Fast Iterative Method Research Articles

Analytical investigations for the design of fast approximation methods for fitting curves and surfaces to scattered data

We present an analytical framework for linear and nonlinear least squares methods and adopt it to the construction of fast iterative methods for fitting curves and surfaces to scattered data. The results are directly applicable to curves and surfaces that have a representation as a linear combination of smooth basis functions associated with the control points. Standard Bézier and B-spline curves / surfaces as well as subdivision schemes have this property. In the global approximation step for the control points our approach couples the standard linear approximation part with the reparameterization to heavily reduce the number of overall steps in the iteration process. This can be formulated in such a way that we have a standard least squares problem in each step. For the local nonlinear parameter corrections our results allow for an optimal choice of the methods used in different stages of the process. Furthermore, regularization terms that express the fairness of the intermediate and / or final result can be added. Adaptivity is easily integrated in our concept. Moreover our approach is well suited for reparameterization occurring in grid generation.

An Improved Adaptive Subspace Tracking Algorithm Based on Approximated Power Iteration

A subspace tracking technique has drawn a lot of attentions due to its wide applications. The main objective of this approach is to estimate signal or noise subspace basis for the sample covariance matrix. In this paper, we focus on providing a fast, stable, and adaptive subspace tracking algorithm that is implemented with low computational complexity. An alternative realization of the fast approximate power iteration (FAPI) method, termed modified FAPI (MFAPI), is also presented. Rather than solving an inverse square root of a matrix employed in the FAPI, the MFAPI applies the matrix product directly to ensure the orthonormality of the subspace basis matrix at each recursion. This approach yields a simpler derivation and is numerically stable while maintaining a similar computational complexity as compared with that of the FAPI. Furthermore, we present a detailed mathematical proof of the numerical stability of our proposed algorithm. Computer simulation results indicate that the MFAPI outperforms many classical subspace tracking algorithms, particularly at the transient-state step.

NEWTON CORRECTION METHODS FOR COMPUTING REAL EIGENPAIRS OF SYMMETRIC TENSORS.

Real eigenpairs of symmetric tensors play an important role in multiple applications. In this paper we propose and analyze a fast iterative Newton-based method to compute real eigenpairs of symmetric tensors. We derive sufficient conditions for a real eigenpair to be a stable fixed point for our method and prove that given a sufficiently close initial guess, the convergence rate is quadratic. Empirically, our method converges to a significantly larger number of eigenpairs compared with previously proposed iterative methods, and with enough random initializations typically finds all real eigenpairs. In particular, for a generic symmetric tensor, the sufficient conditions for local convergence of our Newton-based method hold simultaneously for all its real eigenpairs.

Blind Deconvolution for Image Deblurring Based on Edge Enhancement and Noise Suppression

This paper denotes to obtain an accuracy blur kernel and a shape image. An efficient method that blind deconvolution for image deblurring based on edge enhancement and noise suppression is proposed. First, we exploited an edge detection method to extract the strong edge portion of blurred image. Then, the blurred image was divided into weak edge portion and strong edge portion. At this time, we apply a trilateral filter method to suppress the noise in the weak edge portion. Through mathematical operations for weak edge portion and strong edge portion, we can obtain the new blurred image which as the input of blur kernel estimation. At the phase of the kernel estimation, the problem can be solved via alternate between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> updating. In addition, we utilize improved fast iterative shrinkage thresholding algorithm method to solve the optimization problem. Finally, non-blind deconvolution was employed at the phase of image recovery. A comprehensive evaluation shows that our approach achieves state-of-the-art results for uniformly blurred images.

