Variational Problem Research Articles

A Preconditioned Version of a Nested Primal-Dual Algorithm for Image Deblurring

Variational models for image deblurring problems typically consist of a smooth term and a potentially non-smooth convex term. A common approach to solving these problems is using proximal gradient methods. To accelerate the convergence of these first-order iterative algorithms, strategies such as variable metric methods have been introduced in the literature. In this paper, we prove that, for image deblurring problems, the variable metric strategy proposed in Aleotti et al. (Comput. Optim. Appl., 2024) can be reinterpreted as a right preconditioning method. Consequently, we explore an inexact left-preconditioned version of the same proximal gradient method. We prove the convergence of the new iteration to the minimum of a variational model where the norm of the data fidelity term depends on the preconditioner. The numerical results show that left and right preconditioning are comparable in terms of the number of iterations required to reach a prescribed tolerance, but left preconditioning needs much less CPU time, as it involves fewer evaluations of the preconditioner matrix compared to right preconditioning. The quality of the computed solutions with left and right preconditioning are comparable. Finally, we propose some non-stationary sequences of preconditioners that allow for fast and stable convergence to the solution of the variational problem with the classical ℓ2–norm on the fidelity term.

A MaskFormer EfficientNet instance segmentation approach for crowd counting

In computer vision tasks, object detection is the most significant challenge. Numerous studies on Convolutional neural network based techniques have been extensively utilized in computer vision for the detection of objects. Scale variation, illumination variation or occlusion problems are the most popular challenges in crowd counting. To address this, MaskFormer EfficientNetB7 Instance Segmentation architecture has been proposed that utilized the EfficientNetB7 as the backbone for feature extraction to create efficient and accurate counting of people in challenging scenarios. A basic mask classification model called MaskFormer has been used to predict a series of binary masks, each of which is connected to a single global class for label prediction and EfficientNetB7 has used a compound scaling algorithm that equally scaled each dimension using a predetermined set of scaling coefficients. Experimental findings on the UCF-QNRF, ShanghalTech (Part A and Part B) and Mall datasets have demonstrated that the suggested strategy has provided remarkable outcomes in contrast to existing crowd counting approaches in terms of Mean Absolute Error and Root Mean Squared Error. Therefore, the proposed crowd-counting model has been proven to be more adaptable in different environments or scenarios along with strong generalizability on unseen data.

Lagrange duality and saddle point criteria for semi-infinite variational programming problem with Caputo-Fabrizio fractional derivative

Lagrange duality and saddle point criteria for semi-infinite variational programming problem with Caputo-Fabrizio fractional derivative

Debiased Multimodal Understanding for Human Language Sequences

Human multimodal language understanding (MLU) is an indispensable component of expression analysis (e.g., sentiment or humor) from heterogeneous modalities, including visual postures, linguistic contents, and acoustic behaviours. Existing works invariably focus on designing sophisticated structures or fusion strategies to achieve impressive improvements. Unfortunately, they all suffer from the subject variation problem due to data distribution discrepancies among subjects. Concretely, MLU models are easily misled by distinct subjects with different expression customs and characteristics in the training data to learn subject-specific spurious correlations, limiting performance and generalizability across new subjects. Motivated by this observation, we introduce a recapitulative causal graph to formulate the MLU procedure and analyze the confounding effect of subjects. Then, we propose SuCI, a simple yet effective causal intervention module to disentangle the impact of subjects acting as unobserved confounders and achieve model training via true causal effects. As a plug-and-play component, SuCI can be widely applied to most methods that seek unbiased predictions. Comprehensive experiments on several MLU benchmarks clearly show the effectiveness of the proposed module.

