Variational Solution Research Articles

Variational solutions of stochastic partial differential equations with cylindrical Lévy noise

<p style='text-indent:20px;'>In this article, the existence of a unique solution in the variational approach of the stochastic evolution equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \, \mathrm{d}X(t) = F(X(t)) \, \mathrm{d}t + G(X(t)) \, \mathrm{d}L(t) $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>driven by a cylindrical Lévy process <inline-formula><tex-math id="M1">\begin{document}$ L $\end{document}</tex-math></inline-formula> is established. The coefficients <inline-formula><tex-math id="M2">\begin{document}$ F $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula> are assumed to satisfy the usual monotonicity and coercivity conditions. The noise is modelled by a cylindrical Lévy processes which is assumed to belong to a certain subclass of cylindrical Lévy processes and may not have finite moments.

Topology optimization of thermal problems in a nonsmooth variational setting: closed-form optimality criteria

This paper extends the nonsmooth Relaxed Variational Approach (RVA) to topology optimization, proposed by the authors in a preceding work, to the solution of thermal optimization problems. First, the RVA topology optimization method is briefly discussed and, then, it is applied to a set of representative problems in which the thermal compliance, the deviation of the heat flux from a given field and the average temperature are minimized. For each optimization problem, the relaxed topological derivative and the corresponding adjoint equations are presented. This set of expressions are then discretized in the context of the finite element method and used in the optimization algorithm to update the characteristic function. Finally, some representative (3D) thermal topology optimization examples are presented to asses the performance of the proposed method and the Relaxed Variational Approach solutions are compared with the ones obtained with the level set method in terms of the cost function, the topology design and the computational cost.

A new fractal model for the soliton motion in a microgravity space

PurposeOn a microgravity condition, a motion of soliton might be subject to a microgravity-induced motion. There is no theory so far to study the effect of air density and gravity on the motion property. Here, the author considers the air as discrete molecules and a motion of a soliton is modeled based on He’s fractal derivative in a microgravity space. The variational principle of the alternative model is constructed by semi-inverse method. The variational principle can be used to establish the conservation laws and reveal the structure of the solution. Finally, its approximate analytical solution is found by using two-scale method and homotopy perturbation method (HPM).Design/methodology/approachThe author establishes a new fractal model based on He’s fractal derivative in a microgravity space and its variational principle is obtained via the semi-inverse method. The approximate analytical solution of the fractal model is obtained by using two-scale method and HPM.FindingsHe’s fractal derivative is a powerful tool to establish a mathematical model in microgravity space. The variational principle of the fractal model can be used to establish the conservation laws and reveal the structure of the solution.Originality/valueThe author proposes the first fractal model for the soliton motion in a microgravtity space and obtains its variational principle and approximate solution.

Nonexistence of global characteristics for viscosity solutions

Two different types of generalized solutions, namely viscosity and variational solutions, were introduced to solve the first-order evolutionary Hamilton--Jacobi equation. They coincide if the Hamiltonian is convex in the momentum variable. In this paper we prove that there exists no other class of integrable Hamiltonians sharing this property. To do so, we build for any non-convex non-concave integrable Hamiltonian a smooth initial condition such that the graph of the viscosity solution is not contained in the wavefront associated with the Cauchy problem. The construction is based on a new example for a saddle Hamiltonian and a precise analysis of the one-dimensional case, coupled with reduction and approximation arguments.

Existence of variational solutions to a Cauchy\u2013Dirichlet problem with time-dependent boundary data on metric measure spaces

The objective of this work is an existence proof for variational solutions u to parabolic minimizing problems. Here, the functions being considered are defined on a metric measure space ({mathcal {X}}, d, mu ). For such parabolic minimizers that coincide with Cauchy-Dirichlet data eta on the parabolic boundary of a space-time-cylinder varOmega times (0, T) with an open subset varOmega subset {mathcal {X}} and T > 0, we prove existence in the parabolic Newtonian space L^p(0, T; {mathcal {N}}^{1,p}(varOmega )). In this paper we generalize results from Collins and Herán (Nonlinear Anal 176:56–83, 2018) where only time-independent Cauchy–Dirichlet data have been considered. We argue completely on a variational level.