Estimating Large-Scale Tree Logit Models via a Difference of Strictly Convex Functions

We describe an efficient estimation method for large-scale tree logit models, using a novel change-of-variables transformation that allows us to express the negative log-likelihood as a difference of strictly convex functions. Exploiting this representation, we design a fast iterative method that computes a sequence of parameter estimates. At each iteration, parameters at leaf nodes are updated using a simple formula involving the Lambert-W function, while the parameters at non-leaf nodes are updated simultaneously by minimizing a strictly convex one-dimensional function over the unit interval. No step size or second-order derivative is required. The sequence of parameter estimates yields increasing likelihood values, and we show that every limit point is a stationary point. Numerical results show that our algorithm outperforms state-of-the-art optimization methods, especially for large-scale tree logit models with thousands of nodes.

Monoplane 3D–2D registration of cerebral angiograms based on multi-objective stratified optimization

Registration of 3D pre-interventional to 2D intra-interventional medical images has an increasingly important role in surgical planning, navigation and treatment, because it enables the physician to co-locate depth information given by pre-interventional 3D images with the live information in intra-interventional 2D images such as x-ray. Most tasks during image-guided interventions are carried out under a monoplane x-ray, which is a highly ill-posed problem for state-of-the-art 3D to 2D registration methods. To address the problem of rigid 3D–2D monoplane registration we propose a novel multi-objective stratified parameter optimization, wherein a small set of high-magnitude intensity gradients are matched between the 3D and 2D images. The stratified parameter optimization matches rotation templates to depth templates, first sampled from projected 3D gradients and second from the 2D image gradients, so as to recover 3D rigid-body rotations and out-of-plane translation. The objective for matching was the gradient magnitude correlation coefficient, which is invariant to in-plane translation. The in-plane translations are then found by locating the maximum of the gradient phase correlation between the best matching pair of rotation and depth templates. On twenty pairs of 3D and 2D images of ten patients undergoing cerebral endovascular image-guided intervention the 3D to monoplane 2D registration experiments were setup with a rather high range of initial mean target registration error from 0 to 100 mm. The proposed method effectively reduced the registration error to below 2 mm, which was further refined by a fast iterative method and resulted in a high final registration accuracy (0.40 mm) and high success rate (96%). Taking into account a fast execution time below 10 s, the observed performance of the proposed method shows a high potential for application into clinical image-guidance systems.

Fast Iterative Adaptive Multi-quadric Radial Basis Function Method for Edges Detection of Piecewise Functions—I: Uniform Mesh

In Jung et al. (Appl Numer Math 61:77–91, 2011), an iterative adaptive multi-quadric radial basis function (IAMQ-RBF) method has been developed for edges detection of the piecewise analytical functions. For a uniformly spaced mesh, the perturbed Toeplitz matrices, which are modified by those columns where the shape parameters are reset to zero due to the appearance of edges at the corresponding locations, are created. Its inverse must be recomputed at each iterative step, which incurs a heavy $$O(n^3)$$ computational cost. To overcome this issue of efficiency, we develop a fast direct solver (IAMQ-RBF-Fast) to reformulate the perturbed Toeplitz system into two Toeplitz systems and a small linear system via the Sherman–Morrison–Woodbury formula. The $$O(n^2)$$ Levinson–Durbin recursive algorithm that employed Yule–Walker algorithm is used to find the inverse of the Toeplitz matrix fast. Several classical benchmark examples show that the IAMQ-RBF-Fast based edges detection method can be at least three times faster than the original IAMQ-RBF based one. And it can capture an edge with fewer grid points than the multi-resolution analysis (Harten in J Comput Phys 49:357–393, 1983) and just as good as if not better than the L1PA method (Denker and Gelb in SIAM J Sci Comput 39(2):A559–A592, 2017). Preliminary results in the density solution of the 1D Mach 3 extended shock–density wave interaction problem solved by the hybrid compact-WENO finite difference scheme with the IAMQ-RBF-Fast based shocks detection method demonstrating an excellent performance in term of speed and accuracy, are also shown.

