The geography of symplectic -manifolds with an arbitrary fundamental group

In this article, for each finitely presented group , we construct a family of minimal symplectic -manifolds with which cover most lattice points with large in the region . Furthermore, we show that all these -manifolds admit infinitely many distinct smooth structures.

     

On certain optimization methods with finite-step inner algorithms for convex finite-dimensional problems with inequality constraints

Numerical methods are proposed for solving finite-dimensional convex problems with inequality constraints satisfying the Slater condition. A method based on solving the dual to the original regularized problem is proposed and justified for problems having a strictly uniformly convex sum of the objective function and the constraint functions. Conditions for the convergence of this method are derived, and convergence rate estimates are obtained for convergence with respect to the functional, convergence with respect to the argument to the set of optimizers, and convergence to the g-normal solution. For more general convex finite-dimensional minimization problems with inequality constraints, two methods with finite-step inner algorithms are proposed. The methods are based on the projected gradient and conditional gradient algorithms. The paper is focused on finite-dimensional problems obtained by approximating infinite-dimensional problems, in particular, optimal control problems for systems with lumped or distributed parameters.     

Pseudospectral method with symplectic algorithm for the solution of time-dependent SchrSdinger equations

A pseudospectral method with symplectic algorithm for the solution of time-dependent Schrodinger equations （TDSE） is introduced. The spatial part of the wavefunction is discretized into sparse grid by pseudospectral method and the time evolution is given in symplectic scheme. This method allows us to obtain a highly accurate and stable solution of TDSE. The effectiveness and efficiency of this method is demonstrated by the high-order harmonic spectra of one-dimensional atom in strong laser field as compared with previously published work. The influence of the additional static electric field is also investigated.     

A new algorithm for reconstructing the three-dimensional flow field of the oceanic mesoscale eddy

A new algorithm for reconstructing the three-dimensional flow field of the oceanic mesoscale eddies is proposed in this paper, based on variational method. Firstly, with the numerical differentiation Tikhonov regularizer, we reconstruct the continuous horizontal flow field on discrete grid points at each layer in the oceanic region, in terms of the horizontal flow field observations. Secondly, benefitting from the variational optimization analysis and its improvement, we reconstruct a three-dimensional flow field under the constraint of the horizontal flow and the vertical flow. The results of simulation experiments validate that the relative error of the new algorithm is lower than that of the finite difference method in the case of high grid resolution, which still holds in the case of unknown observational errors or in the absence of vertical velocity boundary conditions. Finally, using the reanalysis horizontal data sourcing from SODA and the proposed algorithm, we reconstruct three-dimensional flow field structure for the real oceanic mesoscale eddy.     

Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces

In this paper we prove that the moduli spaces of framed vector bundles over a surface X, satisfying certain conditions, admit a family of Poisson structures parametrized by the global sections of a certain line bundle on X. This generalizes to the case of framed vector bundles previous results obtained in [B2] for the moduli space of vector bundles over a Poisson surface. As a corollary of this result we prove that the moduli spaces of framed SU(r) – instantons on S⁴ = ℝ⁴ ∪ {∞} admit a natural holomorphic symplectic structure.     

A nonlinear regularization method for the calculation of relaxation spectra

It is well known that the relaxation spectrum characterizing the linear viscoelastic properties of a polymer melt or solution is not directly accessible by an experiment. Therefore, it must be calculated from data for a material function. With Tikhonov regularization the relaxation spectrum in the terminal and plateau region can be calculated from data for a material function in the corresponding region. Serious difficulties arise however, if the spectrum should be determined in a larger range. These difficulties are caused by the considerably different contributions at short and long relaxation times. We show that these difficulties can be avoided by a nonlinear regularization method.     

Regularization,Bayesian Inference,and Machine Learning Methods for Inverse Problems

Classical methods for inverse problems are mainly based on regularization theory, in particular those, that are based on optimization of a criterion with two parts: a data-model matching and a regularization term. Different choices for these two terms and a great number of optimization algorithms have been proposed. When these two terms are distance or divergence measures, they can have a Bayesian Maximum A Posteriori (MAP) interpretation where these two terms correspond to the likelihood and prior-probability models, respectively. The Bayesian approach gives more flexibility in choosing these terms and, in particular, the prior term via hierarchical models and hidden variables. However, the Bayesian computations can become very heavy computationally. The machine learning (ML) methods such as classification, clustering, segmentation, and regression, based on neural networks (NN) and particularly convolutional NN, deep NN, physics-informed neural networks, etc. can become helpful to obtain approximate practical solutions to inverse problems. In this tutorial article, particular examples of image denoising, image restoration, and computed-tomography (CT) image reconstruction will illustrate this cooperation between ML and inversion.     

Estimation of the L-Curve via Lanczos Bidiagonalization

The L-curve criterion is often applied to determine a suitable value of the regularization parameter when solving ill-conditioned linear systems of equations with a right-hand side contaminated by errors of unknown norm. However, the computation of the L-curve is quite costly for large problems; the determination of a point on the L-curve requires that both the norm of the regularized approximate solution and the norm of the corresponding residual vector be available. Therefore, usually only a few points on the L-curve are computed and these values, rather than the L-curve, are used to determine a value of the regularization parameter. We propose a new approach to determine a value of the regularization parameter based on computing an L-ribbon that contains the L-curve in its interior. An L-ribbon can be computed fairly inexpensively by partial Lanczos bidiagonalization of the matrix of the given linear system of equations. A suitable value of the regularization parameter is then determined from the L-ribbon, and we show that an associated approximate solution of the linear system can be computed with little additional work.     

Tensor invariants of natural mechanical systems on compact surfaces and the corresponding integrals

The classification problem for tensor invariants of geodesic flows on compact surfaces is considered. Such invariants include first integrals, multivalued integrals, and symmetry fields. The problem is solved for tensor invariants of arbitrary degrees. It is shown that the invariants under consideration can vanish at some point only in the presence of two-valued symmetry fields and multivalued integrals. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 417–430, September, 1999.     

Asymptotic L-based comparison of Tikhonov regularization and Landweber iteration

The distance between the Tikhonov and Landweber regularized solutions of a linear inverse problem is partly controlled by the L^∞-norm of the difference in their corresponding singular value filters. For large Landweber iteration number, the regularization parameter of the closest Tikhonov filter to a given Landweber filter is determined. This asymptotically computed parameter compares well with the numerically computed value even for moderate sized iteration number. A consequence of the analysis is to determine the range of singular values to which the difference in regularized solutions is most sensitive.