Classical Variational Approach Research Articles

Kalman-based estimation of loading conditions from ultrasonic guided wave measurements

Abstract Ultrasonic guided wave-based structural health monitoring (SHM) of structures can be perturbed by environmental and operations conditions (EOCs) that alter wave propagation. In this work, we present an estimation procedure to reconstruct an EOC-free baseline of the structure from the only available Ultrasonic guided wave measurements. This procedure could typically be used as a prior step to increase the robustness of a more general ultrasonic imaging algorithm or SHM process dedicated to flaw detection. Our approach is model-based, i.e. we use a precise modeling of the wave propagation altered by structure loading conditions. This model is coupled with the acquired data through a data assimilation procedure to estimate the deformation caused by the unknown loading conditions. From a methodological point of view, our approach is original since we have proposed an iterated reduced-order unscented Kalman strategy, which we justify as an alternative to a Levenberg–Marquardt strategy for minimizing the non quadratic least-squares estimation criteria. Therefore, from a data assimilation perspective, we provide a quasi-sequential strategy that can valuably replace more classical variational approaches. Indeed, our resulting algorithm proves to be computationally very effective, allowing us to successfully apply our strategy to realistic 3D industrial SHM configurations.

Reducing Entanglement with Physically Inspired Fermion-To-Qubit Mappings

In electronic structure simulations, fermion-to-qubit mappings represent the initial encoding step from the problem of fermions into a problem of qubits. This work introduces a physically inspired method for constructing mappings that significantly simplify entanglement requirements when one is simulating states of interest. The presence of electronic excitations drives the construction of our mappings, reducing correlations for target states in the qubit space. To benchmark our method, we simulate ground-states of small molecules and observe an enhanced performance when compared with classical and quantum variational approaches from prior research using conventional mappings. In particular, on the quantum side, our mappings require a reduced number of entangling layers to achieve accuracy for LiH, H2, (H2)2, H4≠ stretching, and benzene’s π system using the RY hardware-efficient ansatz. In addition, our mappings also provide an enhanced ground-state simulation performance in the density matrix renormalization group algorithm for the N2 molecule. Published by the American Physical Society 2024

Stable Deep MRI Reconstruction Using Generative Priors.

Data-driven approaches recently achieved remarkable success in magnetic resonance imaging (MRI) reconstruction, but integration into clinical routine remains challenging due to a lack of generalizability and interpretability. In this paper, we address these challenges in a unified framework based on generative image priors. We propose a novel deep neural network based regularizer which is trained in a generative setting on reference magnitude images only. After training, the regularizer encodes higher-level domain statistics which we demonstrate by synthesizing images without data. Embedding the trained model in a classical variational approach yields high-quality reconstructions irrespective of the sub-sampling pattern. In addition, the model shows stable behavior when confronted with out-of-distribution data in the form of contrast variation. Furthermore, a probabilistic interpretation provides a distribution of reconstructions and hence allows uncertainty quantification. To reconstruct parallel MRI, we propose a fast algorithm to jointly estimate the image and the sensitivity maps. The results demonstrate competitive performance, on par with state-of-the-art end-to-end deep learning methods, while preserving the flexibility with respect to sub-sampling patterns and allowing for uncertainty quantification.

On variational principles for polarization responses in electromechanical systems

Classical electrodynamics uses a dielectric constant to describe the polarization response of electromechanical systems to changes in an electric field. We generalize that description to include a wide variety of responses to changes in the electric field, as found in most systems and applications. Electromechanical systems can be found in many physical and biological applications, such as ion transport in membranes, batteries, and dielectric elastomers. We present a unified, thermodynamically consistent, variational framework for modeling electromechanical systems as they respond to changes in the electric field; that is to say, as they polarize. This framework is motivated and developed using the classical energetic variational approach (EnVarA). The coupling between the electric part and the chemo-mechanical parts of the system is described either by Lagrange multipliers or various energy relaxations. The classical polarization and its dielectrics and dielectric constants appear as outputs of this analysis. The Maxwell equations then become universal conservation laws of charge and current, conjoined to an electromechanical description of polarization. Polarization describes the entire electromechanical response to changes in the electric field and can sometimes be approximated as a dielectric constant or dielectric dispersion.