Optimization Based Methods for Solving the Equilibrium Problems with Applications in Variational Inequality Problems and Solution of Nash Equilibrium Models

In this paper, we propose two modified two-step proximal methods that are formed through the proximal-like mapping and inertial effect for solving two classes of equilibrium problems. A weak convergence theorem for the first method and the strong convergence result of the second method are well established based on the mild condition on a bifunction. Such methods have the advantage of not involving any line search procedure or any knowledge of the Lipschitz-type constants of the bifunction. One practical reason is that the stepsize involving in these methods is updated based on some previous iterations or uses a stepsize sequence that is non-summable. We consider the well-known Nash–Cournot equilibrium models to support our well-established convergence results and see the advantage of the proposed methods over other well-known methods.

Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties

<p style="text-indent:20px;">The existence of weak solutions to the obstacle problem for a nonlocal semilinear fourth-order parabolic equation is shown, using its underlying gradient flow structure. The model governs the dynamics of a microelectromechanical system with heterogeneous dielectric properties.

Dynamics of Bose-Einstein Condensates Subject to the Pöschl-Teller Potential through Numerical and Variational Solutions of the Gross-Pitaevskii Equation.

We present for the first time an approach about Bose–Einstein condensates made up of atoms with attractive interatomic interactions confined to the Pöschl–Teller hyperbolic potential. In this paper, we consider a Bose–Einstein condensate confined in a cigar-shaped, and it was modeled by the mean field equation known as the Gross–Pitaevskii equation. An analytical (variational method) and numerical (two-step Crank–Nicolson) approach is proposed to study the proposed model of interatomic interaction. The solutions of the one-dimensional Gross–Pitaevskii equation obtained in this paper confirmed, from a theoretical point of view, the possibility of the Pöschl–Teller potential to confine Bose–Einstein condensates. The chemical potential as a function of the depth of the Pöschl–Teller potential showed a behavior very similar to the cases of Bose–Einstein condensates and superfluid Fermi gases in optical lattices and optical superlattices. The results presented in this paper can open the way for several applications in atomic and molecular physics, solid state physics, condensed matter physics, and material sciences.

Numerical investigation of transonic buffet on supercritical airfoil considering uncertainties in wind tunnel testing

To improve the predictive ability of computational fluid dynamics (CFD) on the transonic buffet phenomenon, NASA SC(2)-0714 supercritical airfoil is numerically investigated by noninstructive probabilistic collocation method for uncertainty quantification. Distributions of uncertain parameters are established according to the NASA wind tunnel report. The effects of the uncertainties on lift, drag, mean pressure and root-mean square pressure are discussed. To represent the stochastic solution, the mean and standard deviation of variation of flow quantities such as lift and drag coefficients are computed. Furthermore, mean pressure distribution and root-mean square pressure distribution from the upper surface are displayed with uncertainty bounds containing 95% of all possible values. It is shown that the most sensitive part of flow to uncertain parameters is near the shock wave motion region. Comparing uncertainty bounds with experimental data, numerical results are reliable to predict the reduced frequency and mean pressure distribution. However, for root-mean square pressure distribution, numerical results are higher than the experimental data in the trailing edge region.

Regular Versus Singular Solutions in a Quasilinear Indefinite Problem with an Asymptotically Linear Potential

AbstractThe aim of this paper is analyzing the positive solutions of the quasilinear problem-(u′/1+(u′)2)′=λ⁢a⁢(x)⁢f⁢(u) in ⁢(0,1),u′⁢(0)=0,u′⁢(1)=0,-\bigl{(}u^{\prime}/\sqrt{1+(u^{\prime})^{2}}\big{)}^{\prime}=\lambda a(x)f(u)% \quad\text{in }(0,1),\qquad u^{\prime}(0)=0,\quad u^{\prime}(1)=0,whereλ∈ℝ{\lambda\in\mathbb{R}}is a parameter,a∈L∞⁢(0,1){a\in L^{\infty}(0,1)}changes sign once in(0,1){(0,1)}and satisfies∫01a⁢(x)⁢𝑑x&lt;0{\int_{0}^{1}a(x)\,dx&lt;0}, andf∈𝒞1⁢(ℝ){f\in\mathcal{C}^{1}(\mathbb{R})}is positive and increasing in(0,+∞){(0,+\infty)}with a potential,F⁢(s)=∫0sf⁢(t)⁢𝑑t{F(s)=\int_{0}^{s}f(t)\,dt}, quadratic at zero and linear at+∞{+\infty}. The main result of this paper establishes that this problem possesses a component of positive bounded variation solutions,𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}, bifurcating from(λ,0){(\lambda,0)}at someλ0&gt;0{\lambda_{0}&gt;0}and from(λ,∞){(\lambda,\infty)}at someλ∞&gt;0{\lambda_{\infty}&gt;0}. It also establishes that𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}consists of regular solutions if and only if∫0z(∫xza(t)dt)-1/2dx=+∞ or ∫z1(∫xza(t)dt)-1/2dx=+∞.\int_{0}^{z}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty\quad\text% {or}\quad\int_{z}^{1}\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}dx=+\infty.Equivalently, the small positive regular solutions of𝒞λ0+{\mathscr{C}_{\lambda_{0}}^{+}}become singular as they are sufficiently large if and only if(∫xza(t)dt)-1/2∈L1(0,z) and (∫xza(t)dt)-1/2∈L1(z,1).\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(0,z)\quad\text{and}% \quad\Biggr{(}\int_{x}^{z}a(t)\,dt\Bigg{)}^{-{1/2}}\in L^{1}(z,1).This is achieved by providing a very sharp description of the asymptotic profile, asλ→λ∞{\lambda\to\lambda_{\infty}}, of the solutions. According to the mutual positions ofλ0{\lambda_{0}}andλ∞{\lambda_{\infty}}, as well as the bifurcation direction, the occurrence of multiple solutions can also be detected.

