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The aim of this paper is the derivation of structure preserving schemes for the solution of the EPDiff equation, with particular emphasis on the two dimensional case. We develop three different schemes based on the Discrete Variational Derivative Method (DVDM) on a rectangular domain discretized with a regular, structured, orthogonal grid. We present numerical experiments to support our claims: we investigate the preservation of energy and linear momenta, the reversibility, and the empirical convergence of the schemes. The quality of our schemes is finally tested by simulating the interaction of singular wave fronts.

Traditionally, stochastic approximation schemes for SVIs have relied on strong monotonicity and Lipschitzian properties of the underlying map. In contrast, we consider monotone stochastic variational inequality (SVI) problems where the strong monotonicity and Lipschitzian assumptions on the mappings are weakened. In the first part of the paper, to address such shortcomings, a regularized smoothed SA (RSSA) scheme is developed wherein the stepsize, smoothing, and regularization parameters are diminishing sequences updated after every iteration. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to a solution in an almost sure sense, extending the results in [16] to the non-Lipschitzian regime. Motivated by the need to develop non-asymptotic rate statements, in the second part of the paper, we develop a variant of the RSSA scheme, denoted by aRSSA$_r$, in which we employ a weighted iterate-averaging, parametrized by a scalar $r$ where $r = 1$ provides us with the standard averaging scheme. We make several contributions in this context: First, we show that the gap function associated with the sequences by the aRSSA$_r$ scheme tends to zero when the parameter sequences are chosen appropriately. Second, we show that the gap function associated with the averaged sequence diminishes to zero at the optimal rate $\cal{O}(1/\sqrt{K})$ after $K$ steps when smoothing and regularization are suppressed and $r < 1$, thus improving the rate statement for the standard averaging which admits a rate of $\cal{O}(\ln(K)/\sqrt{K})$. Third, we develop a window-based variant of this scheme that also displays the optimal rate for $r < 1$. Notably, we prove the superiority of the scheme with $r < 1$ with its counterpart with $r=1$ in terms of the constant factor of the error bound when the size of the averaging window is sufficiently large.

We review variational and quantum Monte Carlo approaches based on (extended) Lang-Firsov transformations of the Hamiltonian. Derivations for one, two and many electrons are given, and results for the Holstein polaron, the Holstein-Hubbard bipolaron, and the spinless Holstein model at finite carrier densities are presented.

We review variational and quantum Monte Carlo approaches based on (extended) Lang-Firsov transformations of the Hamiltonian. Derivations for one, two and many electrons are given, and results for the Holstein polaron, the Holstein-Hubbard bipolaron, and the spinless Holstein model at finite carrier densities are presented.

Following the subtraction procedure for manifolds with boundaries, we calculate by variational methods, the Schwarzschild and Flat space energy difference. The one loop approximation for TT tensors is considered here. An analogy between the computed energy difference in momentum space and the Casimir effect is illustrated. We find a singular behaviour in the UV-limit, due to the presence of the horizon when $r=2m.$ When $r>2m$ this singular behaviour disappears, which is in agreement with various other models previously presented.

29 p. 28 cm

29 p. 28 cm

29 p. 28 cm

Variational inference provides a powerful tool for approximate probabilistic in- ference on complex, structured models. Typical variational inference methods, however, require to use inference networks with computationally tractable proba- bility density functions. This largely limits the design and implementation of vari- ational inference methods. We consider wild variational inference methods that do not require tractable density functions on the inference networks, and hence can be applied in more challenging cases. As an example of application, we treat stochastic gradient Langevin dynamics (SGLD) as an inference network, and use our methods to automatically adjust the step sizes of SGLD, yielding significant improvement over the hand-designed step size schemes