Existence and uniqueness of a weak solution to a non-autonomous time-fractional diffusion equation (of distributed order)

In this contribution, we investigate an initial–boundary value problem for a fractional diffusion equation of distributed order where the coefficients of the elliptic operator are dependent on spatial and time variables. We consider a homogeneous Dirichlet boundary condition. Using a classical variational approach, we establish the existence of a unique weak solution to the problem in C[0,T],H01(Ω)∗∩L∞(0,T),H01(Ω) if the initial data belongs to H01(Ω). The same result is also valid for the fractional diffusion equation with Caputo derivative.

Necessary conditions for stochastic optimal control problems in infinite dimensions

The purpose of this paper is to establish the first and second order necessary conditions for stochastic optimal controls in infinite dimensions. The control system is governed by a stochastic evolution equation, in which both drift and diffusion terms may contain the control variable and the set of controls is allowed to be nonconvex. Only one adjoint equation is introduced to derive the first order necessary optimality condition either by means of the classical variational analysis approach or, under an additional assumption, by using differential calculus of set-valued maps. More importantly, in order to avoid the essential difficulty with the well-posedness of higher order adjoint equations, using again the classical variational analysis approach, only the first and the second order adjoint equations are needed to formulate the second order necessary optimality condition, in which the solutions to the second order adjoint equation are understood in the sense of the relaxed transposition.

On Free Bending Vibrations Of Concrete Beams With Variable Cross Section

The paper consists of an introduction and three sections. The introduction discusses the relevance of issues related to the study of inhomogeneous beams vibrations. The analysis of publications and the results obtained in this area is performed. The first section is devoted to the formulation of the boundary-value problem of finding natural-vibration frequencies for an inhomogeneous beam under the Euler-Bernoulli hypotheses. By introducing new variables, the problem originally formulated in displacements reduces to an identical one, but formulated in terms of bending moment. The next section describes the method of integro-differential relations, which is an alternative to the classical variational approaches. Further, the possibilities of constructing various bilateral energy estimates of the quality of approximate solutions arising from the method of integro-differential relations are investigated. In the third section, using the example of free vibrations of a supported concrete beam, numerical aspects of constructing an approximate solution for boundary problems described by an ordinary differential equation with variable coefficients are studied. The proposed bilateral quality criteria of the approximate solution make it possible to obtain high-precision solutions for mathematical models of small dimension.

Novel computational approach to solve convolutional integral equations: method of sampling for one dimension [Online First

Objective: This paper proposes a new methodology to solve one-dimensional cases of integral equations with difference kernels using Fourier analysis. Methodology: In this study, it was proven that any Fredholm equation of the first kind can be expressed as an extended convolutional problem; consequently, a new approach to solve that problem, using the nonideal instantaneous sampling theory and Fourier analysis, can be developed. Results and Discussion: The proposal was extensively evaluated and compared with the method of moments by considering two benchmarks. The first was a narrowband problem related to a second-order differential equation with specific boundaries. The second was a standard wideband problem related to wire antenna radiation in electrodynamics, known as the Pocklington equation. In both cases, we derived new interpretations and different approaches to solve the problems efficiently. Conclusions: The new proposal generalized the method of moments via new interpretations, strategies and design rules. We found that the techniques based on the method of moments are point-matching procedures independent of the weighting functions; the basis functions can be designed as generalized interpolation functions with more information provided by the original domain; the weighting functions literally represent a sampled linear filter; the unknown continuous function can be approximated without using the classical variational approach; and several new strategies based on the Fourier transform can be used to reduce the computational cost

A Nonstandard Higher-Order Variational Model for Speckle Noise Removal and Thin-Structure Detection

In this work, we propose a multiscale approach for a nonstandard higher-order PDE based on the $p(\cdot)$-Kirchhoff energy. First, we consider a topological gradient approach for a semilinear case in order to detect important object of image. Then, we consider a fully nonlinear $p(\cdot)$-Kirchhoff equation with variables exponent functions that are chosen adaptively based on the map furnished by the topological gradient in order to preserve important features of the image. Then, we consider the split Bregman method for the numerical implementation of our proposed model. We compare our model with other classical variational approaches such that the TVL and biharmonic restoration models. Finally, we present some numerical results to illustrate the effectiveness of our approach.

