Numerical Algorithms最新文献

We investigate numerical methods for computation of polynomial and rational approximations of functions in complex domains based on Faber polynomials and the Lanczos (tau )-method. Our interest is motivated by applications in fractional partial differential equations. We give an overview of previous results related to the basis of Faber polynomials associated with a complex domain, Faber expansion, and the Lanczos (tau )-method. We also collect numerical algorithms for the computational realization of these concepts. Our main new contribution is a (tau )-method for rational approximation in complex domains which uses Faber polynomials in the perturbation term. We realize it via a novel hybrid symbolic-numeric algorithm which can be applied to arbitrary functions satisfying a suitable differential equation. We present some numerical examples, where we use sectors lying in the complex plane as our domains of interest. We compare results for the various polynomial and rational approximation techniques outlined above; in particular, we observe exponential convergence with respect to the rational degree for our proposed method.

In this paper, unified convergence analyses for a class of iterative methods of order three, five, and six are studied to solve the nonlinear systems in Banach space settings. Our analysis gives the number of iterations needed to achieve the given accuracy and the radius of the convergence ball precisely using weaker conditions on the involved operator. Various numerical examples have been taken to illustrate the proposed method, and the theoretical convergence has been validated via these examples.

We suggest a revised form of a classic measure function to be employed in the optimization model of the nonnegative matrix factorization problem. More exactly, using sparse matrix approximations, the revision term is embedded to the model for penalizing the ill-conditioning in the computational trajectory to obtain the factorization elements. Then, as an extension of the Euclidean norm, we employ the ellipsoid norm to gain adaptive formulas for the Dai–Liao parameter in a least-squares framework. In essence, the parametric choices here are obtained by pushing the Dai–Liao direction to the direction of a well-functioning three-term conjugate gradient algorithm. In our scheme, the well-known BFGS and DFP quasi–Newton updating formulas are used to characterize the positive definite matrix factor of the ellipsoid norm. To see at what level our model revisions as well as our algorithmic modifications are effective, we seek some numerical evidence by conducting classic computational tests and assessing the outputs as well. As reported, the results weigh enough value on our analytical efforts.

The (C^1)-(P_5) Bell finite element removes the three degrees of freedom of the edge normal derivatives of the (C^1)-(P_5) Argyris finite element. We call a (C^1)-(P_k) finite element a Bell finite element if it has no edge-degree of freedom and it contains the (P_{k-1}) space locally. We construct three families of odd-degree (C^1)-(P_{2m+1}) Bell finite elements on triangular meshes. Comparing to the (C^1)-(P_{2m}) Argyris finite element, the (C^1)-(P_{2m+1}) Bell finite elements produce same-order solutions with much less unknowns. For example, the second (C^1)-(P_7) Bell element (from the second family) and the (C^1)-(P_6) Argyris element have numbers of local degrees of freedom of 31 and 28 respectively, but oppositely their numbers of global degrees of freedom are 12V and 19V asymptotically, respectively, where V is the number of vertices in a triangular mesh. A numerical example says the new element has about 3/4 number of unknowns, but is about 5 times more accurate. Numerical computations with all three families of elements are performed.

We consider a stationary iteration for solving a linear system of arbitrary order. The method includes, e.g. Kaczmarz iteration, the Landweber iteration and the SOR (Gauss-Seidel) iteration. A study of the behavior of the iterates, both theoretically and experimentally, is performed. In particular we compare the behavior with and without noise in the data. The results give insight into the interplay between noise free and noisy iterates. For comparision we also included a Krylov type method CGLS in the experiments. As expected CGLS works well for noise free data but also tends to amplify the noise faster than the other methods, thus making it more critical when to stop the iterations.

In this paper we describe TRIPs-Py, a new Python package of linear discrete inverse problems solvers and test problems. The goal of the package is two-fold: 1) to provide tools for solving small and large-scale inverse problems, and 2) to introduce test problems arising from a wide range of applications. The solvers available in TRIPs-Py include direct regularization methods (such as truncated singular value decomposition and Tikhonov) and iterative regularization techniques (such as Krylov subspace methods and recent solvers for (ell _p)-(ell _q) formulations, which enforce sparse or edge-preserving solutions and handle different noise types). All our solvers have built-in strategies to define the regularization parameter(s). Some of the test problems in TRIPs-Py arise from simulated image deblurring and computerized tomography, while other test problems model real problems in dynamic computerized tomography. Numerical examples are included to illustrate the usage as well as the performance of the described methods on the provided test problems. To the best of our knowledge, TRIPs-Py is the first Python software package of this kind, which may serve both research and didactical purposes.