Resource-adaptive quantum flow algorithms for quantum simulations of many-body systems: sub-flow embedding procedures

In this study, we utilised the quantum flow (QFlow) method to perform quantum simulations of correlated systems. The QFlow approach allows for sampling large sub-spaces of the Hilbert space by solving coupled variational problems in reduced dimensionality active spaces. Our research demonstrates that the circuits for evaluating the low dimensionality subproblems of the QFlow algorithm on quantum computers are significantly less complex than the parent (large sub-space of the Hilbert space) problem, opening up possibilities for scalable and constant-circuit-depth quantum computing. Our simulations indicate that QFlow can be used to optimise a large number of wave function parameters without a significant increase in the required number of qubits. We were able to showcase that a variation of the QFlow procedure can be employed to optimise 1100 wave function parameters using modest quantum resources. Furthermore, we investigated an adaptive approach known as the sub-flow approach (sub-QFlow), which involves a limited number of active spaces in the quantum flow process. Our findings shed light on the potential of QFlow in efficiently handling correlated systems via quantum simulations.

A New Family of Buckled Rings on the Unit Sphere

Buckled rings, also known as pressurized elastic circles, can be described as critical points for a variational problem, namely the integral of a quadratic polynomial in the geodesic curvature of a curve. Thus, they are a generalization of elastic curves, and they are solitary wave solutions to a flow in a (three-dimensional) filament hierarchy. An example of such a curve is the Kiepert Trefoil, which has three leaves meeting at a central singular point. Such a variational problem can be considered for curves in other surfaces. In particular, researchers have found many examples of such curves in a unit sphere. In this article, we consider a new family of such curves, having a discrete dihedral symmetry about a central singular point. That is, these are spherical analogues of the Kiepert curve. We determine such curves explicitly using the notion of a Killing field, which is a vector field along a curve that is the restriction of an isometry of the sphere. The curvature k of each such curve is given explicitly by an elliptic function. If the curve is centered at the south pole of the sphere and has minimum value ρ, then k−ρ is linear in the height above the pole.

Static solutions for Choquard equations with Coulomb potential and upper critical growth

Abstract This paper focuses on static solutions for the following Choquard equation with zero mass and Coulomb potential $$\begin{aligned} -\Delta u+\left( \frac{1}{4\pi |x|}*u^2\right) u=\mu |u|^{p-2}u+(I_{\alpha }*|u|^{\alpha +3})|u|^{\alpha +1}u, \ \ \ \ x\in {\mathbb {R}}^{3}, \end{aligned}$$ - Δ u + 1 4 π | x | ∗ u 2 u = μ | u | p - 2 u + ( I α ∗ | u | α + 3 ) | u | α + 1 u , x ∈ R 3 , where $$\mu >0$$ μ > 0 , $$\frac{18}{7}<p\le 6$$ 18 7 < p ≤ 6 , $$\alpha \in (0, 3)$$ α ∈ ( 0 , 3 ) , $$\alpha +3$$ α + 3 is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, $$I_{\alpha }{:}\,\mathbb {R}^3\rightarrow \mathbb {R}$$ I α : R 3 → R is the Riesz potential, and $$\frac{1}{4\pi |x|}$$ 1 4 π | x | is the Coulomb potential. By carefully analyzing the intricate interplay between the power and Coulomb terms, we establish three types of variational geometries of the problem and prove the following existence results based on the behavior of p: the existence of two solutions, one being a local minimizer and the other of mountain-pass type, for an explicit range $$0<\mu <\mathrm {Const.}$$ 0 < μ < Const . when $$\frac{18}{7}<p<3$$ 18 7 < p < 3 ; the existence of a positive solution if $$\mu $$ μ takes some particular value when $$p=3$$ p = 3 ; the existence of a ground state solution for all $$\mu >0$$ μ > 0 when $$4<p<6$$ 4 < p < 6 , and for two explicit ranges $$\mu >\mathrm {Const.}$$ μ > Const . when $$3<p<4$$ 3 < p < 4 and $$p=4$$ p = 4 . Furthermore, we obtain a non-existence result for the case $$p=6$$ p = 6 . Particularly, we identify different compactness thresholds for above three cases, and introduce three types of test functions to control the corresponding minimax levels to be less than prescribed thresholds, thereby overcoming the loss of compactness arising from the nonlocal critical term. The derivation of these strict inequalities is a novel contribution and constitutes one of the noteworthy highlights of this work, which is available and new for the limiting Sobolev critical problem as $$\alpha \rightarrow 0$$ α → 0 . We believe that the underlying ideas have potential for future development and can be applied to a broader range of variational problems with critical growth.

