Solution Map Research Articles

Analyzing the role of the Inf-Sup condition for parameter identification in saddle point problems with application in elasticity imaging

We study the inverse problem of parameter identification in general saddle point problems. For saddle point problems, the use of elliptic regularization is an essential component. Saddle point problems, after discretization, lead to a non-invertible system, whereas the regularized saddle point problems result in an invertible system. Regularization methods, in the context of saddle point problems, have also been used to mitigate the role of the Inf-Sup condition, synonymously, also called the Babuska-Brezzi condition. This work aims to analyze the impact of regularizing the saddle point problem on the inverse problem. We investigate the inverse problem by using the output least-squares objective. To exploit the use of regularization fully, we work under the assumption that the solution map is nonempty. We regularize the saddle point problem and consider a family of optimization problems using the output least-squares objective for the regularized saddle point problem where some noise contaminates the whole data set. We give a complete convergence analysis showing that the optimization problems, given for the regularized output least-squares, approximate the original problem 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 present some heuristic numerical results. In the context of the elasticity imaging inverse problem, we conduct detailed numerical experiments on synthetic data (to study the role of the regularization parameter) as well as on phantom data.

Parametric optimization for morphing structures design: application to morphing wings adapting to changing flight conditions

Morphing structures can allow significant improvements in performance by optimally changing shape across varying conditions. A critical barrier to the design of morphing structures is the challenge of determining how optimal shape changes as a function of the many operating conditions that affect optimality. Traditional engineering optimization techniques are able to determine an optimal shape only for one condition or an aggregation over operating conditions (i.e., optimizing average performance). Parametric optimization is an alternative approach that can solve a family of related optimization problems simultaneously. Herein the authors analyze the design of a structurally consistent camber morphing wing for light aircraft applications using parametric optimization techniques. The approach combines rigorous consideration of structural constraints via Class/Shape Transformation (CST) equations and use of a C1-continuous analytical representation of wing outer mold line geometry with the Predictive Parametric Pareto Genetic Algorithm (P3GA), an algorithm for nonlinear multi-parametric optimization. The system is tuned to maximize lift-to-drag ratio, a key metric for aircraft flight range. Kriging-based interpolation is applied to P3GA output to obtain an optimal solution map determining optimal shape variable values as a function of flight conditions (airspeed, angle of attack, and altitude). Solutions obtained from iterated use of traditional optimization techniques are utilized to benchmark the more novel and efficient parametric optimization accuracy. Results show that parametric optimization is useful for optimizing morphing structures across a range of operating conditions.

Upper Semicontinuity of the Solution Map to a Parametric Elliptic Optimal Control Problem

This paper studies solution stability of a parametric optimal control problem governed by semilinear elliptic equations and nonconvex objective function with mixed pointwise constrains in which the controls act both in the domain and on the boundary. We give sufficient conditions under which the solution map is upper semicontinuous and continuous in parameters.

On higher-order proto-differentiability and higher-order asymptotic proto-differentiability of weak perturbation maps in parametric vector optimization

The main purpose of this paper is to study higher-order sensitivity analysis in parametric vector optimization problems. Firstly, the higher-order proto-differentiability/the higher-order asymptotic proto-differentiability of the feasible map of a parametric vector optimization problem are considered. Then, we verify that the weak efficient solution map and the weak perturbation map of a parameterized vector optimization problem are higher-order proto-differentiable/higher-order asymptotic proto-differentiable under some suitable qualification conditions.

The second-order composed radial derivatives of perturbation mappings of parametric set-valued optimization problems

In the paper, we study the generalized differentiability in set-valued optimization, namely stydying the second-order composed radial derivative of a given set-valued mapping. Inspired by the adjacent cone and the higher-order radial con in Anh NLH et al. (2011), we introduce the second-order composed radial derivative. Then, its basic properties are investigated and relationships between the second-order compsoed radial derivative of a given set-valued mapping and that of its profile are obtained. Finally, applications of this derivative to sensitivity analysis are studied. In detail, we work on a parametrized set-valued optimization problem concerning Pareto solutions. Based on the above-mentioned results, we find out sensitivity analysis for Pareto solution mapping of the problem. More precisely, we establish the second-order composed radial derivative for the perturbation mapping (here, the perturbation means the Pareto solution mapping concerning some parameter). Some examples are given to illustrate our results. The obtained results are new and improve the existing ones in the literature.

Solvability and optimal control of fractional differential hemivariational inequalities

ABSTRACT This paper investigates the solvability and the optimal control of fractional differential hemivariational inequalities (FDHVIs, for short). Firstly, we establish several sufficient assumptions for FDHVIs to transform the solution mappings into differential inclusions. Then, we verify the non-emptiness and the compactness of the set of all the mild solutions for FDHVIs. Furthermore, we discuss the optimal control problems of FDHVIs via some appropriate hypotheses. Finally, a typical example is presented to illustrate our main results.

