Hull-White模型中参数校准的正则化方法

基于Hull-White模型,研究由零息债券的市场价格进行参数校准的问题.构造函数将问题转化为正则化问题,并利用正则化方法得到解的存在性,稳定性和所满足的必要条件.最后利用必要条件进行数值计算,给出了数值模拟算例和实证分析,数值结果表明了方法中引入正则项的有效性,且改善了其参数的稳定性,具有实际意义.     

Choice of Regularization Parameter in Adaptive Filtering Problems

Computational Mathematics and Mathematical Physics - The paper proposes a new method for choosing a regularization parameter when solving an integral equation of convolution type in problems of...     

Combined State and Parameter Reduction

This work demonstrates combined state and parameter space reduction for large-scale control systems. Two methods for combined reduction are briefly introduced and tested on two different models for brain connectivity. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularization Parameter Selection for the Low Rank Matrix Recovery

Journal of Optimization Theory and Applications - A popular approach to recover low rank matrices is the nuclear norm regularized minimization (NRM) for which the selection of the regularization...     

Adaptive Choice of the Regularization Parameter in Numerical Differentiation

We investigate a novel adaptive choice rule of the Tikhonov regularization parameter in numerical differentiation which is a classic ill-posed problem. By assuming a general unknown Hölder type error estimate derived for numerical differentiation, we choose a regularization parameter in a geometric set providing a nearly optimal convergence rate with very limited a-priori information. Numerical simulation in image edge detection verifies reliability and efficiency of the new adaptive approach.     

Regularization Parameter Selection in Total Variation Based Image Denoising

A new approach of selecting the regularization parameter in a multiscale total variation model is proposed. The method is combined with a primal–dual algorithm to get a new iteration method for noise removal. Numerical results illustrate significant improvement over other popular methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularization Using QR Factorization and the Estimation of the Optimal Parameter

In this paper we propose a direct regularization method using QR factorization for solving linear discrete ill-posed problems. The decomposition of the coefficient matrix requires less computational cost than the singular value decomposition which is usually used for Tikhonov regularization. This method requires a parameter which is similar to the regularization parameter of Tikhonov's method. In order to estimate the optimal parameter, we apply three well-known parameter choice methods for Tikhonov regularization.This revised version was published online in October 2005 with corrections to the Cover Date.     

Gradient-based Regularization Parameter Selection for Problems With Nonsmooth Penalty Functions

In high-dimensional and/or nonparametric regression problems , regularization (or penalization) is used to control model complexity and induce desired structure. Each penalty has a weight parameter that indicates how strongly the structure corresponding to that penalty should be enforced. Typically, the parameters are chosen to minimize the error on a separate validation set using a simple grid search or a gradient-free optimization method. It is more efficient to tune parameters if the gradient can be determined, but this is often difficult for problems with nonsmooth penalty functions. Here, we show that for many penalized regression problems, the validation loss is actually smooth almost-everywhere with respect to the penalty parameters. We can, therefore, apply a modified gradient descent algorithm to tune parameters. Through simulation studies on example regression problems, we find that increasing the number of penalty parameters and tuning them using our method can decrease the generalization error.     

Selection of the Regularization Parameter in Graphical Models Using Network Characteristics

Gaussian graphical models represent the underlying graph structure of conditional dependence between random variables, which can be determined using their partial correlation or precision matrix. In a high-dimensional setting, the precision matrix is estimated using penalized likelihood by adding a penalization term, which controls the amount of sparsity in the precision matrix and totally characterizes the complexity and structure of the graph. The most commonly used penalization term is the L1 norm of the precision matrix scaled by the regularization parameter, which determines the trade-off between sparsity of the graph and fit to the data. In this article, we propose several procedures to select the regularization parameter in the estimation of graphical models that focus on recovering reliably the appropriate network structure of the graph. We conduct an extensive simulation study to show that the proposed methods produce useful results for different network topologies. The approaches are also applied in a high-dimensional case study of gene expression data with the aim to discover the genes relevant to colon cancer. Using these data, we find graph structures, which are verified to display significant biological gene associations. Supplementary material is available online.     

Heuristic Parameter Choice Rules for Tikhonov Regularization with Weakly Bounded Noise

We study the choice of the regularization parameter for linear ill-posed problems in the presence of noise that is possibly unbounded but only finite in a weaker norm, and when the noise-level is unknown. For this task, we analyze several heuristic parameter choice rules, such as the quasi-optimality, heuristic discrepancy, and Hanke-Raus rules and adapt the latter two to the weakly bounded noise case. We prove convergence and convergence rates under certain noise conditions. Moreover, we analyze and provide conditions for the convergence of the parameter choice by the generalized cross-validation and predictive mean-square error rules.     

On the Monotone Error Rule for Parameter Choice in Iterative and Continuous Regularization Methods

We consider in Hilbert spaces linear ill-posed problems Ax = y with noisy data y satisfying y –y. Regularized approximations x _r to the minimum-norm solution x of Ax = y are constructed by continuous regularization methods or by iterative methods. For the choice of the regularization parameter r (the stopping index n in iterative methods) the following monotone error rule (ME rule) is used: we choose r = r _ME (n = n _ME) as the largest r-value with the guaranteed monotonical decrease of the error x _r – x for r [0, r _ME] (x _n – x <#60; x _n–1 – x for n = 1, 2, ..., n _ME). Main attention is paid to iterative methods of gradient type and to nonstationary implicit iteration methods. As shown, the ME rule leads for many methods to order optimal error bounds. Comparisons with other rules for the choice of the stopping index are made and numerical examples are given.This revised version was published online in October 2005 with corrections to the Cover Date.     

