An accelerated majorization-minimization algorithm with convergence guarantee for non-Lipschitz wavelet synthesis model *

2021 ◽  
Vol 38 (1) ◽  
pp. 015001
Author(s):  
Yanan Zhao ◽  
Chunlin Wu ◽  
Qiaoli Dong ◽  
Yufei Zhao
Keyword(s):  
Minimization Problem ◽  
Stationary Point ◽  
Numerical Experiments ◽  
Complete Convergence ◽  
Convergence Result ◽  
Image Process ◽  
Minimization Algorithm ◽  
Computationally Efficient ◽  
Special Case ◽  
Reconstruction Model

Abstract We consider a wavelet based image reconstruction model with the ℓ p (0 < p < 1) quasi-norm regularization, which is a non-convex and non-Lipschitz minimization problem. For solving this model, Figueiredo et al (2007 IEEE Trans. Image Process. 16 2980–2991) utilized the classical majorization-minimization framework and proposed the so-called Isoft algorithm. This algorithm is computationally efficient, but whether it converges or not has not been concluded yet. In this paper, we propose a new algorithm to accelerate the Isoft algorithm, which is based on Nesterov’s extrapolation technique. Furthermore, a complete convergence analysis for the new algorithm is established. We prove that the whole sequence generated by this algorithm converges to a stationary point of the objective function. This convergence result contains the convergence of Isoft algorithm as a special case. Numerical experiments demonstrate good performance of our new algorithm.

Computation of the Streamfunction and Velocity Potential for Limited and Irregular Domains

2006 ◽  
Vol 134 (11) ◽  
pp. 3384-3394 ◽  
Author(s):  
Zhijin Li ◽  
Yi Chao ◽  
James C. McWilliams
Keyword(s):  
Tikhonov Regularization ◽  
Minimization Problem ◽  
Velocity Potential ◽  
Minimization Algorithm ◽  
Practical Application ◽  
Computationally Efficient ◽  
Irregular Domains ◽  
Coastal Oceans ◽  
Uniqueness Of The Solution ◽  
The Given

Abstract An algorithm is proposed for the computation of streamfunction and velocity potential from given horizontal velocity vectors based on solving a minimization problem. To guarantee the uniqueness of the solution and computational reliability of the algorithm, a Tikhonov regularization is applied. The solution implies that the obtained streamfunction and velocity potential have minimal magnitude, while the given velocity vectors can be accurately reconstructed from the computed streamfunction and velocity potential. Because the formulation of the minimization problem allows for circumventing the explicit specification of separate boundary conditions on the streamfunction and velocity potential, the algorithm is easily applicable to irregular domains. By using an advanced minimization algorithm with the use of adjoint techniques, the method is computationally efficient and suitable for problems with large dimensions. An example is presented for coastal oceans to illustrate the practical application of the algorithm.

scholarly journals Evaluating the Observed Log-Likelihood Function in Two-Level Structural Equation Modeling with Missing Data: From Formulas to R Code

Psych ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 197-232
Author(s):  
Yves Rosseel
Keyword(s):  
Structural Equation ◽  
Structural Equation Models ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Missing At Random ◽  
Equation Modeling ◽  
Computationally Efficient ◽  
Log Likelihood ◽  
Efficient Expression ◽  
Special Case

This paper discusses maximum likelihood estimation for two-level structural equation models when data are missing at random at both levels. Building on existing literature, a computationally efficient expression is derived to evaluate the observed log-likelihood. Unlike previous work, the expression is valid for the special case where the model implied variance–covariance matrix at the between level is singular. Next, the log-likelihood function is translated to R code. A sequence of R scripts is presented, starting from a naive implementation and ending at the final implementation as found in the lavaan package. Along the way, various computational tips and tricks are given.

