There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors. (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

http://sma.epfl.ch/~anchpcommon/publications/tensorsurvey.pdf

A literature survey of low-rank tensor approximation techniques

nonlinear functional analysis and its applications is available in our book collection an online access to it is set as public so you can get it instantly. Our books collection hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the nonlinear functional analysis and its applications is universally compatible with any devices to read.

Nonlinear Functional Analysis And Its Applications

Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders

Hierarchical tensors can be regarded as a generalisation, preserving many crucial features, of the singular value decomposition to higher-order tensors. For a given tensor product space, a recursive decomposition of the set of coordinates into a dimension tree gives a hierarchy of nested subspaces and corresponding nested bases. The dimensions of these subspaces yield a notion of multilinear rank. This rank tuple, as well as quasi-optimal low-rank approximations by rank truncation, can be obtained by a hierarchical singular value decomposition. For fixed multilinear ranks, the storage and operation complexity of these hierarchical representations scale only linearly in the order of the tensor. As in the matrix case, the set of hierarchical tensors of a given multilinear rank is not a convex set, but forms an open smooth manifold. A number of techniques for the computation of hierarchical low-rank approximations have been developed, including local optimisation techniques on Riemannian manifolds as well as truncated iteration methods, which can be applied for solving high-dimensional partial differential equations. This article gives a survey of these developments. We also discuss applications to problems in uncertainty quantification, to the solution of the electronic Schrodinger equation in the strongly correlated regime, and to the computation of metastable states in molecular dynamics.

Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations

In this paper we discuss the construction, analysis and implementation of iterative schemes for the solution of inverse problems based on total variation regularization. Via different approximations of the nonlinearity we derive three different schemes resembling three well-known methods for nonlinear inverse problems in Hilbert spaces, namely iterated Tikhonov, Levenberg–Marquardt and Landweber. These methods can be set up such that all arising subproblems are convex optimization problems, analogous to those appearing in image denoising or deblurring. We provide a detailed convergence analysis and appropriate stopping rules in the presence of data noise. Moreover, we discuss the implementation of the schemes and the application to distributed parameter estimation in elliptic partial differential equations.

https://iopscience.iop.org/article/10.1088/0266-5611/25/10/105004/pdf

Iterative total variation schemes for nonlinear inverse problems

We consider a framework for the construction of iterative schemes for operator equations that combine low-rank approximation in tensor formats and adaptive approximation in a basis. Under fairly general assumptions, we conduct a rigorous convergence analysis, where all parameters required for the execution of the methods depend only on the underlying infinite-dimensional problem, but not on a concrete discretization. Under certain assumptions on the rates for the involved low-rank approximations and basis expansions, we can also give bounds on the computational complexity of the iteration as a function of the prescribed target error. Our theoretical findings are illustrated and supported by computational experiments. These demonstrate that problems in very high dimensions can be treated with controlled solution accuracy.

/pdf/adaptive-near-optimal-rank-tensor-approximation-for-high-2im0owozlu.pdf

Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations

The Atomic Cluster Expansion (Drautz, Phys. Rev. B 99, 2019) provides a framework to systematically derive polynomial basis functions for approximating isometry and permutation invariant functions, particularly with an eye to modelling properties of atomistic systems. Our presentation extends the derivation by proposing a precomputation algorithm that yields immediate guarantees that a complete basis is obtained. We provide a fast recursive algorithm for efficient evaluation and illustrate its performance in numerical tests. Finally, we discuss generalisations and open challenges, particularly from a numerical stability perspective, around basis optimisation and parameter estimation, paving the way towards a comprehensive analysis of the convergence to a high-fidelity reference model.

Atomic Cluster Expansion: Completeness, Efficiency and Stability

We consider the linear elliptic equation − div( a ∇ u ) = f on some bounded domain D , where a has the form a = exp( b ) with b a random function defined as b ( y ) = ∑ j ≥ 1 y j ψ j where y = ( y j ) ∈ ℝ N are i.i.d. standard scalar Gaussian variables and ( ψ j ) j ≥ 1 is a given sequence of functions in L ∞ ( D ). We study the summability properties of Hermite-type expansions of the solution map y  →  u ( y ) ∈  V  :=  H 0 1 ( D ) , that is, expansions of the form u ( y ) = ∑ ν ∈ ℱ u ν H ν ( y ), where H ν ( y ) = ∏ j ≥1 H ν j ( y j ) are the tensorized Hermite polynomials indexed by the set ℱ of finitely supported sequences of nonnegative integers. Previous results [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826] have demonstrated that, for any 0 p ≤ 1, the l p summability of the sequence ( j ∥ ψ j ∥ L ∞ ) j ≥ 1 implies l p summability of the sequence (∥ u ν ∥ V ) ν ∈ ℱ . Such results ensure convergence rates n − s with s = (1/ p )−(1/2) of polynomial approximations obtained by best n -term truncation of Hermite series, where the error is measured in the mean-square sense, that is, in L 2 (ℝ N ,V,γ ) , where γ is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the l p summability of (∥ u ν ∥ V ) ν ∈ ℱ expressed in terms of the pointwise summability properties of the sequence (| ψ j |) j ≥ 1 . This leads to a refined analysis which takes into account the amount of overlap between the supports of the ψ j . For instance, in the case of disjoint supports, our results imply that, for all 0 p l p summability of (∥ u ν ∥ V ) ν ∈ ℱ follows from the weaker assumption that (∥ ψ j ∥ L ∞ ) j ≥ 1 is l q summable for q  := 2 p /(2− p ) . In the case of arbitrary supports, our results imply that the l p summability of (∥ u ν ∥ V ) ν ∈ ℱ follows from the l p summability of ( j β ∥ ψ j ∥ L ∞ ) j ≥ 1 for some β >1/2 , which still represents an improvement over the condition in [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826]. We also explore intermediate cases of functions with local yet overlapping supports, such as wavelet bases. One interesting observation following from our analysis is that for certain relevant examples, the use of the Karhunen−Loeve basis for the representation of b might be suboptimal compared to other representations, in terms of the resulting summability properties of (∥ u ν ∥ V ) ν ∈ ℱ . While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.

/pdf/sparse-polynomial-approximation-of-parametric-elliptic-pdes-i7z9krficq.pdf

Markus Bachmayr

Papers

Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations

Iterative total variation schemes for nonlinear inverse problems

Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations

Atomic Cluster Expansion: Completeness, Efficiency and Stability

Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients