvelet method for numerical solution of partial differential equations

This paper proposes a fast curvelet based finite difference method for numerical solutions of partial differential equations (PDEs). The method uses finite difference approximations for differential operators involved in the PDEs. After the approximation, the curvelet is used for the compression of the finite difference matrices and subsequently for computing the dyadic powers of these matrices required for solving the PDE in a fast and efficient manner. As a prerequisite, compression and reconstruction errors for the curvelet have been tested against different parameters. The developed method has been applied on five test problems of different nature. For each test problem the convergence of the method is examined. Moreover, to measure the performance of the proposed method the computational time taken by the proposed method is compared to that of the finite difference method. It is observed that the proposed method is computationally very efficient.

Ground rolls attenuation via gradient flow regularization

We propose a novel method for seismic ground rolls attenuation by employing the dips differences between ground rolls and reflective signals. The dips differences are modeled by gradient-direction differences, and the overall gradient directions are termed gradient flow. Typically, there are frequency overlap between the ground rolls and reflective signals. But, the gradient flow of a signal needs only the direction information of the gradients, not the exact scale information, and thus, can be extracted using only part of the signal. Firstly, gradient flows for the desired ground rolls and reflective signals are calculated according to the low-frequency part of the ground rolls and high-frequency part of the reflective signals via the structure tensor method. Then, an optimization model is built, which regularizes the separated ground rolls and reflective signals to have the desired gradient flows. An efficient algorithm is designed by employing the specific structure of the finite difference matrices, which are used for calculating the gradients. The optimization problem is strictly convex, and is guaranteed to converge with arbitrary initialization. The method we propose is easy to use, as only two weighting parameters of the regularization terms are set manually to tune the energy distribution of the ground rolls and reflective signals. We demonstrate effectiveness of the proposed method with two field data tests. Compared with the traditional high-pass filtering method and two improved methods, the method we propose attenuates more ground rolls and eliminates less reflective signals, and increases the lateral coherence of the restored reflective signals at the same time.

Discretization error estimates in maximum norm for convergent splittings of matrices with a monotone preconditioning part

For finite difference matrices that are monotone, a discretization error estimate in maximum norm follows from the truncation errors of the discretization. It enables also discretization error estimates for derivatives of the solution. These results are extended to convergent operator splittings of the difference matrix where the major, preconditioning part is monotone but the whole operator is not necessarily monotone.

A fast adaptive diffusion wavelet method for Burger’s equation

A fast adaptive diffusion wavelet method is developed for solving the Burger’s equation. The diffusion wavelet is developed in 2006 (Coifman and Maggioni, 2006) and its most important feature is that it can be constructed on any kind of manifold. Classes of operators which can be used for construction of the diffusion wavelet include second order finite difference differentiation matrices. The efficiency of the method is that the same operator is used for the construction of the diffusion wavelet as well as for the discretization of the differential operator involved in the Burger’s equation. The diffusion wavelet is used for the construction of an adaptive grid as well as for the fast computation of the dyadic powers of the finite difference matrices involved in the numerical solution of Burger’s equation. In this paper, we have considered one dimensional and two dimensional Burger’s equation with Dirichlet and periodic boundary conditions. For each test problem the CPU time taken by fast adaptive diffusion wavelet method is compared with the CPU time taken by finite difference method and observed that the proposed method takes lesser CPU time. We have also verified the convergence of the given method.

Infimal convolution regularizations with discrete ℓ1-type functionals

As first demonstrated by Chambolle and Lions the staircasing effect of the RudinOsher-Fatemi model can be reduced by using infimal convolutions of functionals containing higher order derivatives. In this paper, we examine a modification of such infimal convolutions in a general discrete setting. For the special case of finite difference matrices, we show the relation of our approach to the continuous total generalized variation approach recently developed by Bredies, Kunisch and Pock. We present splitting methods to compute the minimizers of the l 2 (modified) infimal convolution functionals which are superior to previously applied second order cone programming methods. Moreover, we illustrate the differences between the ordinary and the modified infimal convolution approach by numerical examples.

COMPARISONS OF THE PARALLEL PRECONDITIONERS FOR LARGE NONSYMMETRIC SPARSE LINEAR SYSTEMS ON A PARALLEL COMPUTER

In this paper we compare various parallel preconditioners for solving large sparse nonsymmetric linear systems. They are Block Jacobi, Point-SSOR, ILU(0) in the wavefront order, ILU(0) in the multi-color order, SPAI(SParse Approximate Inverse), and Multi-Color Block SOR. The Block Jacobi and Point-SSOR are well-known, and ILU(0) is one of the most popular preconditioners, but it is inherently serial. ILU(0) in the wavefront order maximizes the parallelism, and ILU(0) in the multi-color order achieves the parallelism of order (N), where N is the order of the matrix. The SPAI tries to capture the approximate inverse in sparse form, which, then, is expected to be a scalable preconditioner. Finally, we implemented the Multi-Color Block SOR preconditioner combined with direct sparse matrix solver. For the Laplacian matrix the SOR method is known to have a non-deteriorating rate of convergence when used with Multi-Color ordering. Since most of the time is spent on the diagonal inversion, which is done on each processor, we expect it to be a good scalable preconditioner. Finally, due to the blocking effect, it will be effective for ill-conditioned problems. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to 1024×1024, and for an ill-conditioned matrix from the shell problem from the Harwell–Boeing collection. CRAY-T3E with 128 nodes was used. MPI library was used for interprocess communications. The results show that Multi-Color Block SOR and ILU(0) with Multi-Color ordering give the best performances for the finite difference matrices and for the shell problem only the Multi-Color Block SOR and Block Jacobi converges. Based on this we recommend that the Multi-Color Block SOR is the most robust preconditioner out of the preconditioners considered.