Efficient Numerical Solution of Inverse Heat Conduction Problems

1995 ◽  
Author(s):  
Hans-Jürgen Reinhardt ◽  
Dinh Nho Hao
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Numerical Methods ◽  
Stationary Point ◽  
Forthcoming Paper ◽  
Inverse Heat Conduction ◽  
Piecewise Constant ◽  
Numerical Tests ◽  
Inverse Heat Conduction Problems ◽  
Special Case

Abstract In this contribution we propose new numerical methods for solving inverse heat conduction problems. The methods are constructed by considering the desired heat flux at the boundary as piecewise constant (in time) and then deriving an explicit expression for the solution of the equation for a stationary point of the minimizing functional. In a very special case the well-known Beck method is obtained. For the time being, numerical tests could not be included in this contribution but will be presented in a forthcoming paper.

scholarly journals Virtual Element Method for the Laplace-Beltrami equation on surfaces

2018 ◽  
Vol 52 (3) ◽  
pp. 965-993 ◽  
Author(s):  
Massimo Frittelli ◽  
Ivonne Sgura
Keyword(s):  
Numerical Experiments ◽  
Beltrami Equation ◽  
Convergence Result ◽  
Point Of View ◽  
Polygonal Approximation ◽  
Virtual Element Method ◽  
Element Space ◽  
First Order ◽  
Virtual Element ◽  
Element Method

We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equation on a surface in ℝ3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) (Dziuk, Eliott, G. Dziuk and C.M. Elliott., Acta Numer. 22 (2013) 289–396.) and the recent VEM (Beirão da Veiga et al., Math. Mod. Methods Appl. Sci. 23 (2013) 199–214.) in order to allow for a general polygonal approximation of the surface. We account for the error arising from the geometry approximation and in the case of polynomial order k = 1 we extend to surfaces the error estimates for the interpolation in the virtual element space. We prove existence, uniqueness and first order H1 convergence of the numerical solution.We highlight the differences between SVEM and VEM from the implementation point of view. Moreover, we show that the capability of SVEM of handling nonconforming and discontinuous meshes can be exploited in the case of surface pasting. We provide some numerical experiments to confirm the convergence result and to show an application of mesh pasting.

scholarly journals Non-static surfaces in MCNPX: The chopper extension1

2020 ◽  
Vol 22 (2-3) ◽  
pp. 191-198
Author(s):  
Kyle B. Grammer ◽  
Franz X. Gallmeier ◽  
Erik B. Iverson
Keyword(s):  
Single Crystal ◽  
Stationary Point ◽  
Direct Simulation ◽  
Cold Neutrons ◽  
Special Case ◽  
Rotating Objects ◽  
Common Components

Rotating objects, such as choppers, are common components of a neutron beamline, and the motion of these components is not described in the static geometry of an MCNPX model. The special case of non-static surfaces for rotation about a stationary point in space has been developed for MCNPX. In addition, velocity dependent kinematics due to the motion of the medium have been implemented. This implementation allows for the simulation of rotating objects at speeds comparable to the velocity of cold neutrons. Applications of the chopper extension will be discussed, including the direct simulation of a bandwidth chopper system, the thermalization of neutrons inside a spinning material, and the discussion of the implementation of a spinning single crystal.

scholarly journals Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 156 ◽  
Author(s):  
Chanjuan Pan ◽  
Yuanheng Wang
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Minimization Problem ◽  
Numerical Experiments ◽  
Splitting Method ◽  
Splitting Methods ◽  
Strong Convergence Theorem ◽  
Accretive Operators ◽  
Convex Minimization Problem ◽  
Inertial Extrapolation

In this article, we study a modified viscosity splitting method combined with inertial extrapolation for accretive operators in Banach spaces and then establish a strong convergence theorem for such iterations under some suitable assumptions on the sequences of parameters. As an application, we extend our main results to solve the convex minimization problem. Moreover, the numerical experiments are presented to support the feasibility and efficiency of the proposed method.