Analysis of Dynamic Behavior of Beams with Variable Cross-section

A formulation of a boundary value problem to find natural frequencies of an inhomogeneous beam in the framework of the Euler–Bernoulli hypotheses are represented. Questions related to various classical variational formulations for a spectral problem arising in the beam theory are discussed. Particularities of the application of the Hamiltonian principles to boundary-value problems are considered. The method of integro-differential relations, which is an alternative to the classical variational approaches is discussed. Various bilateral energy quality estimates for approximate solutions that follow from the method of integro-differential relations are proposed. In the final part of the paper advantages of the variational technique in problems of free vibrations of inhomogeneous beams are discussed based on a numerical example.

Semi-analytical solution for buckling of SMA thin plates with linearly distributed loads

Buckling analysis of shape memory alloy (SMA) rectangular plates subjected to uniform and linearly distributed in-plane loads is the main objective in the present paper. Brinson\'s model is developed to express the constitutive characteristics of SMA plate. Using the classical plate theory and variational approach, stability equations are derived. In addition to external in-plane mechanical loads, the plate is subjected to the pre-stresses caused by the recovery stresses that are generated during martensitic phase transformation. Ritz method is used for solving the governing stability equations. Finally, the effects of conditions on the edges, thickness, aspect ratio, temperature and pre-strains on the critical buckling loads of SMA plate are investigated in details.

Existence and Uniqueness of Entropy Solution for an Elliptic Problem with Nonlinear Boundary Conditions and Measure-data

In this paper, we study a class of elliptic equations on a bounded domain with nonlinear boundary conditions of type graph and measure - data. First of all, give some spaces and basic assumptions. Next, we apply the classical variational approach. So, we need an essentially bounded estimate on the solution, which is not evident to obtain directly in our problem. The obstacles which we encounter is that we cannot get rid of the non-linear term evaluated as a zero gradient and it appear at the boundary, for the part of the measure-data, a term which cannot vanish, when one uses the integration by parts formula. To overcome this difficulties, we first redefine and extend the function which appears in the third Leray-Lions-type conditions and we add a penalization term on the boundary. Secondly, we consider a smooth domain in order to work with the Sobolev spaces that are the closure of indefinitely differentiable and null functions on the bounary, and to going back later to the classical Sobolev space. Then, we assume that the domain is extensible. Next, we obtain a priori estimates and convergence results in the approach problem, which allow us to delete the penalization term. To finish, we introduce a notion of entropy solution for our main problem and prove that it is the limit of the solution obtained in the variational case.

The maximum principles for partially observed risk-sensitive optimal controls of Markov regime-switching jump-diffusion system

ABSTRACTThis paper studies partially observed risk-sensitive optimal control problems with correlated noises between the system and the observation. It is assumed that the state process is governed by a continuous-time Markov regime-switching jump-diffusion process and the cost functional is of an exponential-of-integral type. By virtue of a classical spike variational approach, we obtain two general maximum principles for the aforementioned problems. Moreover, under certain convexity assumptions on both the control domain and the Hamiltonian, we give a sufficient condition for the optimality. For illustration, a linear-quadratic risk-sensitive control problem is proposed and solved using the main results. As a natural deduction, a fully observed risk-sensitive maximum principle is also obtained and applied to study a risk-sensitive portfolio optimization problem. Closed-form expressions for both the optimal portfolio and the corresponding optimal cost functional are obtained.

First and second order necessary conditions for stochastic optimal controls

The main purpose of this paper is to establish the first and second order necessary optimality conditions for stochastic optimal controls using the classical variational analysis approach. The control system is governed by a stochastic differential equation, in which both drift and diffusion terms may contain the control variable and the set of controls is allowed to be nonconvex. Only one adjoint equation is introduced to derive the first order necessary condition; while only two adjoint equations are needed to state the second order necessary conditions for stochastic optimal controls.

The topological gradient method for semi-linear problems and application to edge detection and noise removal

The goal of this paper is to apply the topological gradient method to edge detection and noise removal for images degraded by various noises and blurs. First applied to edge detection for images degraded by a Gaussian noise, we propose here to extend the method to blurred images contaminated either by a multiplicative noise of gamma law or to blurred Poissonian images. We compute, both for perforated and cracked domains, the topological gradient for each noise model. Then we present an edge detection/restoration algorithm based on this notion and we apply it to the two degradation models previously described. We compare our method with other classical variational approaches such that the Mumford-Shah and TV restoration models and with the classical Canny edge detector. Some experimental results showing the efficiency, the robustness and the rapidity of the method are presented.