In this article we consider problems of interval estimation of a set of solutions to point and interval (partially interval) systems of nonlinear equations. Most developed interval methods are intended only for estimating solutions of point nonlinear systems in some given interval box. And methods for estimating solution sets of nonlinear interval systems are not yet very developed, since the solution sets of such systems geometrically represent a rather complex structure. Here we conducted a general analysis on existing classical interval methods to test their applicability for interval systems. In this case, we chose the methods of Newton and Krawczyk. The results of the analysis show that these and similar other iterative methods are generally not applicable for interval systems due to the limited admissible area. Based on the results of the analysis, a new combined vertex method for outer estimation of solution sets of interval nonlinear systems is proposed, which includes these classical interval methods. Numerical experiments have shown that the proposed method is more efficient and gives more accurate estimates in feasible regions than the direct application of Newton, Krawczyk or Hansen-Sengupta interval methods for interval systems.

In this paper, incorporating the quaternion matrix framework, the logarithmic norm of quaternion matrices is employed to approximate rank. Unlike conventional sparse representation techniques for matrices, which treat RGB channels separately, quaternion-based methods maintain image structure by representing color images within a pure quaternion matrix. Leveraging the logarithmic norm, factorization and truncation techniques can be applied for proficient image recovery. Optimization of these approaches is facilitated through an alternate minimization framework, supplemented by meticulous mathematical scrutiny ensuring convergence. Finally, some numerical examples are used to demonstrate the effectiveness of the proposed algorithms.

Since the seminal work in 1986, the treecode algorithm has been widely used in a variety of science and engineering problems, such as the electrostatic and magnetostatic fields calculations. With the continuous advancements of science exploration and engineering applications, efficient numerical simulations for problems defined on complex domains have become increasingly necessary. In this paper, based on a hierarchy geometry tree, an efficient implementation of the treecode algorithm is described in detail for the numerical solution of a Poisson equation defined on a general domain. The features of our algorithm include: i) with the hierarchy geometry tree, the neighbor and non-neighbor patches for a given element can be generated efficiently, ii) no restriction on the geometry of the domain, which means that our algorithm can be applied for general problem, iii) the desired computational complexity ({varvec{mathcal {O}}}(varvec{N},varvec{log },{varvec{N}})) can be observed well, where (varvec{N}) denotes the number of degrees of freedom in the domain, and iv) very friendly to the parallel computing, i.e., an ideal speedup can be observed successfully from numerical results with OpenMP technique. It is believed that our solution potentially is a quality candidate for implementing the treecode algorithm for problems defined on general domains with unstructured grids.

This paper studies the numerical solutions of Caputo–Hadamard fractional stochastic differential equations. Firstly, we construct an Euler–Maruyama (EM) scheme for the equations, and the corresponding convergence rate is investigated. Secondly, we propose a fast EM scheme based on the sum-of-exponentials approximation to decrease the computational cost of the EM scheme. More concretely, the fast EM scheme reduces the computational cost from (O(N^2)) to (O(Nlog ^2 N)) and the storage from O(N) to (O(log ^2 N)) when the final time (Tapprox e), where N is the total number of time steps. Moreover, considering the statistical errors from Monte Carlo path approximations, multilevel Monte Carlo (MLMC) techniques are utilized to reduce computational complexity. In particular, for a prescribed accuracy (varepsilon >0), the EM scheme and the fast EM scheme, integrated with the MLMC technique, respectively reduce the standard EM scheme’s computational complexity from (O(varepsilon ^{-2-frac{2}{widetilde{alpha }}})) to (O(varepsilon ^{-frac{2}{widetilde{alpha }}})) and the fast EM scheme’s complexity to (O(varepsilon ^{-frac{1}{widetilde{alpha }}}left|log varepsilon right|^3)), where (0<widetilde{alpha }=alpha -frac{1}{2}<frac{1}{2}). Finally, numerical examples are included to verify the theoretical results and demonstrate the performance of our methods.