Injectivity and weak*-to-weak continuity suffice for convergence rates in ℓ1-regularization

2018 ◽  
Vol 26 (1) ◽  
pp. 85-94 ◽  
Author(s):  
Jens Flemming ◽  
Daniel Gerth
Keyword(s):  
Exact Solution ◽  
Convergence Rate ◽  
Convergence Rates ◽  
Weak Continuity ◽  
Source Type ◽  
Ill Posed ◽  
Sparse Solutions ◽  
ℓ1 Regularization

AbstractWe show that the convergence rate of {\ell^{1}}-regularization for linear ill-posed equations is always {{\mathcal{O}}(\delta)} if the exact solution is sparse and if the considered operator is injective and weak*-to-weak continuous. Under the same assumptions convergence rates in case of non-sparse solutions are proven. The results base on the fact that certain source-type conditions used in the literature for proving convergence rates are automatically satisfied.

scholarly journals Tikhonov regularization with ℓ0-term complementing a~convex penalty: ℓ1-convergence under sparsity constraints

2019 ◽  
Vol 27 (4) ◽  
pp. 575-590 ◽  
Author(s):  
Wei Wang ◽  
Shuai Lu ◽  
Bernd Hofmann ◽  
Jin Cheng
Keyword(s):  
Convergence Rates ◽  
Regularization Parameter ◽  
A Priori ◽  
Convex Functional ◽  
Important Special Case ◽  
Bounded Linear ◽  
Sparsity Constraints ◽  
Ill Posed ◽  
Sparse Solutions ◽  
Special Case

Abstract Measuring the error by an {\ell^{1}} -norm, we analyze under sparsity assumptions an {\ell^{0}} -regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed operator equations {Ax=y} with an injective and bounded linear operator A mapping between {\ell^{2}} and a Banach space Y are regularized. For sparse solutions, error estimates as well as linear and sublinear convergence rates are derived based on a variational inequality approach, where the regularization parameter can be chosen either a priori in an appropriate way or a posteriori by the sequential discrepancy principle. To further illustrate the balance between the {\ell^{0}} -term and the complementing convex penalty, the important special case of the {\ell^{2}} -norm square penalty is investigated showing explicit dependence between both terms. Finally, some numerical experiments verify and illustrate the sparsity promoting properties of corresponding regularized solutions.

Convex Tikhonov regularization in Banach spaces: New results on convergence rates

2016 ◽  
Vol 24 (3) ◽  
Author(s):  
Stefan Kindermann
Keyword(s):  
Exact Solution ◽  
Variational Inequality ◽  
Banach Spaces ◽  
Tikhonov Regularization ◽  
Convergence Rates ◽  
Operator Equations ◽  
Ill Posed ◽  
Fidelity Term ◽  
The Difference ◽  
Regularized Solution

AbstractTikhonov regularization in Banach spaces with convex penalty and convex fidelity term for linear ill-posed operator equations is studied. As a main result, convergence rates in terms of the Bregman distance of the regularized solution to the exact solution is proven by imposing a generalization of the established variational inequality conditions on the exact solution. This condition only involves a decay rate of the difference of the penalty functionals in terms of the residual.

scholarly journals On ℓ1-Regularization in Light of Nashed's Ill-Posedness Concept

2015 ◽  
Vol 15 (3) ◽  
pp. 279-289 ◽  
Author(s):  
Jens Flemming ◽  
Bernd Hofmann ◽  
Ivan Veselić
Keyword(s):  
Inverse Problems ◽  
Variational Inequalities ◽  
Operator Equation ◽  
Convergence Rates ◽  
Cesàro Operator ◽  
Infinite Dimensional ◽  
Ill Posed ◽  
Cesaro Operator ◽  
ℓ1 Regularization

AbstractBased on the powerful tool of variational inequalities, in recent papers convergence rates results on ℓ1-regularization for ill-posed inverse problems have been formulated in infinite dimensional spaces under the condition that the sparsity assumption slightly fails, but the solution is still in ℓ1. In the present paper, we improve those convergence rates results and apply them to the Cesáro operator equation in ℓ2 and to specific denoising problems. Moreover, we formulate in this context relationships between Nashed's types of ill-posedness and mapping properties like compactness and strict singularity.

