A new kernel-based approach for linear system identification

For Example: Modelling Processes in Porous Media with Differential Equations.- For the Beginning: The Finite Difference Method for the Poisson Equation.- The Finite Element Method for the Poisson Equation.- The Finite Element Method for Linear Elliptic Boundary Value Problems of Second Order.- Grid Generation and a posteriori Error Estimation.- Iterative Methods for Systems of Linear Equations.- The Finite Volume Method.- Discretization Methods for Parabolic Initial Boundary Value Problems.- Iterative Methods for Nonlinear Equations.- Discretization Methods for Convection-Dominated Problems.- Appendices.

https://webs.um.es/eliseo/um/uploads/Main/Knabner.pdf

Numerical Methods for Elliptic and Parabolic Partial Differential Equations

A large number of learning algorithms, for example, spectral clustering, kernel Principal Components Analysis and many manifold methods are based on estimating eigenvalues and eigenfunctions of operators defined by a similarity function or a kernel, given empirical data. Thus for the analysis of algorithms, it is an important problem to be able to assess the quality of such approximations. The contribution of our paper is two-fold: 1. We use a technique based on a concentration inequality for Hilbert spaces to provide new much simplified proofs for a number of results in spectral approximation. 2. Using these methods we provide several new results for estimating spectral properties of the graph Laplacian operator extending and strengthening results from von Luxburg et al. (2008).

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On Learning with Integral Operators

We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and computer sciences. The main focus is put onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.

Geometrical structure of Laplacian eigenfunctions

Introduction.- Operators on Networks.- Function Spaces on Networks.- Operator Semigroups.- And Now Something Completely Different: A Crash Course in Cortical Modeling.- Sequilinear Froms and Analytic Semigroups.- Evolution Equations Associated With Self-adjoint Operators.- Symmetry Properties.- Index.

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Semigroup Methods for Evolution Equations on Networks

Sobolev spaces on domains

The problem of boundedness of the fractional maximal operator M α, 0 ≤ α < n, in general local Morrey-type spaces is reduced to the problem of boundedness of the supremal operator in weighted L p -spaces on the cone of non-negative non-decreasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.

Boundedness of the fractional maximal operator in local Morrey-type spaces

The problem of the boundedness of the Riesz potential I α , 0 < α < n, in local Morrey-type spaces is reduced to the boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for the boundedness in local Morrey-type spaces for all admissible values of the parameters. Moreover, for a certain range of the parameters, these sufficient conditions coincide with the necessary ones.

Necessary and Sufficient Conditions for the Boundedness of the Riesz Potential in Local Morrey-type Spaces

Spectral stability of the Neumann Laplacian

The problem of boundedness of the Riesz potential in local Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p -spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.

Victor Burenkov

Papers

Sobolev spaces on domains

Boundedness of the fractional maximal operator in local Morrey-type spaces

Necessary and Sufficient Conditions for the Boundedness of the Riesz Potential in Local Morrey-type Spaces

Spectral stability of the Neumann Laplacian

Boundedness of the Riesz Potential in Local Morrey-Type Spaces