Variational inference provides a powerful tool for approximate probabilistic in- ference on complex, structured models. Typical variational inference methods, however, require to use inference networks with computationally tractable proba- bility density functions. This largely limits the design and implementation of vari- ational inference methods. We consider wild variational inference methods that do not require tractable density functions on the inference networks, and hence can be applied in more challenging cases. As an example of application, we treat stochastic gradient Langevin dynamics (SGLD) as an inference network, and use our methods to automatically adjust the step sizes of SGLD, yielding significant improvement over the hand-designed step size schemes

Variational inference provides a powerful tool for approximate probabilistic in- ference on complex, structured models. Typical variational inference methods, however, require to use inference networks with computationally tractable proba- bility density functions. This largely limits the design and implementation of vari- ational inference methods. We consider wild variational inference methods that do not require tractable density functions on the inference networks, and hence can be applied in more challenging cases. As an example of application, we treat stochastic gradient Langevin dynamics (SGLD) as an inference network, and use our methods to automatically adjust the step sizes of SGLD, yielding significant improvement over the hand-designed step size schemes

Variational inference provides a powerful tool for approximate probabilistic in- ference on complex, structured models. Typical variational inference methods, however, require to use inference networks with computationally tractable proba- bility density functions. This largely limits the design and implementation of vari- ational inference methods. We consider wild variational inference methods that do not require tractable density functions on the inference networks, and hence can be applied in more challenging cases. As an example of application, we treat stochastic gradient Langevin dynamics (SGLD) as an inference network, and use our methods to automatically adjust the step sizes of SGLD, yielding significant improvement over the hand-designed step size schemes

Inference methods are often formulated as variational approximations: these approximations allow easy evaluation of statistics by marginalization or linear response, but these estimates can be inconsistent. We show that by introducing constraints on covariance, one can ensure consistency of linear response with the variational parameters, and in so doing inference of marginal probability distributions is improved. For the Bethe approximation and its generalizations, improvements are achieved with simple choices of the constraints. The approximations are presented as variational frameworks; iterative procedures related to message passing are provided for finding the minima.

An interdisciplinary research group in modeling, analysis and simulation was established at Florida A&M University (FAMU). This group set as its scientific goal the development of robust computational algorithms that are well suited for nonconvex variational principles, and other nonlinear problems that arise in the study of highly nonlinear materials. The equally important second goal of the group was to enhance the participation of under-represented minorities in applied disciplines connected to emerging technologies that are important to the Air Force Office of Scientific Research. Mechanisms used for meeting these goals include the strengthening of the infrastructure for research at FAMU, strengthening of the existing relationship between AFOSR and FAMU, and enhancing the existing relationship between FAMU, Carnegie Mellon University, University of Minnesota, and North Carolina University. This report describes the background and overall research goals of this project, and delineates the accomplishments made thus far.

Variational methods for the approximate solution of subsonic and transonic flows of a compressible fluid before the occurrence of shock waves have been carried out in previous papers. The methods fail as soon as the shock waves occur as the flow behind the shock waves now becomes rotational and has variable entropy. Since most transonic flows are accompanied by shock waves, a method which allows for shock waves and variable entropy is necessary for the study of such flows. By modifying Bateman's variational principle for irrotational flows, it is shown that a variational principle for flows with rotation and variable entropy can be obtained. By applying this variational principle to the regions of flow behind shock waves and Bateman's original principle to the other regions in the fluid, shock equations can be directly obtained. A procedure for computing numerical solutions for such flows is suggested, and a numerical example is carried out. At high Mach number above a certain limiting value, the results show that irrotational flow fails. However, by inserting shock waves and allowing a part of the flow to be rotational, computation indicates that solution exists again.

Variational methods for the approximate solution of subsonic and transonic flows of a compressible fluid before the occurrence of shock waves have been carried out in previous papers. The methods fail as soon as the shock waves occur as the flow behind the shock waves now becomes rotational and has variable entropy. Since most transonic flows are accompanied by shock waves, a method which allows for shock waves and variable entropy is necessary for the study of such flows. By modifying Bateman's variational principle for irrotational flows, it is shown that a variational principle for flows with rotation and variable entropy can be obtained. By applying this variational principle to the regions of flow behind shock waves and Bateman's original principle to the other regions in the fluid, shock equations can be directly obtained. A procedure for computing numerical solutions for such flows is suggested, and a numerical example is carried out. At high Mach number above a certain limiting value, the results show that irrotational flow fails. However, by inserting shock waves and allowing a part of the flow to be rotational, computation indicates that solution exists again.