Quantum Elastica

This paper presents a quantum method to tackle optimization challenges. Departing from the typical applications of quantum theory in particle physics, we demonstrate our approach using the elastica problem as a concrete example. The elastica, a classic variational problem extensively studied by mathematicians, serves as an ideal test case. Within quantum theory, our central innovation lies in the way we handle boundary conditions by combining forward and backward propagating wave solutions, a concept inspired by the superposition of forward and backward time-traveling particle waves in quantum mechanics. This approach not only provides a novel solution method for the elastica problem but also opens new pathways for applying quantum mathematical techniques to classical optimization challenges in other domains.

Extended truncated exponential method for solving tempered fractional variational problems

In this paper, we propose a novel method, that is, extended truncated exponential (ETE) method for solving tempered fractional variational problems (TFVPs) with integral constraints. The method leverages the extended truncated exponential function (ETEF) and the tempered Caputo fractional derivative (TCFD) to transform TFVPs into solvable algebraic systems. We derive the operational matrix of derivatives for the ETEF and apply it to approximate the solutions of TFVPs efficiently. The method demonstrates superior performance in terms of computational efficiency and accuracy compared to existing approaches. Several numerical examples validate the effectiveness of our method.

On a class of obstacle problems with (p, q)-growth and explicit u-dependence

Abstract In this paper, we establish the higher differentiability of the gradient of solutions to variational obstacle problems of the type min ⁡ { ∫ Ω F ⁢ ( x , u , D ⁢ u ) ⁢ 𝑑 x : u ∈ 𝒦 ψ ⁢ ( Ω ) } . \min\Biggl{\{}\int_{\Omega}F(x,u,Du)\,dx:u\in\mathcal{K}_{\psi}(\Omega)\Biggr{% \}}. Here Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} is a bounded open set, ψ ∈ W 0 1 , q ⁢ ( Ω ) {\psi\in W^{1,q}_{0}(\Omega)} is a fixed function called obstacle, and 𝒦 ψ ⁢ ( Ω ) {\mathcal{K}_{\psi}(\Omega)} is the class of admissible functions. The main feature of the energy densities under consideration here is that they satisfy non-standard growth conditions with respect to the gradient variable and that they explicitly depend on the pair ( x , u ) {(x,u)} . Assuming that ψ ∈ L loc ∞ ⁢ ( Ω ) ∩ W loc 2 , 2 ⁢ q - p ⁢ ( Ω ) {\psi\in L^{\infty}_{\mathrm{loc}}(\Omega)\cap W^{2,2q-p}_{\mathrm{loc}}(% \Omega)} , we are able to prove a second order regularity result for the solution.

Method of measuring precession details on a coordinate measuring machine

The article focuses on the development and implementation of an effective methodology for measuring high-precision parts on coordinate measuring machines (CMM). The proposed approach addresses the challenges associated with complex geometry measurements under variable environmental conditions by combining advanced mathematical modeling techniques with adaptive error compensation algorithms. The mathematical foundation is based on the application of tensor formalism in Riemannian space, which allows for more precise model-ing of geometric errors using Christoffel symbols and covariant derivatives of the error po-tential. This approach significantly improves the accuracy of spatial positioning. A refined stochastic error model, incorporating Stratonovich integrals and fractional Brownian motion, provides a more accurate description of random processes occurring in the measurement sys-tem. To enhance overall measurement accuracy, an adaptive correction algorithm is pro-posed, based on Itô stochastic differential equations with Fourier-Bessel series expansion, which ensures efficient compensation of systematic errors. In addition, the measurement tra-jectory optimization is formulated as a variational problem, taking into account holonomic and non-holonomic constraints, enabling the optimal positioning strategy. A thermoelastic deformation model based on sixth-rank tensors and Green’s functions is developed to account for temperature-induced deformations, ensuring the reliability of measurements under ther-mal instability. Experimental verification confirmed the effectiveness of the proposed method-ology, demonstrating a 15-20% reduction in systematic errors, a 10-15% decrease in random errors, and an approximately 20% improvement in the accuracy of measurement uncertainty estimation. The proposed methodology combines theoretical advancements with practical so-lutions, providing a robust tool for improving the accuracy and reliability of coordinate measurements in industrial metrology applications.