On the weak convergence of U-statistic processes, and of the empirical process

1978 ◽  
Vol 83 (2) ◽  
pp. 269-272 ◽  
Author(s):  
R. M. Loynes
Keyword(s):  
Weak Convergence ◽  
Wiener Process ◽  
Empirical Process ◽  
Convergence Result ◽  
The Other ◽  
Direct Argument ◽  
Random Functions ◽  
Martingale Theorem ◽  
The Relationship ◽  
Special Case

1. Summary and introductionIn (5) a weak convergence result for U-statistics was obtained as a special case of a reverse martingale theorem; in (7) Miller and Sen obtained another such result for U-statistics by a direct argument. As they stand these results are not very closely connected, since one is concerned with U-statistics Uk for k ≥ n, while the other deals with Uk for k ≤ n, but if one instead thinks of k as unrestricted and transforms the random functions Xn which enter into one of these results into new functions Yn by setting Yn(t) = tXn(t−1) one finds that the Yn are (aside from variations in interpolated values) just the functions with which the other result is concerned. As the limiting Wiener process W is well-known to have the property that tW(t−1) is another Wiener process it is not too surprising that both results should hold, and part of the purpose of this paper is to provide a general framework within which the relationship between these results will become clear. A second purpose is to illustrate the simplification that the martingale property brings to weak convergence studies; this is shown both in the U-statistic example and in a new proof of the convergence of the empirical process.

A finite-difference method for the numerical solution of the Sal’nikov thermokinetic oscillator problem

1995 ◽  
Vol 449 (1936) ◽  
pp. 255-271
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Stationary Point ◽  
Difference Method ◽  
Numerical Experiments ◽  
Euler Method ◽  
Initial Value ◽  
First Order ◽  
Highly Nonlinear ◽  
Implicit Finite Difference

A finite-difference method is developed for solving two coupled, ordinary differential equations that model a sequence of chemical reactions. The initial-value problem is highly nonlinear and involves three parameters. Various types of theoretical solution of this problem (the Sal’nikov thermokinetic oscillator problem) may be found, depending on these parameters; this is because the stationary point is surrounded by up to two limit cycles. The well-known, first-order, explicit Euler method and an implicit finite difference method of the same order are used to compute the solution. It is shown that this implicit method may, in fact, be used explicitly and extensive numerical experiments are made to confirm the superior stability properties of the alternative method.

scholarly journals Wavepath eikonal traveltime inversion: Theory

Geophysics ◽  
1993 ◽  
Vol 58 (9) ◽  
pp. 1314-1323 ◽  
Author(s):  
Gerard T. Schuster ◽  
Aksel Quintus‐Bosz
Keyword(s):  
Ray Tracing ◽  
Weighting Factor ◽  
Back Projection ◽  
Computationally Efficient ◽  
Traveltime Inversion ◽  
Traveltime Tomography ◽  
Inversion Theory ◽  
Order Of Magnitude ◽  
Band Limited ◽  
Special Case

We present a general formula for the back projection of traveltime residuals in traveltime tomography. For special choices of an arbitrary weighting factor this formula reduces to the asymptotic back‐projection term in ray‐tracing tomography (RT), the Woodward‐Rocca method, wavepath eikonal traveltime inversion (WET), and wave‐equation traveltime inversion (WT). This unification provides for an understanding of the differences and similarities among these traveltime tomography methods. The special case of the WET formula leads to a computationally efficient inversion scheme in the space‐time domain that is, in principle, almost as effective as WT inversion yet is an order of magnitude faster. It also leads to an analytic formula for the fast computation of wavepaths. Unlike ray‐tracing tomography, WET partially accounts for band‐limited source and shadow effects in the data. Several numerical tests of the WET method are used to illustrate its properties.

scholarly journals Gauss-Newton Method for Segmentation assisted Deformable Registration

2014 ◽  
Author(s):  
Miro Jurisic ◽  
Tobias Fechter ◽  
Frida Hauler ◽  
Hugo Furtado ◽  
Ursula Nestle ◽  
...  
Keyword(s):  
Newton Method ◽  
Minimization Problem ◽  
Energy Minimization ◽  
Numerical Experiments ◽  
High Accuracy ◽  
Energy Model ◽  
Deformable Registration ◽  
Data Sets ◽  
Preliminary Study ◽  
Ct Data

In this work, we try to develop a fast converging method for segmentation assisted deformable registration. The segmentation step consists of a piece-wise constant Mumford-Shah energy model while reg- istration is driven by the sum of squared distances of both initial images and segmented mask with a diffusion regularization. In order to solve this energy minimization problem, a second order Gauss-Newton opti- mization method is used. For the numerical experiments we used CT data sets from the EMPIRE10 challenge. In this preliminary study, we show high accuracy of our algorithm.

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