Inexact Implicit Method with Variable Parameter for Mixed Monotone Variational Inequalities

In this paper, we focus on a useful modification of the implicit method by Noor (Ref. 1) for mixed variational inequalities. Experience on applications has shown that the number of iterations of the original method depends significantly on the penalty parameter. One of the contributions of the proposed method is that we allow the penalty parameter to be variable. By introducing a self-adaptive rule, we find that our method is more flexible and efficient than the original one. Another contribution is that we require only an inexact solution of the nonlinear equations at each iteration. A detailed convergence analysis of our method is also included.     

基于混沌粒子群算法的Tikhonov正则化参数选取

Tikhonov正则化方法是求解不适定问题最为有效的方法之一,而正则化参数的最优选取是其关键.本文将混沌粒子群优化算法与Tikhonov正则化方法相结合,基于Morozov偏差原理设计粒子群的适应度函数,利用混沌粒子群优化算法的优点,为正则化参数的选取提供了一条有效的途径.数值实验结果表明,本文方法能有效地处理不适定问题,是一种实用有效的方法.     

State-Space Regularization: Geometric Theory

Regularization of nonlinear ill-posed inverse problems is analyzed for a class of problems that is characterized by mappings which are the composition of a well-posed nonlinear and an ill-posed linear mapping. Regularization is carried out in the range of the nonlinear mapping. In applications this corresponds to the state-space variable of a partial differential equation or to preconditioning of data. The geometric theory of projection onto quasi-convex sets is used to analyze the stabilizing properties of this regularization technique and to describe its asymptotic behavior as the regularization parameter tends to zero. Accepted 26 April 1996     

Optimal Control Problems with Integral Functional and Phase Constraints: Reduction to Optimal Consistency Parameter Problems

An iteration method to solve a class of optimal control problems with integral functional and phase constraints is developed in the paper.     

Proximity Frames and Regularization

It is well known that the category KHaus of compact Hausdorff spaces is dually equivalent to the category KRFrm of compact regular frames. By de Vries duality, KHaus is also dually equivalent to the category DeV of de Vries algebras, and so DeV is equivalent to KRFrm, where the latter equivalence can be described constructively through Booleanization. Our purpose here is to lift this circle of equivalences and dual equivalences to the setting of stably compact spaces. The dual equivalence of KHaus and KRFrm has a well-known generalization to a dual equivalence of the categories StKSp of stably compact spaces and StKFrm of stably compact frames. Here we give a common generalization of de Vries algebras and stably compact frames we call proximity frames. For the category PrFrm of proximity frames we introduce the notion of regularization that extends that of Booleanization. This yields the category RPrFrm of regular proximity frames. We show there are equivalences and dual equivalences among PrFrm, its subcategories StKFrm and RPrFrm, and StKSp. Restricting to the compact Hausdorff setting, the equivalences and dual equivalences among StKFrm, RPrFrm, and StKSp yield the known ones among KRFrm, DeV, and KHaus. The restriction of PrFrm to this setting provides a new category StrInc whose objects are frames with strong inclusions and whose morphisms and composition are generalizations of those in DeV. Both KRFrm and DeV are subcategories of StrInc that are equivalent to StrInc. For a compact Hausdorff space X, the category StrInc not only contains both the frame of open sets of X and the de Vries algebra of regular open sets of X, these two objects are isomorphic in StrInc, with the second being the regularization of the first. The restrictions of these categories are considered also in the setting of spectral spaces, Stone spaces, and extremally disconnected spaces.     

Moving Coframes: II. Regularization and Theoretical Foundations

The primary goal of this paper is to provide a rigorous theoretical justification of Cartans method of moving frames for arbitrary finite-dimensional Lie group actions on manifolds. The general theorems are based a new regularized version of the moving frame algorithm, which is of both theoretical and practical use. Applications include a new approach to the construction and classification of differential invariants and invariant differential operators on jet bundles, as well as equivalence, symmetry, and rigidity theorems for submanifolds under general transformation groups. The method also leads to complete classifications of generating systems of differential invariants, explicit commutation formulae for the associated invariant differential operators, and a general classification theorem for syzygies of the higher order differentiated differential invariants. A variety of illustrative examples demonstrate how the method can be directly applied to practical problems arising in geometry, invariant theory, and differential equations.     

Regularization,GSVD and truncatedGSVD

The purpose of this paper is to analyze Tikhonov regularization in general form by means of generalized SVD (GSVD) in the same spirit as SVD is used to analyze standard-form regularization. We also define a truncated GSVD solution which is of interest in its own right and which sheds light on regularization as well. In addition, our analysis gives insight into a particular numerical method for solving the general-form problem via a transformation to standard form.Part of this work was carried out while visiting the Mathematical Sciences Section, Oak Ridge National Laboratory, Tennessee, during the Numerical Linear Algebra Year 1987–88, and was supported by the Danish Natural Science Foundation.     

Regularization of currents and entropy

Let T be a positive closed (p,p)-current on a compact Kähler manifold X. We prove the existence of smooth positive closed (p,p)-forms and such that weakly. Moreover, where cX>0 is a constant independent of T. We also extend this result to positive pluriharmonic currents. Then we study the wedge product of positive closed (1,1)-currents having continuous potential with positive pluriharmonic currents. As an application, we give an estimate for the topological entropy of meromorphic maps on compact Kähler manifolds.     

Elastic-Net Regularization: Iterative Algorithms and Asymptotic Behavior of Solutions

We derive strongly convergent algorithms to solve inverse problems involving elastic-net regularization. Moreover, using functional analysis techniques, we provide a rigorous study of the asymptotic properties of the regularized solutions that allows to cast in a unified framework ?¹, elastic-net and classical Tikhonov regularization.