New Extrapolation Modulus-based Economical Cascadic Multigrid Method for Variational Inequality Problems with Nonlinear Source Terms

New Extrapolation Modulus-based Economical Cascadic Multigrid Method for Variational Inequality Problems with Nonlinear Source Terms

Sparse identification of nonlocal interaction kernels in nonlinear gradient flow equations via partial inversion

We address the inverse problem of identifying nonlocal interaction potentials in nonlinear aggregation–diffusion equations from noisy discrete trajectory data. Our approach involves formulating and solving a regularized variational problem, which requires minimizing a quadratic error functional across a set of hypothesis functions, further augmented by a sparsity-enhancing regularizer. We employ a partial inversion algorithm, akin to the CoSaMP and subspace pursuit algorithms, to solve the basis pursuit problem. A key theoretical contribution is our novel stability estimate for the PDEs, validating the error functional ability in controlling the 2-Wasserstein distance between solutions generated using the true and estimated interaction potentials. Our work also includes an error analysis of estimators caused by discretization and observational errors in practical implementations. We demonstrate the effectiveness of the methods through various 1D and 2D examples showcasing collective behaviors.

Asymptotic Analysis of Elastic Elliptic Membrane Shells in Frictional Contact: Exploring Wear Phenomena

We consider a family of linearly elastic shells, all sharing the same middle surface, with thickness 2 ε , clamped along their entire lateral face, which upon deformation may enter in frictional contact with a moving foundation along its lower face. As a result of friction, material might be removed from the interface, thus causing wear. We focus in the case of an elliptic membrane, for which the orders of applied body force density, surface tractions density, and compliance functions with respect to the small parameter ε , representing thickness, are O ( 1 ) , O ( ε ) , and O ( ε ) , respectively. We show that the solution pair ( u ( ε ) , w ( ε ) ) of displacements and wear fields of the three-dimensional scaled variational contact problem converges to a pair of limit functions, ( u , w ) , which can be identified with the solution pair of a limit two-dimensional variational problem, since u = ( u i ) is independent of the transverse variable, x 3 . Besides, not all the convergences happen in the same topologies, since u α ( ε ) → u α in C ( [ 0 , T ] ; H 1 ( Ω ) ) , u 3 ( ε ) → u 3 in C ( [ 0 , T ] ; L 2 ( Ω ) ) , and w ( ε ) → w in C ( [ 0 , T ] ; L 2 ( ω ) ) as ε → 0 , where ω is a domain in R 2 and Ω = ω × [ − 1 , 1 ] .

Carrot John domains in variational problems

In this paper, we explore carrot John domains within variational problems, dividing our examination into two distinct sections. The initial part is dedicated to establishing the lower semicontinuity of the (optimal) John constant with respect to Hausdorff convergence for bounded John domains. This result holds promising implications for both shape optimization problems and Teichmüller theory. In the subsequent section, we demonstrate that an unbounded open set satisfying the carrot John condition with a center at \(\infty\), appearing in the Mumford–Shah problem, can be covered by a uniformly finite number of unbounded John domains (defined conventionally through cigars). These domains, in particular, support Sobolev–Poincaré inequalities.

Absence of Barren Plateaus and Scaling of Gradients in the Energy Optimization of Isometric Tensor Network States

Vanishing gradients can pose substantial obstacles for high-dimensional optimization problems. Here we consider energy minimization problems for quantum many-body systems with extensive Hamiltonians and finite-range interactions, which can be studied on classical computers or in the form of variational quantum eigensolvers on quantum computers. Barren plateaus correspond to scenarios where the average amplitude of the energy gradient decreases exponentially with increasing system size. This occurs, for example, for quantum neural networks and for brickwall quantum circuits when the depth increases polynomially in the system size. Here we prove that the variational optimization problems for matrix product states, tree tensor networks, and the multiscale entanglement renormalization ansatz are free of barren plateaus. The derived scaling properties for the gradient variance provide an analytical guarantee for the trainability of randomly initialized tensor network states (TNS) and motivate certain initialization schemes. In a suitable representation, unitary tensors that parametrize the TNS are sampled according to the uniform Haar measure. We employ a Riemannian formulation of the gradient based optimizations which simplifies the analytical evaluation.

