Preface Introduction 1. Fluid field equations and modal analysis in rigid containers 2. Linear forced sloshing 3. Viscous damping and sloshing suppression devices 4. Weakly nonlinear lateral sloshing 5. Equivalent mechanical models 6. Parametric sloshing (Faraday's waves) 7. Dynamics of liquid sloshing impact 8. Linear interaction of liquid sloshing with elastic containers 9. Nonlinear interaction under external and parametric excitations 10. Interactions with support structures and tuned sloshing absorbers 11. Dynamics of rotating fluids 12. Microgravity sloshing dynamics Bibliography Index.

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Liquid Sloshing Dynamics

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

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.

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The Exponentially Convergent Trapezoidal Rule

We give a short introduction to methods for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods, as the inverses of partial differential operators or as solutions of control problems. The result of the approximation will be so-called hierarchical matrices (or short H-matrices). These matrices form a subset of the set of all matrices and have a data-sparse representation. The essential operations for these matrices (matrix-vector and matrix – matrix multiplication, addition and inversion) can be performed in, up to logarithmic factors, optimal complexity. We give a review of specialised variants of H-matrices, especially of H 2 -matrices, and finally consider applications of the different methods to problems from integral equations, partial differential equations and control theory. q 2003 Elsevier Science Ltd. All rights reserved.

https://www.mis.mpg.de/preprints/2002/preprint2002_18.pdf

Introduction to Hierarchical Matrices with Applications

In the present paper, we discuss the novel concept of super-compressed tensor-structured data formats in high-dimensional applications. We describe the multifolding or quantics-based tensor approximation method of O(dlog N)-complexity (logarithmic scaling in the volume size), applied to the discrete functions over the product index set {1,…,N}⊗d, or briefly N-d tensors of size Nd, and to the respective discretized differential-integral operators in ℝd. As the basic approximation result, we prove that a complex exponential sampled on an equispaced grid has quantics rank 1. Moreover, a Chebyshev polynomial, sampled over a Chebyshev Gauss–Lobatto grid, has separation rank 2 in the quantics tensor format, while for the polynomial of degree m over a Chebyshev grid the respective quantics rank is at most 2m+1. For N-d tensors generated by certain analytic functions, we give a constructive proof of the O(dlog Nlog e−1)-complexity bound for their approximation by low-rank 2-(dlog N) quantics tensors up to the accuracy e>0. In the case e=O(N−α), α>0, our approach leads to the quantics tensor numerical method in dimension d, with the nearly optimal asymptotic complexity O(d/αlog 2e−1). From numerical examples presented here, we observe that the quantics tensor method has proved its value in application to various function related tensors/matrices arising in computational quantum chemistry and in the traditional finite element method/boundary element method (FEM/BEM). The tool apparently works.

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O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling

Collocation methods for Volterra integral and related functional equations.

/pdf/book-review-variational-analysis-in-sobolev-and-bv-spaces-17vtowrsoc.pdf

Book Review: Variational analysis in Sobolev and BV spaces

The class of H -matrices allows an approximate matrix arithmetic with almost linear complexity. In the present paper, we apply the H-matrix technique combined with the Kronecker tensor-product approximation (ef. [2. 20]) to represent the inverse of a discrete elliptic operator in a hypercube 0.1 d = Kd on the ease of a high spatial dimension d. In this data-sparse format, we also represent the operator exponential, the fractional power of an elliptic operator as well as the solution operator of the matrix Lyapunov-Sylvester equation. The complexity of our approximations can be estimated by C(dn log2 n) where N - n2 is the discrete problem size.

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Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems

In earlier papers we developed a method for the data-sparse approximation of the solution operators for elliptic, parabolic, and hyperbolic PDEs based on the Dunford-Cauchy representation to the operator-valued functions of interest combined with the hierarchical matrix approximation of the operator resolvents. In the present paper, we discuss how these techniques can be applied to approximate a hierarchy of the operator-valued functions generated by an elliptic operator £.

http://www.sam.math.ethz.ch/~hiptmair/Seminars/SS04/Papers/GHK03.pdf

Data-sparse approximation to a class of operator-valued functions

We develop a data-sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly P-positive operator [4].

https://www.mis.mpg.de/preprints/2000/preprint2000_42.pdf

Ivan P. Gavrilyuk

Papers

Collocation methods for Volterra integral and related functional equations.

Book Review: Variational analysis in Sobolev and BV spaces

Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems

Data-sparse approximation to a class of operator-valued functions

H-matrix approximation for the operator exponential with applications