Qualitative properties of the minimum sum-of-squares clustering problem

Fundamental qualitative properties of the minimum sum-of-squares clustering problem are established in this paper. We prove that the problem always has a global solution and, under a mild condition, the global solution set is finite. Moreover, the components of each global solution can be computed by an explicit formula. Based on a new concept of non-trivial local solution, we get necessary conditions for a system of centroids to be such a local solution. Interestingly, these necessary conditions are also sufficient ones. Finally, it is proved that the optimal value function is locally Lipschitz, the global solution map is locally upper Lipschitz, and the local solution map has the Aubin property, provided that the original data points are distinct. The obtained complete characterizations of the non-trivial local solutions allow one to understand better the performance of not only the k-means algorithm, but also of other solution methods for the problem in question.

Bilinear Minimax Optimal Control Problems for a von Kárman System with Long Memory

In the present article, we consider a von Kárman equation with long memory. The goal is to study a quadratic cost minimax optimal control problems for the control system governed by the equation. First, we show that the solution map is continuous under a weak assumption on the data. Then, we formulate the minimax optimal control problem. We show the first and twice Fréchet differentiabilities of the nonlinear solution map from a bilinear input term to the weak solution of the equation. With the Fréchet differentiabilities of the control to solution mapping, we prove the uniqueness and existence of an optimal pair and find its necessary optimality condition.

Hadamard directional differentiability of the optimal value of a linear second-order conic programming problem

<p style='text-indent:20px;'>In this paper, we consider perturbation properties of a linear second-order conic optimization problem and its Lagrange dual in which all parameters in the problem are perturbed. We prove the upper semi-continuity of solution mappings for the pertured problem and its Lagrange dual problem. We demonstrate that the optimal value function can be expressed as a min-max optimization problem over two compact convex sets, and it is proven as a Lipschitz continuous function and Hadamard directionally differentiable.</p>

Refined existence and regularity results for a class of semilinear dissipative SPDEs

We prove the existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the [Formula: see text]th moment of solutions depends on the integrability of the initial datum, in the whole range [Formula: see text]. Lipschitz continuity of the solution map in [Formula: see text]th moment is established, under a Lipschitz continuity assumption on the diffusion coefficient, in the even larger range [Formula: see text]. A key role is played by an Itô formula for the square of the norm in the variational setting for processes satisfying minimal integrability conditions, which yields pathwise continuity of solutions. Moreover, we show how the regularity of the initial datum and of the diffusion coefficient improves the regularity of the solution and, if applicable, of the invariant measures.

The Lipschitz Properties of the Value Function and the Solution Map to a Parametric Discrete Optimal Control Problem

Motivated by the works of Toan and Kien (J. Nonlinear Convex Anal.12:635–650, 2011) and Thuy and Toan (Acta Math. Vietnam.43:175–199, 2018) on the stability and the solution sensitivity in parametric discrete optimal control problems, this paper is devoted to the study of the Lipschitz property of the value function and the Aubin property of the solution map to a parametric dynamic programming problem with linear constraints. By establishing abstract results on the Lipschitz property of the value function to a parametric programming and the Aubin property of the solution map to a parametric variational inequality, we obtain the Lipschitz property of the value function and the Aubin property of the solution map to a parametric discrete optimal control problem.

Well-Posedness for Set Optimization Problems Involving Set Order Relations

In this paper, we investigate set optimization problems with three types of set order relations. Various kinds of well-posedness for these problems and their relationship are concerned. Then, sufficient conditions for set optimization problems to be well-posed are established. Moreover, Kuratowski measure of noncompactness is applied to survey characterizations of well-posedness for set optimization problems. Furthermore, approximating solution maps and their stability are researched to propose the link between stability of the approximating problem and well-posedness of the set optimization problem.

Nonlocal stochastic functional differential equations driven by G‐Brownian motion and mean random dynamical systems

In this paper, we consider a class of nonlocal stochastic functional differential equations driven by G‐Brownian motion (GNSFDEs) at phase space whose coefficients are dependent on the p‐th moment. Existence and uniqueness of solutions for GNSFDEs are investigated by virtue of theory of nonlinear expectation. Furthermore, we show that the solution map of GNSFDEs can generate a p‐mean random dynamical system in nonlinear expectation framework. In addition, two examples are provided to illustrate the effectiveness of the obtained results

Existence, iteration procedures and directional differentiability for parabolic QVIs

We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities. Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.

On the lower semicontinuity of the solution map- ping for parametric vector mixed quasivariational inequality problem of the Minty type

In this paper, we study a class of parametric vector mixed quasivariational inequality problem of the Minty type (in short, (MQVIP)). Afterward, we establish some sufficient conditions for the stability properties such as the inner-openness, lower semicontinuity and Hausdorff lower semicontinuity of the solution mapping for this problem. The results presented in this paper is new and wide to the corresponding results in the literature

LEARNING METHODS OF DECISION MAKING SKILLS: DISCUSSION, ASSIGNMENT, AND PRACTICE Case Study in Islamic Boarding School of Jagad 'Alimussirry, Indonesia