Recovering both the space-dependent heat source and the initial temperature by using a fast convergent iterative method

ABSTRACTIn this paper, we solve two types of inverse heat source problems: one recovers an unknown space-dependent heat source without using initial value, and another recovers both the unknown space-dependent heat source and the initial value. Upon inserting the adjoint Trefftz test functions into Green’s second identity, we can retrieve the unknown space-dependent heat source by an expansion method whose expansion coefficients are derived in closed form. We assess the stability of the closed-form expansion coefficients method by using the condition numbers of coefficients matrices. Then, numerical examples are performed, which demonstrates that the closed-form expansion coefficient method is effective and stable even when it imposes a large noise on the final time data. Next, we develop a coupled iterative scheme to recover the unknown heat source and initial value simultaneously, under two over specified temperature data at two different times. A simple regularization technique is derived to overcome the highly ill-posed behavior of the second inverse problem, of which the convergence rate and stability are examined. This results in quite accurate numerical results against large noise.

Ball convergence of some iterative methods for nonlinear equations in Banach space under weak conditions

The aim of this paper is to expand the applicability of a fast iterative method in a Banach space setting. Moreover, we provide computable radius of convergence, error bounds on the distances involved and a uniqueness of the solution result based on Lipschitz-type functions not given before. Furthermore, we avoid hypotheses on high order derivatives which limit the applicability of the method. Instead, we only use hypotheses on the first derivative. The convegence order is determined using the computational order of convergence or the approximate order of convergence. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.

Erratum to: Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations

The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.

Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations

Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations

A 3-D Hybrid Cell Method for Induction Heating Problems

A novel hybrid approach for solving induction heating is presented. The electro-thermal problem is discretized by the Cell Method and coupled to the Boundary Element Method to avoid the air region meshing. A symmetric indefinite linear system, solved by a fast TFQMR iterative method, is obtained for computing time-harmonic eddy currents. The transient thermal problem, weakly coupled to the electrical one, is solved by the 0-method under convection and radiation boundary conditions.

Low-Rank Spatial Channel Estimation for Millimeter Wave Cellular Systems

The tremendous bandwidth available in the millimeter wave frequencies above 10 GHz have made these bands an attractive candidate for next-generation cellular systems. However, reliable communication at these frequencies depends critically on beamforming with very high-dimensional antenna arrays. Estimating the channel sufficiently accurately to perform beamforming can be challenging due to both low coherence time and a large number of antennas. Also, the measurements used for channel estimation may need to be made with analog beamforming, where the receiver can “look” in only one direction at a time. This paper presents a novel method for estimation of the receive-side spatial covariance matrix of a channel from a sequence of power measurements made in different angular directions. It is shown that maximum likelihood estimation of the covariance matrix reduces to a non-negative matrix completion problem. We show that the non-negative nature of the covariance matrix reduces the number of measurements required when the matrix is low-rank. The fast iterative methods are presented to solve the problem. Simulations are presented for both single-path and multi-path channels using models derived from real measurements in New York City at 28 GHz.

A Fast New Algorithm for Solving a Nonlinear Beam Equation under Nonlinear Boundary Conditions

Abstract For the problem of a nonlinear beam equation under nonlinear boundary conditions of moments, a fast iterative method is developed by transforming the ordinary differential equation into an integral one. The sinusoidal functions are used subtly as test functions as well as the bases of numerical solution in the calculation. Due to the orthogonality of the sinusoidal functions, the expansion coefficients of numerical solution in closed form can be found easily. Hence, the iterative scheme converges very fast to find numerical solutions with high accuracy.

Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non-smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles.

3D gravity data-space inversion with sparseness and bound constraints

One of the most remarkable basis of the gravity data inversion is the recognition of sharp boundaries between an ore body and its host rocks during the interpretation step. Therefore, in this work, it is attempted to develop an inversion approach to determine a 3D density distribution that produces a given gravity anomaly. The subsurface model consists of a 3D rectangular prisms of known sizes and positions and unknown density contrasts that are required to be estimated. The proposed inversion scheme incorporates the Cauchy norm as a model norm that imposes sparseness and the depth weighting of the solution. A physicalbound constraint is enforced using a generic transformation of the model parameters. The inverse problem is posed in the data space, leading to a smaller dimensional linear system of equations to be solved and a reduction in the computation time. For more efficiency, the low-dimensional linear system of equations is solved using a fast iterative method such as Lanczos Bidiagonalization. The tests carried out on the synthetic data show that the sparse data-space inversion produces blocky and focused solutions. The results obtained for the 3D inversion of the field gravity data (Mobrun gravity data) indicate that the sparse data-space inversion could produce the density models consistent with the true structures.