Fractional Calculus of Variations for Composed Functionals with Generalized Derivatives

This paper extends the fractional calculus of variations to include generalized fractional derivatives with dependence on a given kernel, encompassing a wide range of fractional operators. We focus on variational problems involving the composition of functionals, deriving the Euler–Lagrange equation for this generalized case and providing optimality conditions for extremal curves. We explore problems with integral and holonomic constraints and consider higher-order derivatives, where the fractional orders are free.

On projected solutions for quasi equilibrium problems with non-self constraint map

In a normed space setting, this paper studies the conditions under which the projected solutions to a quasi equilibrium problem with non-self constraint map exist. Our approach is based on an iterative algorithm which gives rise to a sequence such that, under the assumption of asymptotic regularity, its limit points are projected solutions. Finally, as a particular case, we discuss the existence of projected solutions to a quasi variational inequality problem.

Ground states of a non‐local variational problem and Thomas–Fermi limit for the Choquard equation

AbstractWe study non‐negative optimisers of a Gagliardo–Nirenberg‐type inequality which involves the non‐local Riesz energy with , , and . For , the equivalent problem has been studied in connection with the Keller–Segel diffusion–aggregation models in the past few decades. The general case considered here appears in the study of Thomas–Fermi limit regime for the Choquard equations with local repulsion. We establish, for the first time, optimal ranges of parameters for the validity of the above interpolation inequality, discuss the existence and qualitative properties of the non‐negative maximisers and in some special cases estimate the optimal constant. For , it is known that the maximisers are Hölder continuous and compactly supported on a ball. We show that for , the maximisers are smooth functions supported on , while for , the maximisers consist of a characteristic function of a ball and a non‐constant non‐increasing Hölder continuous function supported on the same ball. We use these qualitative properties of the maximisers to establish the validity of the Thomas–Fermi approximations for the Choquard equations with local repulsion. The results are verified numerically with extensive examples.

Lateral Si/Si1‐xGex/Si Channel Heterostructure Charge Plasma Nanowire JLFET to Eliminate the Effects of Variation of Geometrical Dimensions

ABSTRACTIn this article, a charge plasma (CP) based doping‐less (DL) nanowire junctionless field effect transistor (NW JLFET) has been investigated for better immunity against geometrical dimension variation from a low power application perspective. SiGe source/drain and Si/SiGe/Si heterostructure channel have been used to improve the electrostatics in the channel to reduce the leakage current. With this doping‐less structure, the concept of charge plasmas has been incorporated by selecting electrodes with appropriate work functions. In addition to a low thermal budget, the doping‐less devices are easier to fabricate, have a reduced random fluctuation effect, and offer a low cost per unit. The doping‐less structure also offers improved mobility and higher current flow. The proposed device is compared with the conventional SiGe nanowire junctionless FET. When both devices are compared, lateral Si/SiGe/Si CP DL NW JLFET shows fewer changes in geometrical dimension variation in terms of germanium content x, nanowire thickness (tsi) and doping profile (Nd) on the drain current (IDS), ION/IOFF ratio, threshold voltage (Vth), drain‐induced barrier lowering (DIBL), and subthreshold slope (SS). A drain current model for lateral Si/SiGe/Si CP DL NW JLFET has also been developed in this paper, which includes the impact of the charge plasma technique. The impact of geometrical dimension variation on the analog characteristics of both devices has been studied in terms of like transconductance (gm) and transconductance gain factor (TGF) (gm/IDS). Thus, in the lateral Si/SiGe/Si CP DL NW JLFET, the charge plasma technique along with channel engineering solves the problem of geometrical dimension variation without affecting the inherited properties of junctionless devices.