By means of a simple structural problem, an important requirement often overlooked in practice on the basis functions used in Rayleigh-Ritz-Galerkin type methods is brought into focus. The problem of the static deformation of a uniformly loaded beam is solved variationally by expanding the beam displacement in a Fourier Cosine series. The potential energy functional is rendered stationary subject to the geometric boundary conditions. It is demonstrated that the variational approach does not converge to the true solution. The object is to resolve this paradox, and in so doing, indicate the practical implications of norms and completeness in an appropriate inner product space.

We consider the problem of scale detection in images where a region of interest is present together with a measurement tool (e.g. a ruler). For the segmentation part, we focus on the graph based method by Flenner and Bertozzi which reinterprets classical continuous Ginzburg-Landau minimisation models in a totally discrete framework. To overcome the numerical difficulties due to the large size of the images considered we use matrix completion and splitting techniques. The scale on the measurement tool is detected via a Hough transform based algorithm. The method is then applied to some measurement tasks arising in real-world applications such as zoology, medicine and archaeology.

The scientific goals of this research project are: 1. To explore the factors (e.g. uncertainties in initial conditions versus those in surface forcing and boundary conditions) that limit the predictability of the circulation in regional ocean models in a variety of dynamical regimes; 2. To compare two state-of-the-art variational data assimilation strategies (4DVAR and IOM) and gain experience using both in regional ocean models; 3. To develop ensemble prediction techniques for regional ocean models; 4. Demonstrate the utility of the ROMS data assimilation framework in a real-time, sea-going environment for prediction studies in the Intra-Americas Sea (IAS) with particular emphasis in the Caribbean Sea. As such, we will demonstrate, as a proof of concept, the utility of adjoint modeling and 4DVAR data assimilation in a real-time operational setting, at sea.

The scientific goals of this research project are: 1. To explore the factors (e.g. uncertainties in initial conditions versus those in surface forcing and boundary conditions) that limit the predictability of the circulation in regional ocean models in a variety of dynamical regimes; 2. To compare two state-of-the-art variational data assimilation strategies (4DVAR and IOM) and gain experience using both in regional ocean models; 3. To develop ensemble prediction techniques for regional ocean models; 4. Demonstrate the utility of the ROMS data assimilation framework in a real-time, sea-going environment for prediction studies in the Intra-Americas Sea (IAS) with particular emphasis in the Caribbean Sea. As such, we will demonstrate, as a proof of concept, the utility of adjoint modeling and 4DVAR data assimilation in a real-time operational setting, at sea.

We consider structure-preserving methods for conservative systems, which rigorously replicate the conservation property yielding better numerical solutions. There, corresponding to the skew-symmetry of the differential operator, that of difference operators is essential to the discrete conservation law. Unfortunately, however, when we employ the standard central difference operator, the simplest one, the numerical solutions often suffer from undesirable spatial oscillations. In this letter, we propose a novel "average-difference method," which is tougher against such oscillations, and combine it with an existing conservative method. Theoretical and numerical analysis in the linear case show the superiority of the proposed method.

Three fundamental variational principles used for solving elastodynamic eigenvalue problems are studied within the context of elastic wave propagation in periodic composites (phononics). We study the convergence of the eigenvalue problems resulting from the displacement Rayleigh quotient, the stress Rayleigh quotient and the mixed quotient. The convergence rates of the three quotients are found to be related to the continuity and differentiability of the density and compliance variation over the unit cell. In general, the mixed quotient converges faster than both the displacement Rayleigh and the stress Rayleigh quotients, however, there exist special cases where either the displacement Rayleigh or the stress Rayleigh quotient shows the exact same convergence as the mixed-method. We show that all methods converge faster for smoother material property variations, but when density variation is rough, the difference between the mixed quotient and stress Rayleigh quotient is higher and similarly, when compliance variation is rough, the difference between the mixed quotient and displacement Rayleigh quotient is higher. Since eigenvalue problems such as those considered in this paper tend to be highly computationally intensive, it is expected that these results will lead to fast and efficient algorithms in the areas of phononics and photonics.

