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The variational techniques (e.g. the total variation based method) are well established and effective for image restoration, as well as many other applications, while the wavelet frame based approach is relatively new and came from a different school. This paper is designed to establish a connection between these two major approaches for image restoration. The main result of this paper shows that when spline wavelet frames of are used, a special model of a wavelet frame method, called the analysis based approach, can be viewed as a discrete approximation at a given resolution to variational methods. A convergence analysis as image resolution increases is given in terms of objective functionals and their approximate minimizers. This analysis goes beyond the establishment of the connections between these two approaches, since it leads to new understandings for both approaches. First, it provides geometric interpretations to the wavelet frame based approach as well as its solutions. On the other hand, for any given variational model, wavelet frame based approaches provide various and flexible discretizations which immediately lead to fast numerical algorithms for both wavelet frame based approaches and the corresponding variational model. Furthermore, the built-in multiresolution structure of wavelet frames can be utilized to adaptively choose proper differential operators in different regions of a given image according to the order of the singularity of the underlying solutions. This is important when multiple orders of differential operators are used in various models that generalize the total variation based method. These observations will enable us to design new methods according to the problems at hand, hence, lead to wider applications of both the variational and wavelet frame based approaches. Links of wavelet frame based approaches to some more general variational methods developed recently will also be discussed.

In this paper, we deal with a class of planar Schrödinger–Poisson systems, namely, - Δ u + V ( x ) u + γ 2 π ( log ( | · | ) ∗ | u | 2 ) u = b | u | p - 2 u in R 2 , where γ > 0 , b ≥ 0 , p > 2 and V ∈ C ( R 2 , R ) is an unbounded potential function with inf R 2 V > 0 . Suppose moreover that the potential V satisfies { x ∈ R 2 : V ( x ) ≤ M } < ∞ for every M > 0 , we establish the existence of ground state solutions for this system via variational methods. Furthermore, we also explore the minimax characterization of ground state solutions. Our main results can be viewed as a counterpart of the results from Molle and Sardilli (Proc Edinb Soc 65:1133–1146, 2022), where the authors studied the existence of ground state solutions for the above planar Schrödinger–Poisson system in the case where b > 0 and p > 4 .

In this paper, we study multi-bump solutions of the following Kirchhoff type equation: - M ∫ R 2 | ∇ u | 2 d x Δ u + μ V ( x ) + h ( x ) u = λ f ( u ) in R 2 , where M is continuous with inf R + M > 0 , V ≥ 0 and its zero set has several disjoint bounded components, μ , λ are positive parameters, f has exponential critical growth. When V decays to zero at infinity, we use variational methods to obtain the existence and concentration behavior of multi-bump solutions.

The accelerated progress in manufacturing noisy, intermediate-scale quantum (NISQ) computing hardware has opened the possibility of exploring its application in transforming approaches to solving computationally challenging problems. The important limitations common among all NISQ computing technologies are the absence of error correction and the short coherence time, which limit the computational power of these systems. Shortening the required time of a single run of a quantum algorithm is essential for reducing environment-induced errors and for the efficiency of the computation. We have investigated the ability of a variational version of adiabatic state preparation (ASP) to generate an accurate state more efficiently compared to existing adiabatic methods. The standard ASP method uses a time-dependent Hamiltonian, connecting the initial Hamiltonian with the final Hamiltonian. In the current approach, a navigator Hamiltonian is introduced which has a non-zero amplitude only in the middle of the annealing process. Both the initial and navigator Hamiltonians are determined using variational methods. A Hermitian cluster operator, inspired by coupled-cluster theory and truncated to single and double excitations/de-excitations, is used as a navigator Hamiltonian. A comparative study of our variational algorithm (VanQver) with that of standard ASP, starting with a Hartree-Fock Hamiltonian, is presented. The results indicate that the introduction of the navigator Hamiltonian significantly improves the annealing time required to achieve chemical accuracy by two to three orders of magnitude. The efficiency of the method is demonstrated in the ground-state energy estimation of molecular systems, namely, H2, P4, and LiH.

In this paper, we study the existence of multiple normalized solutions for a Choquard equation involving fractional p-Laplacian in RN. With the help of variational methods, minimization techniques, and the Lusternik–Schnirelmann category, the existence of multiple normalized solutions is obtained for the above problem.

In this article, we use approximation techniques and variational methods to study a class of nonlocal equations with variable exponents and mixed criticality. We prove the existence of the ground state nontrivial solutions with the least energy. Our results are applied to a specific Schrödinger-Poisson type system.

We have established a coherent framework for applying variational methods to partial differential equations on hypergraphs, which includes the propositions of calculus and function spaces on hypergraphs. Several results related to the maximum principle on hypergraphs have also been proven. As applications, we demonstrated how these can be used to study partial differential equations on hypergraphs.

In this paper, we deal with the planar Schrödinger–Poisson system -Δu+V(x)u+ϕu=b|u|p-2uinR2,Δϕ=u2inR2,where b≥0, p>2 and V∈C(R2,R) is a potential function with infR2V>0. Suppose moreover that V exhibits a bounded potential well in the sense that lim|x|→∞V(x) exists and is equal to supR2V. By using the variational methods, we obtain the existence of ground state solutions for this system in the case where p≥3. Furthermore, we also present a minimax characterization of ground state solutions. The main feature of this work is that we do not assume any periodicity or symmetry condition on the external potential V, which is essential to establish the compactness condition of Cerami sequences.

This paper studies a singular anisotropic system of coupled quasilinear elliptic equations. The system features anisotropic diffusion operators with variable exponents p i and q i , singular terms of the form v − γ 1 and u − γ 2 , and nonlinear source terms. By employing variational methods and an approximation problem, we prove the existence of positive solutions under suitable conditions on the nonlinearities.

In this paper, we study the existence and asymptotic behaviour of solutions of the nonhomogeneous quasilinear Schrödinger–Poisson system where and are positive parameters, is a continuous and coercive potential function with positive infimum, and is a Carathéodory function defined on and satisfying the classic Ambrosetti–Rabinowitz condition. Under some suitable assumptions on , , and , we obtain the existence of two different energy nontrivial solutions by use of variational methods and truncation technique for sufficiently small and fixed . Moreover, the asymptotic behaviour of these solutions is studied whenever and , respectively, tend to zero.