Zeros of orthogonal polynomials

Linear Regression Analysis

The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic PDEs Abstractly, it can be viewed as determining a “good” n-dime

/pdf/convergence-rates-for-greedy-algorithms-in-reduced-basis-bziqevzpme.pdf

Convergence Rates for Greedy Algorithms in Reduced Basis Methods

Preface 1. Introduction 2. Background results in representation theory 3. Representations of SO(3) and harmonic analysis on S2 4. Background results in probability and graphical methods 5. Spectral representations 6. Characterizations of isotropy 7. Limit theorems for Gaussian subordinated random fields 8. Asymptotics for the sample power spectrum 9. Asymptotics for sample bispectra 10. Spherical needlets and their asymptotic properties 11. Needlets estimation of power spectrum and bispectrum 12. Spin random fields Appendix Bibliography Index.

/pdf/random-fields-on-the-sphere-representation-limit-theorems-2c4bpjjfjs.pdf

Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications

SUMMARY
In this paper, we propose a method for the numerical solution of linear systems of equations in low rank tensor format. Such systems may arise from the discretisation of PDEs in high dimensions, but our method is not limited to this type of application. We present an iterative scheme, which is based on the projection of the residual to a low dimensional subspace. The subspace is spanned by vectors in low rank tensor format which—similarly to Krylov subspace methods—stem from the subsequent (approximate) application of the given matrix to the residual. All calculations are performed in hierarchical Tucker format, which allows for applications in high dimensions. The mode size dependency is treated by a multilevel method. We present numerical examples that include high-dimensional convection–diffusion equations.Copyright © 2012 John Wiley & Sons, Ltd.

/pdf/a-projection-method-to-solve-linear-systems-in-tensor-format-4nj1d67lh0.pdf

A projection method to solve linear systems in tensor format

The easy path wavelet transform (EPWT) has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data, in particular from image data. The EPWT is a locally adaptive wavelet transform. It works along pathways through the array of function values and exploits the local correlations of the given data in a simple appropriate manner. However, the EPWT suffers from its adaptivity costs that arise from the storage of path vectors. In this paper, we propose a new hybrid method for image approximation that exploits the advantages of the usual tensor product wavelet transform for the representation of smooth images and uses the EPWT for an efficient representation of edges and texture. Numerical results show the efficiency of this procedure.

/pdf/a-new-hybrid-method-for-image-approximation-using-the-easy-31qai1ho9t.pdf

A New Hybrid Method for Image Approximation Using the Easy Path Wavelet Transform

Aims. A method for constructing new uniform grids on the sphere is given. Methods. We define a bijection in R 2 , which maps squares onto discs and preserves areas. Then we use this bijection, combined with Lambert azimuthal projection, for lifting uniform grids from the square to the sphere. Results. We can obtain uniform spherical grids that allow a hierarchical data manipulation and have an isolatitudinal distribution of cells. Compared with HEALPix grids, nowadays the most used in astronomy and astrophysics, our grids have the advantage of allowing easier implementation, and in addition one can move approximating functions from the square to the sphere by a simple technique.

/pdf/new-uniform-grids-on-the-sphere-3vwzx9jgxl.pdf

New uniform grids on the sphere

Wavelet transform on manifolds: Old and new approaches

In computational work, data sets must often be represented on the surface of a sphere or inside a ball, requiring uniform grids. We construct a new volume-preserving projection between a cube and the set of unit quaternions. The projection consists of two steps: an equal-volume mapping from the cube to the unit ball, followed by an inverse generalized Lambert projection to either of the two unit quaternion hemispheres. The new projection provides a one-to-one mapping between a grid in the cube and elements of the special orthogonal group SO(3), i.e., 3D rotations. We provide connections to other rotation representation schemes, including the Rodrigues–Frank vector and the homochoric parameterizations, and illustrate the new mapping through example applications relevant to texture analysis.

https://iopscience.iop.org/article/10.1088/0965-0393/22/7/075013/pdf

A new method of constructing a grid in the space of 3D rotations and its applications to texture analysis

http://na.math.uni-goettingen.de/pdf/cubsphere3.pdf

Daniela Roşca

Papers

A New Hybrid Method for Image Approximation Using the Easy Path Wavelet Transform

New uniform grids on the sphere

Wavelet transform on manifolds: Old and new approaches

A new method of constructing a grid in the space of 3D rotations and its applications to texture analysis

Uniform spherical grids via equal area projection from the cube to the sphere