On some multilinear type integral systems

In this paper we prove some Liouville theorems for systems of integral equations related to Beckner and Stein–Weiss inequalities on RN. We consider both variational and non-variational solutions. In order to study the first type of solutions we applied Rellich type identities.

Local-in-Time Error in Variational Quantum Dynamics.

The McLachlan "minimum-distance" principle for optimizing approximate solutions of the time-dependent Schrödinger equation is revisited, with a focus on the local-in-time error accompanying the variational solutions. Simple, exact expressions are provided for this error, which are then evaluated in illustrative cases, notably the widely used mean-field approach and the adiabatic quantum molecular dynamics. Based on these findings, we demonstrate the rigorous formulation of an adaptive scheme that resizes on the fly the underlying variational manifold and, hence, optimizes the overall computational cost of a quantum dynamical simulation. Such adaptive schemes are a crucial requirement for devising and applying direct quantum dynamical methods to molecular and condensed-phase problems.

Soluções variacionais e numéricas da Equação de Schrödinger 1D submetida ao potencial de Pöschl-Teller

&lt;p&gt;This paper presents the numerical and variational solutions of the 1D Schrödinger Equation submitted to the Pöschl-Teller potential. The methods used were the Variational Method and the Finite Difference Method. They were presented in a didactic and detailed way with the purpose of instructing both undergraduate and graduate students, about the applicability and effectiveness of the aforementioned methods. We use the Pöschl-Teller potential due to the fact that it is little explored in the books of Quantum Mechanics used in undergraduation courses and also because of its diverse applications, such as in Bose-Einstein condensates, waveguides, topological defects in field theory and so on. We conclude this paper comparing the variational and numerical solutions with the analytical solution and present the advantages of each method.&lt;/p&gt;

Convergent numerical approximation of the stochastic total variation flow

We study the stochastic total variation flow (STVF) equation with linear multiplicative noise. By considering a limit of a sequence of regularized stochastic gradient flows with respect to a regularization parameter varepsilon we obtain the existence of a unique variational solution of the STVF equation which satisfies a stochastic variational inequality. We propose an energy preserving fully discrete finite element approximation for the regularized gradient flow equation and show that the numerical solution converges to the solution of the unregularized STVF equation. We perform numerical experiments to demonstrate the practicability of the proposed numerical approximation.

CARTESIAN VECTOR DIRECTION COSINES AS THE MULTI-OPTIONAL HYBRID FUNCTIONS OPTIMAL DISTRIBUTION

It is made an attempt to discover an explainable plausible reason for the existence of the conditions of optimality for Cartesian vector direction cosines, having importance in energy mechanical engineering, with the help of the multi-optional hybrid functions entropy conditional optimality doctrine. Substantiation is made in terms of the calculus of variations theory with the help of the special hybrid-optional effectiveness functions uncertainty measure, which includes the hybrid functions entropy of the traditional Shannon’s style. In the studied cases, the simplest variational problems solutions, which are the numbers known as the direction cosines of a Cartesian vector, are stipulated by the specified natural logarithmic quadratic forms. It is proposed to evaluate the uncertainty/certainty degree of the magnitude and direction of a Cartesian vector with the use of the objective functional. This is a new insight into the scientific explanation of the well-known dependency derived in another way. The developed theoretical contemplations and mathematical derivations are finalized with a simplest numerical example for the variated value of the multi-optional hybrid function resulting in the objective functional.