Variational methods (VM) sensitivity analysis employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

We introduce a new approach to highly correlated systems which generalizes the Fermi Hypernetted Chain and Correlated Basis Function techniques. While the latter approaches can only be applied to systems for which a nonrelativistic wave function can be defined, the new approach is based on the variation of a trial hamiltonian within a path integral framework and thus can also be applied to relativistic and field theoretical problems. We derive a diagrammatic scheme for the new approach and show how a particular choice of the trial hamiltonian corresponds exactly to the use of a Jastrow correlated ansatz for the wave function in the Fermi Hypernetted Chain approach. We show how our new approach can be used to find upper bounds to ground state energies in systems which the FHNC cannot handle, including those described by an energy-dependent effective hamiltonian. We demonstrate our approach by applying it to a quantum field theoretical system of interacting pions and nucleons.

The calculation of vibrations, and in particular, resonances from bounded elastic shells can be quite tedious and time consuming when using the exact elastodynamic equations. Thus, a popular approach has been to employ various dynamic assumptions about the motion of the shell surface when subjected to disturbances. This can be done using variational considerations in which energy is minimized when various constraints are imposed. We exploit the technique using various assumptions which give rise to several shell theories. We can use the resulting expressions to calculate resonances over a frequency range and compare then with the exact results. We may then rank the various approximations in order of their agreement to the exact results. Limitations of each of the methods can then be outlined as well as those of shell methods in general.... Acoustic scattering, Shallow water, Waveguide propagation.

The calculation of vibrations, and in particular, resonances from bounded elastic shells can be quite tedious and time consuming when using the exact elastodynamic equations. Thus, a popular approach has been to employ various dynamic assumptions about the motion of the shell surface when subjected to disturbances. This can be done using variational considerations in which energy is minimized when various constraints are imposed. We exploit the technique using various assumptions which give rise to several shell theories. We can use the resulting expressions to calculate resonances over a frequency range and compare then with the exact results. We may then rank the various approximations in order of their agreement to the exact results. Limitations of each of the methods can then be outlined as well as those of shell methods in general.... Acoustic scattering, Shallow water, Waveguide propagation.

This research paper introduces and compares the efficacy of two techniques, namely the Laplace Variational Iteration Method (LVIM) and the Laplace Homotopy Permutation Method (LHPM), in obtaining approximate solutions for fractional differential equations (FDEs) incorporating the Atangana-Baleanu fractional derivative in the Caputo sense (ABCO). The Atangana-Baleanu fractional derivative, a specialized form of fractional calculus, offers a more accurate representation of anomalous diffusion processes. The LVIM and LHPM are employed to tackle fractional differential equations, which frequently arise in diverse scientific and engineering disciplines. The LVIM combines the advantages of the Laplace transform and the variational iteration technique to yield a reliable and efficient solution strategy. On the other hand, the LHPM integrates the Laplace transform and the homotopy permutation technique to address the same class of equations.This paper provides a detailed exposition of both methodologies, including their algorithmic frameworks and computational procedures. The comparison of the two techniques involves assessing their performance in approximating solutions to fractional differential equations with the Atangana-Baleanu fractional derivative. The obtained results underscore the success of both methods in yielding accurate and practical approximate solutions for this specific type of fractional differential equation.

