Data Approximation Problems Research Articles

On Snapshot-Based Model Reduction Under Compatibility Conditions for a Nonlinear Flow Problem on Networks

This paper is on the construction of structure-preserving, online-efficient reduced models for the barotropic Euler equations with a friction term on networks. The nonlinear flow problem finds broad application in the context of gas distribution networks. We propose a snapshot-based reduction approach that consists of a mixed variational Galerkin approximation combined with quadrature-type complexity reduction. Its main feature is that certain compatibility conditions are assured during the training phase, which make our approach structure-preserving. The resulting reduced models are locally mass conservative and inherit an energy bound and port-Hamiltonian structure. We also derive a wellposedness result for them. In the training phase, the compatibility conditions pose challenges, we face constrained data approximation problems as opposed to the unconstrained training problems in the conventional reduction methods. The training of our model order reduction consists of a principal component analysis under a compatibility constraint and, notably, yields reduced models that fulfill an optimality condition for the snapshot data. The training of our quadrature-type complexity reduction involves a semi-definite program with combinatorial aspects, which we approach by a greedy procedure. Efficient algorithmic implementations are presented. The robustness and good performance of our structure-preserving reduced models are showcased at the example of gas network simulations.

Scattered data fitting of Hermite type by a weighted meshless method

In many practical problems, it is often desirable to interpolate not only the function values but also the values of derivatives up to certain order, as in the Hermite interpolation. The Hermite interpolation method by radial basis functions is used widely for solving scattered Hermite data approximation problems. However, sometimes it makes more sense to approximate the solution by a least squares fit. This is particularly true when the data are contaminated with noise. In this paper, a weighted meshless method is presented to solve least squares problems with noise. The weighted meshless method by Gaussian radial basis functions is proposed to fit scattered Hermite data with noise in certain local regions of the problem’s domain. Existence and uniqueness of the solution is proved. This approach has one parameter which can adjust the accuracy according to the size of the noise. Another advantage of the weighted meshless method is that it can be used for problems in high dimensions with nonregular domains. The numerical experiments show that our weighted meshless method has better performance than the traditional least squares method in the case of noisy Hermite data.

Inverse-free recursive multiresolution algorithms for a data approximation problem

We present inverse-free recursive multiresolution algorithms for data approximation problems based on energy functionals minimization. During the multiresolution process a linear system needs to be solved at each different resolution level, which can be solved with direct or iterative methods. Numerical results are reported, using the sparse Cholesky factorization, for two applications: one concerning the localization of regions in which the energy of a given surface is mostly concentrated, and another one regarding noise reduction of a given dataset. In addition, for large-scale data approximation problems that require a very fine resolution, we discuss the use of the Preconditioned Conjugate Gradient (PCG) iterative method coupled with a specialized monolithic preconditioner, for which one preconditioner is built for the highest resolution level and then the corresponding blocks of that preconditioner are used as preconditioners for the forthcoming lower levels.

Using Gaussian Eigenfunctions to Solve Boundary Value Problems

AbstractKernel-based methods are popular in computer graphics, machine learning, and statistics, among other fields; because they do not require meshing of the domain under consideration, higher dimensions and complicated domains can be managed with reasonable effort. Traditionally, the high order of accuracy associated with these methods has been tempered by ill-conditioning, which arises when highly smooth kernels are used to conduct the approximation. Recent advances in representing Gaussians using eigenfunctions have proven successful at avoiding this destabilization in scattered data approximation problems. This paper will extend these techniques to the solution of boundary value problems using collocation. The method of particular solutions will also be considered for elliptic problems, using Gaussian eigenfunctions to stably produce an approximate particular solution.

Some computational aspects of smooth approximation

The paper is devoted to the problem of smooth approximation of data. We are concerned with the exact interpolation of the data at nodes and, at the same time, with the smoothness of the interpolating curve and its derivatives. The same procedure is applied to the fitting of data (smoothing of data), too. The approximating curve is defined as the solution of a variational problem with constraints (interpolation conditions at nodes, data fitting property of the curve) the existence of whose unique solution we prove. Except for the constraints, the formulation of the problem of data approximation is not unique in general as our requirements on the behavior of the approximating curve between the nodes can be very subjective. The smooth approximation of this kind can lead, e.g., to the cubic spline interpolation.We discuss the choice of basis systems for this way of approximation and present results of several 1D numerical examples that show some advantages and drawbacks of smooth approximation.

How effective is the nuclear norm heuristic in solving data approximation problems?

The question in the title is answered empirically by solving instances of three classical problems: fitting a straight line to data, fitting a real exponent to data, and system identification in the errors-in-variables setting. The results show that the nuclear norm heuristic performs worse than alternative problem dependant methods—ordinary and total least squares, Kung's method, and subspace identification. In the line fitting and exponential fitting problems, the globally optimal solution is known analytically, so that the suboptimality of the heuristic methods is quantified.

Marcinkiewicz–Zygmund Inequalities and Polynomial Approximation from Scattered Data on SO(3)

We consider scattered data approximation problems on SO(3). To this end, we construct a new operator for polynomial approximation on the rotation group. This operator reproduces Wigner-D functions up to a given degree and has uniformly bounded L p -operator norm for all 1 ≤ p ≤ ∞. The operator provides a polynomial approximation with the same approximation degree of the best polynomial approximation. Moreover, the operator together with a Markov type inequality for Wigner-D functions enables us to derive scattered data L p -Marcinkiewicz–Zygmund inequalities for these functions for all 1 ≤ p ≤ ∞. As a major application of such inequalities, we consider the stability of the weighted least squares approximation problem on SO(3).