A variational autoencoder solution for road traffic forecasting systems: Missing data imputation, dimension reduction, model selection and anomaly detection

Efforts devoted to mitigate the effects of road traffic congestion have been conducted since 1970s. Nowadays, there is a need for prominent solutions capable of mining information from messy and multidimensional road traffic data sets with few modeling constraints. In that sense, we propose a unique and versatile model to address different major challenges of traffic forecasting in an unsupervised manner. We formulate the road traffic forecasting problem as a latent variable model, assuming that traffic data is not generated randomly but from a latent space with fewer dimensions containing the underlying characteristics of traffic. We solve the problem by proposing a variational autoencoder (VAE) model to learn how traffic data are generated and inferred, while validating it against three different real-world traffic data sets. Under this framework, we propose an online unsupervised imputation method for unobserved traffic data with missing values. Additionally, taking advantage of the low dimension latent space learned, we compress the traffic data before applying a prediction model obtaining improvements in the forecasting accuracy. Finally, given that the model not only learns useful forecasting features but also meaningful characteristics, we explore the latent space as a tool for model and data selection and traffic anomaly detection from the point of view of traffic modelers.

An assessment of refined hierarchical kinematic models for the bending and free vibration analyses of laminated and functionally graded sandwich cylindrical panels

An effort is made in this study to systematically develop and evaluate two-dimensional (2D) displacement-based higher order theories for stress and vibration analyses of moderately thick laminated and functionally graded (FG) sandwich cylindrical shell panels. Comparison of various displacement models is presented in order to assess the contribution of thickness stretching effects in laminated and sandwich shells. The equivalent single layer (ESL) approach is followed and a refined thickness criterion is adopted to enhance the accuracy of present formulation for moderately thick to thick cylindrical shell panels. Contribution of membrane and bending deformations is examined using the hierarchical kinematic models. Governing set of equations is obtained using energy minimization based on a variational approach and solution is obtained subsequently with Navier’s method for cylindrical shells with all edges on diaphragm (simple) supports. Effective FG material properties are obtained using Voigt’s rule of mixture with power law gradation. Solutions are compared with the available three dimensional (3D) solutions and a comparative assessment is done for different higher order theories. The present study accentuates the necessity of including transverse normal strain in the theory for FG sandwich panels.

Pullback attraction in H_{0}^{1} for semilinear heat equation in expanding domains

In this article, we consider the pullback attraction in H_{0}^{1} of pullback attractor for semilinear heat equation with domains expanding in time. Firstly, we establish higher-order integrability of difference about variational solutions; then, we prove the continuity of variational solution in H_{0}^{1}(O_{t}). As application of continuity, we obtain the pullback mathscr{D}_{lambda _{1}} attraction in H_{0}^{1}-norm.

Analytical Solution of an Induced Fiber Grating Relative Humidity Sensor

In our previous work, a distributed moisture sensor based on the induced long-period fiber grating in the presence of moisture is proposed. The proposed structure is analyzed numerically (Opt Laser Technol 117:126–133, 2019). In this article, due to the axial symmetry and periodic structure of the proposed sensor, the Love’s potential and Fourier series are employed to obtain an analytical solution for the refractive index variation along the sensor versus moisture density. The analytical method reduces the computation time relative to our numerical calculation drastically. The analytical results are in good agreement with those that are obtained by numerical calculations. The analytical method is employed for the analysis and design of distributed moisture fiber sensor.

Stability in probability and inverse optimal control of evolution systems driven by Levy processes

This paper first develops a Lyapunov-type theorem to study global well-posedness &#x0028 existence and uniqueness of the strong variational solution &#x0029 and asymptotic stability in probability of nonlinear stochastic evolution systems &#x0028 SESs &#x0029 driven by a special class of Levy processes, which consist of Wiener and compensated Poisson processes. This theorem is then utilized to develop an approach to solve an inverse optimal stabilization problem for SESs driven by Levy processes. The inverse optimal control design achieves global well-posedness and global asymptotic stability of the closed-loop system, and minimizes a meaningful cost functional that penalizes both states and control. The approach does not require to solve a Hamilton-Jacobi-Bellman equation &#x0028 HJBE &#x0029. An optimal stabilization of the evolution of the frequency of a certain genetic character from the population is included to illustrate the theoretical developments.