A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation

A fast preconditioned policy iteration method is proposed for the Hamilton–Jacobi–Bellman (HJB) equation involving tempered fractional order partial derivatives, governing the valuation of American options whose underlying asset follows exponential Lévy processes. An unconditionally stable upwind finite difference scheme with shifted Grünwald approximation is first developed to discretize the established HJB equation under the tempered fractional diffusion models. Next, the policy iteration method as an outer iterative method is utilized to solve the discretized HJB equation and proven to be convergent in finite steps to its numerical solution. Given the Toeplitz-like structure of the coefficient matrix in each policy iteration, the resulting linear system can be fast solved by the Krylov subspace method as an inner iterative method via fast Fourier transform (FFT). Furthermore, a novel preconditioner is proposed to speed up the convergence rate of the inner Krylov subspace iteration with theoretical analysis to ensure the linear system can be solved in O(NlogN) operations under some mild conditions, where N is the number of spatial node points. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed fast preconditioned policy method.

Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation

In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of time-space fractional convection-diffusion equations (TSFCDEs). To start with, an implicit difference method based on two-sided weighted shifted Gr\"{u}nwald formulae is proposed with a discussion of the stability and convergence. We construct an implicit difference scheme (IDS) and show that it converges with second order accuracy in both time and space. Then, we develop fast solution methods for handling the resulting system of linear equation with the Toeplitz matrix. The fast Krylov subspace solvers with suitable circulant preconditioners are designed to deal with the resulting Toeplitz linear systems. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from $\mathcal{O}(N^2)$ to $\mathcal{O}(N)$ and the computational complexity from $O(N^3)$ to $O(N\log N)$ in each iterative step, where $N$ is the number of grid nodes. Extensive numerical example runs show the utility of these methods over the traditional direct solvers of the implicit difference methods, in terms of computational cost and memory requirements.

A Fast Shift-Splitting Iteration Method for Nonsymmetric Saddle Point Problems

AbstractBased on the shift-splitting technique and the idea of Hermitian and skew-Hermitian splitting, a fast shift-splitting iteration method is proposed for solving nonsingular and singular nonsymmetric saddle point problems in this paper. Convergence and semi-convergence of the proposed iteration method for nonsingular and singular cases are carefully studied, respectively. Numerical experiments are implemented to demonstrate the feasibility and effectiveness of the proposed method.

Higher order operator splitting methods for an image de-noising model

This paper is concerned with fast iterative methods with development of Euler-Lagrange equation which results from the minimization of Rudin-Osher-Fatemi (ROF) model. There are many applications of image de-noising in field of medical and astronomy. We can classify the image de-noising models into additive and multiplicative noise removal models. In case of additive noise, we have an image u corrupted with additive Gaussian noise η, the main task is to recover u from the image formation model u0 = u+η. This paper mainly focus on additive noise removal. Here semi-implicit (SIM), additive operator splitting (AOS) and additive multiplicative operator splitting (AMOS) type schemes are developed. The quality in AOS is, it treats with all coordinate axes in an equal manner. We develop a new AMOS scheme for the solution of Euler-Lagrange equation arisen from minimization of image additive noise removal model. Comparison of AMOS with SIM and AOS is also presented. Experimental results shows that by using AMOS, additive noisy image can be de-noised with best results. Numerical examples are given to show gain in CPU timing and fast convergence of AMOS-based algorithm. https://doi.org/10.28919/jmcs/3312