Eigenstructures of Spatial Design Matrices

In estimating the variogram of a spatial stochastic process, we use a spatial design matrix. This matrix is the key to Matheron's variogram estimator. We show how the structure of the matrix for any dimension is based on the one-dimensional spatial design matrix, and we compute explicit eigenvalues and eigenvectors for all dimensions. This design matrix involves Kronecker products of second order finite difference matrices, with cosine eigenvectors and eigenvalues. Using the eigenvalues of the spatial design matrix, the statistics of Matheron's variogram estimator are determined. Finally, a small simulation study is performed.

Optimal Order Preconditioning of Finite Difference Matrices

A new multilevel preconditioner is proposed for the iterative solution of linear systems whose coefficient matrix is a symmetric M-matrix arising from the discretization of a second-order elliptic PDE. It is based on a recursive block incomplete factorization of the matrix partitioned in a two-by-two block form, in which the submatrix related to the fine grid nodes is approximated by a modified incomplete (MILU) factorization and the Schur complement computed from a diagonal approximation of the same submatrix. A general algebraic analysis proves optimal order convergence under mild assumptions related to the quality of the approximations of the Schur complement on the one hand and of the fine grid submatrix on the other hand. This analysis does not require a red-black ordering of the fine grid nodes, nor a transformation of the matrices in hierarchical form. Considering more specifically 5 point finite difference approximations of two-dimensional problems, we prove that the spectrum of the preconditioned system is contained in the interval [0.46, 3], independently of possible (large) jumps or anisotropy in the PDE coefficients, as long as the latter are piecewise constant on the coarsest mesh. Numerical results illustrate the efficiency and the robustness of the proposed method.

Spectral and structural analysis of high precision finite difference matrices for elliptic operators

In this paper we study the structural properties of matrices coming from high-precision Finite Difference (FD) formulae, when discretizing elliptic (or semielliptic) differential operators L(a,u) of the form (−) k d k dx k a(x) d k dx k u(x) . Strong relationships with Toeplitz structures and Linear Positive Operators (LPO) are highlighted. These results allow one to give a detailed analysis of the eigenvalues localisation/distribution of the arising matrices. The obtained spectral analysis is then used to define optimal Toeplitz preconditioners in a very compact and natural way and, in addition, to prove Szegö-like and Widom-like ergodic theorems for the spectra of the related preconditioned matrices. A wide numerical experimentation, confirming the theoretical results, is also reported.

Boundary Conditions in Chebyshev and Legendre Methods

We discuss two different ways of treating non-Dirichlet boundary conditions in Chebyshev and Legendre collocation methods for second order differential problems. An error analysis is provided. The effect of preconditioning the corresponding spectral operators by finite difference matrices is also investigated.

Paper 3: Recent Developments in Fluid Film Lubrication Theory

Developments in high-speed electronic computers have greatly influenced the progress in fluid film lubrication over the past ten years. Static and dynamic oil film parameters have been computed for a wide range of finite geometries, for hydrostatic and hydrodynamic bearings lubricated by compressible and incompressible lubricants. These are either sufficient in themselves or else act as a yardstick against which approximate formulas may be tested. Much use has been made of iterative finite difference schemes, which are particularly well suited to digital computers, and these methods are now more fully understood. Other methods of solution include direct inversion of finite difference matrices and solution by expression of the pressure by some infinite series, a finite number of terms of which give adequate representation.Besides the increase in design data available, there has been substantial progress through a re-examination of the effects of modifying some of the assumptions inherent in most of the available solutions of the Reynolds equation. These include the assumption of constant lubricant viscosity, of rigid surfaces and of laminar flow.Major progress has been witnessed in two fields. The interaction of the lubricant film with elastic boundaries has been shown to be of prime importance in highly loaded contacts such as gears. This has led to the development of the special topic of elastohydrodynamic lubrication theory. The applicability of gas bearings in such growing industries as computers, space vehicles and nuclear reactors has resulted in great activity and progress in this field.

On the Stability of Finite Difference Matrices

The stability theory of finite difference approximations to linear partial differential equations is largely based on that for the special case in which the coefficients of the equations are constant and the boundary conditions are periodic. In that case one can employ a Fourier analysis to reduce the problem to the study of families of matrices G, the amplification matrices (see Lax and Richtmyer [6]). If vv is the Fourier transform of the solution vector at the vth time step, then vv = Gvvo and the stability of the difference scheme is equivalent to the uniform boundedness of Gv for vAt ^ 1, where At is the time step. A rather more convenient and equivalent form of this condition is that there should exist a constant a > 0 such that the matrices A = eaAtG satisfy