Stability Analysis of the Nanjung Water Diversion Twin Tunnels based on Convergence Measurement

2019 ◽  
Vol 1 (1) ◽  
pp. 49-60
Author(s):  
Simon Heru Prassetyo ◽  
Ganda Marihot Simangunsong ◽  
Ridho Kresna Wattimena ◽  
Made Astawa Rai ◽  
Irwandy Arif ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Convergence Rate ◽  
Critical Strain ◽  
Convergence Rates ◽  
Strain Analysis ◽  
Water Diversion ◽  
Tunnel Face ◽  
Tunnel Stability ◽  
The Stability ◽  
Twin Tunnels

This paper focuses on the stability analysis of the Nanjung Water Diversion Twin Tunnels using convergence measurement. The Nanjung Tunnel is horseshoe-shaped in cross-section, 10.2 m x 9.2 m in dimension, and 230 m in length. The location of the tunnel is in Curug Jompong, Margaasih Subdistrict, Bandung. Convergence monitoring was done for 144 days between February 18 and July 11, 2019. The results of the convergence measurement were recorded and plotted into the curves of convergence vs. day and convergence vs. distance from tunnel face. From these plots, the continuity of the convergence and the convergence rate in the tunnel roof and wall were then analyzed. The convergence rates from each tunnel were also compared to empirical values to determine the level of tunnel stability. In general, the trend of convergence rate shows that the Nanjung Tunnel is stable without any indication of instability. Although there was a spike in the convergence rate at several STA in the measured span, that spike was not replicated by the convergence rate in the other measured spans and it was not continuous. The stability of the Nanjung Tunnel is also confirmed from the critical strain analysis, in which most of the STA measured have strain magnitudes located below the critical strain line and are less than 1%.

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

scholarly journals KERNEL CONVERGENCE ESTIMATES FOR DIFFUSIONS WITH CONTINUOUS COEFFICIENTS

2011 ◽  
Vol 14 (07) ◽  
pp. 979-1004
Author(s):  
CLAUDIO ALBANESE
Keyword(s):  
Convergence Rate ◽  
Convergence Rates ◽  
Fourier Transforms ◽  
Uniform Continuity ◽  
Time Step ◽  
Uniform Bounds ◽  
Kernel Convergence ◽  
Discretization Schemes ◽  
A New Technique ◽  
Derivatives Of

Bidirectional valuation models are based on numerical methods to obtain kernels of parabolic equations. Here we address the problem of robustness of kernel calculations vis a vis floating point errors from a theoretical standpoint. We are interested in kernels of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in the limit as h → 0. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step so small that the Courant condition is satisfied. We find uniform bounds for the convergence rate as a function of the degree of smoothness. We conjecture these bounds are indeed sharp. The bounds also apply to the time derivatives of the kernel and its first two space derivatives. The proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. We make the simplifying assumption of time-independence and use longitudinal Fourier transforms in the time direction. Convergence rates depend on the degree of smoothness and Hölder differentiability of the coefficients. We find that the fastest convergence rate is of order O(h2) and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of Hölder differentiability except that the convergence rate is slower. Hölder continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.

scholarly journals Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation

2018 ◽  
Vol 39 (4) ◽  
pp. 2096-2134 ◽  
Author(s):  
Charles-Edouard Bréhier ◽  
Jianbo Cui ◽  
Jialin Hong
Keyword(s):  
Convergence Rate ◽  
Strong Convergence ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Auxiliary Problem ◽  
Splitting Scheme ◽  
Cahn Equation ◽  
Strong Convergence Rate

Abstract This article analyses an explicit temporal splitting numerical scheme for the stochastic Allen–Cahn equation driven by additive noise in a bounded spatial domain with smooth boundary in dimension $d\leqslant 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. When $d=1$ and the driving noise is a space–time white noise we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity we then prove that under very mild assumptions on the initial data this scheme achieves the optimal strong convergence rate $\mathcal{O}(\delta t^{\frac 14})$. When $d\leqslant 3$ and the driving noise possesses some regularity in space we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\mathcal{O}(\delta t)$.

A comparison of convergence rates for three models in the theory of dams

1997 ◽  
Vol 34 (01) ◽  
pp. 74-83
Author(s):  
Robert Lund ◽  
Walter Smith
Keyword(s):  
Convergence Rate ◽  
Convergence Rates ◽  
Sample Path ◽  
Jump Process ◽  
Limiting Distribution ◽  
Regenerative Processes ◽  
Input Process ◽  
The Moment ◽  
Finite Dam ◽  
General Conditions

This paper compares the convergence rate properties of three storage models (dams) driven by time-homogeneous jump process input: the infinitely high dam, the finite dam, and the infinitely deep dam. We show that the convergence rate of the infinitely high dam depends on the moment properties of the input process, the finite dam always approaches its limiting distribution exponentially fast, and the infinitely deep dam approaches its limiting distribution exponentially fast under very general conditions. Our methods make use of rate results for regenerative processes and several sample path orderings.

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