This research paper introduces and compares the efficacy of two techniques, namely the Laplace Variational Iteration Method (LVIM) and the Laplace Homotopy Permutation Method (LHPM), in obtaining approximate solutions for fractional differential equations (FDEs) incorporating the Atangana-Baleanu fractional derivative in the Caputo sense (ABCO). The Atangana-Baleanu fractional derivative, a specialized form of fractional calculus, offers a more accurate representation of anomalous diffusion processes. The LVIM and LHPM are employed to tackle fractional differential equations, which frequently arise in diverse scientific and engineering disciplines. The LVIM combines the advantages of the Laplace transform and the variational iteration technique to yield a reliable and efficient solution strategy. On the other hand, the LHPM integrates the Laplace transform and the homotopy permutation technique to address the same class of equations.This paper provides a detailed exposition of both methodologies, including their algorithmic frameworks and computational procedures. The comparison of the two techniques involves assessing their performance in approximating solutions to fractional differential equations with the Atangana-Baleanu fractional derivative. The obtained results underscore the success of both methods in yielding accurate and practical approximate solutions for this specific type of fractional differential equation.

This research paper introduces and compares the efficacy of two techniques, namely the Laplace Variational Iteration Method (LVIM) and the Laplace Homotopy Permutation Method (LHPM), in obtaining approximate solutions for fractional differential equations (FDEs) incorporating the Atangana-Baleanu fractional derivative in the Caputo sense (ABCO). The Atangana-Baleanu fractional derivative, a specialized form of fractional calculus, offers a more accurate representation of anomalous diffusion processes. The LVIM and LHPM are employed to tackle fractional differential equations, which frequently arise in diverse scientific and engineering disciplines. The LVIM combines the advantages of the Laplace transform and the variational iteration technique to yield a reliable and efficient solution strategy. On the other hand, the LHPM integrates the Laplace transform and the homotopy permutation technique to address the same class of equations.This paper provides a detailed exposition of both methodologies, including their algorithmic frameworks and computational procedures. The comparison of the two techniques involves assessing their performance in approximating solutions to fractional differential equations with the Atangana-Baleanu fractional derivative. The obtained results underscore the success of both methods in yielding accurate and practical approximate solutions for this specific type of fractional differential equation.

This research paper introduces and compares the efficacy of two techniques, namely the Laplace Variational Iteration Method (LVIM) and the Laplace Homotopy Permutation Method (LHPM), in obtaining approximate solutions for fractional differential equations (FDEs) incorporating the Atangana-Baleanu fractional derivative in the Caputo sense (ABCO). The Atangana-Baleanu fractional derivative, a specialized form of fractional calculus, offers a more accurate representation of anomalous diffusion processes. The LVIM and LHPM are employed to tackle fractional differential equations, which frequently arise in diverse scientific and engineering disciplines. The LVIM combines the advantages of the Laplace transform and the variational iteration technique to yield a reliable and efficient solution strategy. On the other hand, the LHPM integrates the Laplace transform and the homotopy permutation technique to address the same class of equations.This paper provides a detailed exposition of both methodologies, including their algorithmic frameworks and computational procedures. The comparison of the two techniques involves assessing their performance in approximating solutions to fractional differential equations with the Atangana-Baleanu fractional derivative. The obtained results underscore the success of both methods in yielding accurate and practical approximate solutions for this specific type of fractional differential equation.

This research paper introduces and compares the efficacy of two techniques, namely the Laplace Variational Iteration Method (LVIM) and the Laplace Homotopy Permutation Method (LHPM), in obtaining approximate solutions for fractional differential equations (FDEs) incorporating the Atangana-Baleanu fractional derivative in the Caputo sense (ABCO). The Atangana-Baleanu fractional derivative, a specialized form of fractional calculus, offers a more accurate representation of anomalous diffusion processes. The LVIM and LHPM are employed to tackle fractional differential equations, which frequently arise in diverse scientific and engineering disciplines. The LVIM combines the advantages of the Laplace transform and the variational iteration technique to yield a reliable and efficient solution strategy. On the other hand, the LHPM integrates the Laplace transform and the homotopy permutation technique to address the same class of equations.This paper provides a detailed exposition of both methodologies, including their algorithmic frameworks and computational procedures. The comparison of the two techniques involves assessing their performance in approximating solutions to fractional differential equations with the Atangana-Baleanu fractional derivative. The obtained results underscore the success of both methods in yielding accurate and practical approximate solutions for this specific type of fractional differential equation.