Necessary Optimality Conditions for Weak Local Minima in Stochastic Control

Abstract: This paper is devoted to the first and second order necessary optimality conditions for stochastic optimal control problems of the Bolza type. The control system is governed by a stochastic differential equation whose drift and diffusion terms are control dependent and the set of controls may be nonconvex. The optimal controls under consideration are those providing the weak local minima. The derived first order necessary condition involves one adjoint equation and a pointwise variational inequality. They are very similar to the known necessary conditions of the deterministic optimal control theory. The second order necessary condition is stated in the integral form and involves two adjoint equations. For sufficiently regular singular weak local minimizers this second order integral inequality implies a pointwise condition. To obtain these results we use the classical variational approach reinforced by the set-valued analysis and the Malliavin calculus. The proofs of our results being quite long and technical will appear elsewhere.

Stochastic Maximum Principle for Stochastic Recursive Optimal Control Problem Under Volatility Ambiguity

We study a stochastic recursive optimal control problem in which the cost functional is described by the solution of a backward stochastic differential equation driven by $G$-Brownian motion. Some of the economic and financial optimization problems with volatility ambiguity can be formulated as such problems. Unlike the classical variational approach, we establish the maximum principle by the linearization and weak convergence methods.

The inverse problem of Lagrangian mechanics for a non-material volume

The appropriate consideration of non-material volumes at the level of analytical mechanics is an ongoing research field. In the present paper, we aim at demonstrating the principle of stationary action that is able to yield the proper form of Lagrange’s equation in the context, namely the Lagrange’s equation in the form derived by Irschik and Holl (Acta Mech 153(3–4):231–248, 2002). Such issue will here be interpreted as being the inverse problem of Lagrangian mechanics for a non-material volume. The classical method of Darboux (Lecons sur la Theorie Generale des Surfaces. Gauthier-Villars, Paris, 1891) will be used as the solution technique. This means that our discussion will be restricted to the case of a single degree of freedom. Having such principle of stationary action at hand, the corresponding Hamiltonian formalism will be written in accordance with the classical theory. Furthermore, a conservation law will be demonstrated for the time-independent case. At last, two simple examples will be addressed in order to illustrate the applicability of the proposed formulation. The reader may find some mathematical analogies between the upcoming content and that discussed by Casetta and Pesce (Acta Mech, 2013. doi:10.1007/s00707-013-1004-1) in considering the inverse problem of Lagrangian mechanics for Meshchersky’s equation. The mathematical formulation which will be outlined in the present paper is thus expected to consistently situate non-material volumes within the classical variational approach of mechanics.

A mixed problem of linear elastodynamics

This paper is devoted to a semigroup approach to an initial-boundary value problem of linear elastodynamics in the case where the boundary condition is a regularization of the genuine mixed displacement-traction boundary condition. More precisely, it is a smooth linear combination of displacement and traction boundary conditions, but is not equal to the pure traction boundary condition. Some previous results with mixed displacement-traction boundary condition are due to Inoue and Ito. The crucial point in our semigroup approach is to generalize the classical variational approach to the degenerate case, by using the theory of fractional powers of analytic semigroups.

A frame-indifferent model for a thermo-elastic material beyond the three-dimensional Eulerian and Lagrangian descriptions

The covariance principle of differential geometry within a four-dimensional (4D) space-time ensures the validity of any equations and physical relations through any changes of frame of reference, due to the definition of the 4D space-time and the use of 4D tensors, operations and operators. This enables to separate covariance (i.e. frame-indifference) and material objectivity (i.e. material-indifference). We propose here a method to build a constitutive relation for thermo-elastic materials using such a 4D formalism. A 4D generalization of the classical variational approach is assumed leading to a model for a general thermo-elastic material. The isotropy of the relation can be ensured by the use of the invariants of variables, which offers new possibilities for the construction of constitutive relations. It is then possible to build a general frame-indifferent but not necessarily material-indifferent constitutive relation. It encompasses both the 3D Eulerian and Lagrangian thermo-elastic isotropic relations for finite transformations.