Generally, the learning in Islamic boarding schools use classic methods, ie. rote, bandongan, or sorogan. Decision-making skills is one of the skills needed in the 21st century. This study aims to describe the learning method of decision-making skills. This research is a case study in islamic boarding school of Jagad ‘Alimussirry, Indonesia. It is survey research. The data were obtained through a questionnaire that consists of 9 items of the statement by a 4-point Likert scale. The questionnaire were given to 46 students and 5 lecturers after learning. This study also conducted observations during the learning in 5 courses. The observations were made by 2 observers. The results showed that decision-making skills have been taught with good enough category. The learning methods of decision-making skills is discussion, assignment, and practice. The learning concept are: 1) discuss the expectations and the reality to identify problems with good enough category, 2) assign students to gather relevant information by good enough category, 3) discuss the strategies of mapping for all alternative solutions by good less category, and 4) practice to choose the right decision by good enough category. The strategies of mapping the alternative solutions need to be repaired.

Well-posedness and continuity properties of the new shallow-water model with cubic nonlinearity

This paper is devoted to the new shallow-water model (also called the modified Camassa–Holm–Novikov equation) with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the Fokas–Olver–Rosenau–Qiao equation and the Novikov equation as two special cases. It is shown that the Cauchy problem of the modified Camassa–Holm–Novikov equation for the periodic and the nonperiodic case is well-posed in Sobolev spaces in the sense of Hadamard, that is, the data-to-solution map is continuous. However, the solution map is not uniformly continuous.

Phenomenological soot modeling with solution mapping optimization of biodiesel-diesel blends in diesel engines

It is widely accepted that the particulate matter emissions of a diesel engine may be substantially reduced when a blend of diesel and biodiesel fuels is used. Here we propose a phenomenological model aimed at evaluating the extent of soot formation and oxidation in a diesel engine fueled with diesel–biodiesel blends and providing a better understanding of these processes. These blends are represented by three hydrocarbon functional groups: aromatic, aliphatic, and ester fuels, in different proportions.In a previous study, the phenomenological soot model of Foster and Reitz and colleagues was modified to capture the influence of the diesel–biodiesel molecular composition on soot formation and emissions. In the present work, this model was implemented in computational fluid dynamics (CFD) diesel engine combustion simulations. These CFD simulations were performed using the finite volume method commercial code ANSYS Fluent 16.1. The solution mapping optimization method developed by Frenklach and colleagues was used to calibrate this model for the engine. Validation was performed by comparing the following simulated variables with experimental results: pressure, heat release rate, averaged soot density inside the piston bowl, maximal temperature inside the piston bowl, and exhaust soot emissions. Reasonable agreement was obtained between our computational results and the published experimental results of Mancaruso and colleagues. The soot evolution in the piston bowl in these simulations was to a certain extent qualitatively and quantitatively similar to that in the experiments. The maximal experimental in-cylinder two-color soot density was 44 g/m3 and its computational equivalent was 60 g/m3. The maximal space-averaged two-color soot densities in the experiments and the computations were 9.3 and 7.73, respectively. The experimental soot reduction potentials of biodiesel blending for B20, B50, and biodiesel were 12.0%, 27.3%, and 47.2%, respectively. The respective computational soot reduction potentials were 34.8%, 36.2%, and 43.4%.

Elliptic quasi-variational inequalities under a smallness assumption: uniqueness, differential stability and optimal control

We consider a quasi-variational inequality governed by a moving set. We employ the assumption that the movement of the set has a small Lipschitz constant. Under this requirement, we show that the quasi-variational inequality has a unique solution which depends Lipschitz-continuously on the source term. If the data of the problem is (directionally) differentiable, the solution map is directionally differentiable as well. We also study the optimal control of the quasi-variational inequality and provide necessary optimality conditions of strongly stationary type.

Viral diffusion and cell-to-cell transmission: Mathematical analysis and simulation study

We propose a general model to investigate the joint impact of viral diffusion and cell-to-cell transmission on viral dynamics. The mathematical challenge lies in the fact that the model system is partially degenerate and the solution map is not compact. While the simpler cases with only indirect transmission mode or weak cell-to-cell transmission mode have been extensively studied in the literature, it remains an open problem to understand the local and global dynamics of fully coupled viral infection model with partial degeneracy. In this paper, we identify the basic reproduction number as the spectral radius of the sum of two linear operators corresponding to direct and indirect transmission modes. It is well-known that viral mobility may induce infection in low-risk regions. However, as diffusion coefficient increases, we prove that the basic reproduction number actually decreases, which indicates that faster viral movements may result in a lower level of viral infection. By an innovative construction of Lyapunov functionals, we further demonstrate that the basic reproduction number is the threshold parameter which determines global picture of viral dynamics. In addition to the traditional dichonomy results of extinction and persistence as obtained in earlier works for many simpler models, we are able to prove global asymptotic stability of infection-free steady state and global attractiveness (as well as uniqueness) of chronic-infection steady state, depending on whether the basic reproduction number is smaller or greater than one. Numerical simulation supports our theoretical results and suggests an interesting phenomenon: boundary layer and internal layer may occur when the diffusion parameter tends to zero.