We use variational methods to calculate quasilocal energy quantum corrections. A comparison with the effective potential calculated at quadratic order is made by means of gaussian wave functionals. The method is a particular case of the effective action for composite operators used in quantum field theory. In pure gravity the method is applied for the first time. Implications on the foam-like scenario are discussed.

For the scalar Wick-Cutkosky model in the particle representation we perform a similar variational calculation for the 2-point function as was done by Feynman for the polaron problem. We employ a quadratic nonlocal trial action with a retardation function for which several ans\"atze are used. The variational parameters are determined by minimizing the variational function and in the most general case the nonlinear variational equations are solved numerically. We obtain the residue at the pole, study analytically and numerically the instability of the model at larger coupling constants and calculate the width of the dressed particle.

For the scalar Wick-Cutkosky model in the particle representation we perform a similar variational calculation for the 2-point function as was done by Feynman for the polaron problem. We employ a quadratic nonlocal trial action with a retardation function for which several ans\"atze are used. The variational parameters are determined by minimizing the variational function and in the most general case the nonlinear variational equations are solved numerically. We obtain the residue at the pole, study analytically and numerically the instability of the model at larger coupling constants and calculate the width of the dressed particle.

A variational method is presented for solving eigenvalue problems which arise in connection with the analysis of convective heat transfer in the thermal entrance region of ducts. Consideration is given, to both situations where the temperature profile depends upon one cross-sectional coordinate (e.g. circular tube) or upon two cross-sectional coordinates (e.g. rectangular duct). The variational method is illustrated and verified by application to laminar heat transfer in a circular tube and a parallel-plate channel, and good agreement with existing numerical solutions is attained. Then, application is made to laminar heat transfer in a square duct as a check, an alternate computation for the square duct is made using a method indicated by Misaps and Pohihausen. The variational method can, in principle, also be applied to problems in turbulent heat transfer.

A variational method is presented for solving eigenvalue problems which arise in connection with the analysis of convective heat transfer in the thermal entrance region of ducts. Consideration is given, to both situations where the temperature profile depends upon one cross-sectional coordinate (e.g. circular tube) or upon two cross-sectional coordinates (e.g. rectangular duct). The variational method is illustrated and verified by application to laminar heat transfer in a circular tube and a parallel-plate channel, and good agreement with existing numerical solutions is attained. Then, application is made to laminar heat transfer in a square duct as a check, an alternate computation for the square duct is made using a method indicated by Misaps and Pohihausen. The variational method can, in principle, also be applied to problems in turbulent heat transfer.

Quantum resources of quantum and classical variational methods by Eliska Geplova @QTMLConference

Following the position of moving objects based on the information delivered by single or multiple optical sensors (e.g., video cameras) is the objective of visual tracking. The need for visual tracking is ubiquitous and a multitude of approaches exist for the solution of this tracking problem. Increasingly, computer vision algorithms are required to provide additional information beyond a simple track point, and more complex methodologies are needed to produce the desired information. For noisy, cluttered, and/or dynamic scenes the ability to provide a smooth and faithful signal is essential which leads of course to the entire issue of filtering. In this research program, we have developed a novel visual tracking approach, using statistical variational methods. In particular, we have developed a geometric particle filter for controlled active vision. This has been applied to various tracking problems including tracking through turbulence and UAVs flying in formation.

Contents: Nonlinear vibrations; Viscosity solutions and applications; Wavelets and applications; Variational methods for image segmentation.

Contents: Nonlinear vibrations; Viscosity solutions and applications; Wavelets and applications; Variational methods for image segmentation.

Contents: Nonlinear vibrations; Viscosity solutions and applications; Wavelets and applications; Variational methods for image segmentation.

