Contingent derivatives and regularization for noncoercive inverse problems

Abstract

We study the inverse problem of parameter identification in noncoercive variational problems that commonly appear in applied models. We examine the differentiability of the set-valued parameter-to-solution map using the first-order and the second-order contingent derivatives. We explore the inverse problem using the output least-squares and the modified output least-squares objectives. By regularizing the noncoercive variational problem, we obtain a single-valued regularized parameter-to-solution map and investigate its smoothness and boundedness. We also consider optimization problems using the output least-squares and the modified output least-squares objectives for the regularized variational problem. We give a complete convergence analysis showing that for the output least-squares and the modified output least-squares, the regularized minimization problems approximate the original optimization problems suitably. We also provide the first-order and the second-order adjoint method for the computation of the first-order and the second-order derivatives of the output least-squares objective. We provide discrete formulas for the gradient and the Hessian calculation and present numerical results.     

Highly edge-connected factors using given lists on degrees

Let G be a 2k-edge-connected graph with and let for every . A spanning subgraph F of G is called an L-factor, if for every . In this article, we show that if for every , then G has a k-edge-connected L-factor. We also show that if and for every , then G has a k-edge-connected L-factor.     

Highly edge-connected regular graphs without large factorizable subgraphs

We construct highly edge-connected $r$-regular graphs of even order which do not contain $r?2$ pairwise disjoint perfect matchings. When $r$ is a multiple of 4, the result solves a problem of Thomassen [4].     

基于正弦控制因子的Lateral变异鲨鱼优化算法

针对传统鲨鱼优化算法在求解高维目标函数时,易早熟收敛,陷入局部最优的缺陷.提出一种基于正弦控制因子的Lateral变异鲨鱼优化算法.通过正弦曲线的特性和自适应惯性权重,改善了传统鲨鱼优化算法中由于随机选取控制因子数值大小可能导致算法在迭代后期全局搜索能力降低的问题,提高了算法在迭代后期的全局收敛能力,并对最佳鲨鱼位置引入Lateral变异策略,加强了算法跳出局部最优的可能性.改进后的算法对多个shifted单峰,多峰以及固定维测试函数进行求解,实验结果表明,对比多种不同优化算法而言,本文所提LSSO算法具有更高的收敛精度和搜索速度.     

Finite deformation swelling of a temperature-sensitive hydrogel cylinder under combined extension-torsion

The swelling behavior of a temperature-sensitive poly-N-isopropylacrylamide(PNIPAM) hydrogel circular cylinder is studied subjected to combined extension-torsion and varied temperature. In this regard, a semi-analytical solution is proposed for general combined loading. A finite element(FE) analysis is conducted, subjecting a hydrogel cylinder to the combined extension-torsion and the varied temperature to evaluate the validity and accuracy of the solution. A user-defined UHYPER subroutine is developed and verified under free and constrained swelling conditions. The FE results illustrate excellent agreement with the semi-analytical solution. Due to the complexity of the problem, some compositions and applied loading factors are analyzed. It is revealed that for larger cross-linked density and larger ending temperature, the cylinder yields higher stresses and smaller radial swelling deformation. Besides, the radial and hoop stresses increase by applying larger twist and axial stretch. The hoop stresses intersect at approximately R/Rout = 0.58, where the hoop stress vanishes. Besides, the axial force has direct and inverse relationships with the axial stretch and the twist, respectively. However, the resultant torsional moment behavior is complex, and the position of the maximum point varies significantly by altering the axial stretch and the twist.     

Limit theorems for a class of critical superprocesses with stable branching

We consider a critical superprocess $\{X;{\mathbf{P}}_{\mu }\}$ with general spatial motion and spatially dependent stable branching mechanism with lowest stable index ${\gamma }_0>1$. We first show that, under some conditions, ${\mathbf{P}}_{\mu }(\left|X_t\right|\ne 0)$ converges to 0 as $t\to \infty$ and is regularly varying with index ${({\gamma }_0-1)}^{-1}$. Then we show that, for a large class of non-negative testing functions $f$, the distribution of $\{X_t\left(f\right);{\mathbf{P}}_{\mu }\left(\cdot \right|6X_{t}6\ne 0)\}$, after appropriate rescaling, converges weakly to a positive random variable ${\mathbf{z}}^{({\gamma }_0-1)}$ with Laplace transform $E\left[e^{-u{\mathbf{z}}^{({\gamma }_0-1)}}\right]=1-{(1+u^{-({\gamma }_0-1)})}^{-1∕({\gamma }_0-1)}.$     

There is no (95, 40, 12, 20) strongly regular graph

We show that there is no (95, 40, 12, 20) strongly regular graph and, consequently, there is no (96, 45, 24, 18) strongly regular graph, no nontrivial regular two‐graph on 96 vertices, and no partial geometry pg(4, 9, 2). The main idea of the result is based on the star complement technique and requires a moderate amount of computation.