Contents: Nonlinear vibrations; Viscosity solutions and applications; Wavelets and applications; Variational methods for image segmentation.

In this paper we prove the existence of a nontrivial non-negative radial solution for a quasilinear elliptic problem. Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.

The well known maximum-entropy principle due to Jaynes, which states that given mean parameters, the maximum entropy distribution matching them is in an exponential family, has been very popular in machine learning due to its "Occam's razor" interpretation. Unfortunately, calculating the potentials in the maximum-entropy distribution is intractable \cite{bresler2014hardness}. We provide computationally efficient versions of this principle when the mean parameters are pairwise moments: we design distributions that approximately match given pairwise moments, while having entropy which is comparable to the maximum entropy distribution matching those moments. We additionally provide surprising applications of the approximate maximum entropy principle to designing provable variational methods for partition function calculations for Ising models without any assumptions on the potentials of the model. More precisely, we show that in every temperature, we can get approximation guarantees for the log-partition function comparable to those in the low-temperature limit, which is the setting of optimization of quadratic forms over the hypercube. \cite{alon2006approximating}

Numerical homogenization tries to approximate the solutions of elliptic partial differential equations with strongly oscillating coefficients by functions from modified finite element spaces. We present in this paper a class of such methods that are very closely related to the method of M{\aa}lqvist and Peterseim [Math. Comp. 83, 2014]. Like the method of M{\aa}lqvist and Peterseim, these methods do not make explicit or implicit use of a scale separation. Their compared to that in the work of M{\aa}lqvist and Peterseim strongly simplified analysis is based on a reformulation of their method in terms of variational multiscale methods and on the theory of iterative methods, more precisely, of additive Schwarz or subspace decomposition methods.

Two complementary approaches have been extensively used in signal and image processing leading to novel results, the sparse representation methodology and the variational strategy. Recently, a new sparsity based model has been proposed, the cosparse analysis framework, which may potentially help in bridging sparse approximation based methods to the traditional total-variation minimization. Based on this, we introduce a sparsity based framework for solving overparameterized variational problems. The latter has been used to improve the estimation of optical flow and also for general denoising of signals and images. However, the recovery of the space varying parameters involved was not adequately addressed by traditional variational methods. We first demonstrate the efficiency of the new framework for one dimensional signals in recovering a piecewise linear and polynomial function. Then, we illustrate how the new technique can be used for denoising and segmentation of images.

This paper studies the solutions of variational methods for random ordinary (partial) dierential equations in L2−space. These methods are called Galerkin method, Petrov-Galerkin method, Least-Squares method and Collocation method. Some basic properties of these methods where applying on random problems will be shown throughout some numerical example

This paper presents a comparison of two reduced-order, sequential and variational data assimilation methods: the SEEK filter and the R-4D-Var. A hybridization of the two, combining the variational framework and the sequential evolution of covariance matrices, is also preliminarily investigated and assessed in the same experimental conditions. The comparison is performed using the twin-experiment approach on a model of the Tropical Pacific domain. The assimilated data are simulated temperature profiles at the locations of the TAO/TRITON array moorings. It is shown that, in a quasi-linear regime, both methods produce similarly good results. However the hybrid approach provides slightly better results and thus appears as potentially fruitful. In a more non-linear regime, when Tropical Instability Waves develop, the global nature of the variational approach helps control model dynamics better than the sequential approach of the SEEK filter. This aspect is probably enhanced by the context of the experiments in that there is a limited amount of assimilated data and no model error.

It is known that the variational methods are the most powerful tool for studying the Coulomb three-body bound state problem. However, they often suffer from loss of stability when the number of basis functions increases. This problem can be cured by applying the multiprecision package designed by D.H. Bailey. We consider the variational basis functions of the type exp(-a_n r_1 - b_n r_2 - g_n r_12) with complex exponents. The method yields the best available energies for the ground states of the helium atom and the positive hydrogen ion as well as many other known atomic and molecular systems.