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Register a new userIterative representing set selection for nested cross approximation
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A new fast algebraic method for obtaining an $mathcal{H}^2$-approximation of a matrix from its entries is presented. The main idea behind the method is based on the nested representation and the maximum-volume principle to select submatrices in low-rank matrices. A special iterative approach for the computation of so-called representing sets is established. The main advantage of the method is that it uses only the hierarchical partitioning of the matrix and does not require special proxy surfaces to be selected in advance. The numerical experiments for the electrostatic problem and for the boundary integral operator confirm the effectiveness and robustness of the approach. The complexity is linear in the matrix size and polynomial in the ranks. The algorithm is implemented as an open-source Python package that is available online.
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Gaussian processes are powerful, yet analytically tractable models for supervised learning. A Gaussian process is characterized by a mean function and a covariance function (kernel), which are determined by a model selection criterion. The functions to be compared do not just differ in their parametrization but in their fundamental structure. It is often not clear which function structure to choose, for instance to decide between a squared exponential and a rational quadratic kernel. Based on the principle of approximation set coding, we develop a framework for model selection to rank kernels for Gaussian process regression. In our experiments approximation set coding shows promise to become a model selection criterion competitive with maximum evidence (also called marginal likelihood) and leave-one-out cross-validation.
We propose to differentiate a general curvature functional with two different approaches. In the first one we compute the derivative with the tools of shape optimization and in the second one we compute the derivative of a volumic approximation of the functional with respect to a level set function. We show that the two previous approaches give the same result.
The analysis of linear ill-posed problems often is carried out in function spaces using tools from functional analysis. However, the numerical solution of these problems typically is computed by first discretizing the problem and then applying tools from (finite-dimensional) linear algebra. The present paper explores the feasibility of applying the Chebfun package to solve ill-posed problems. This approach allows a user to work with functions instead of matrices. The solution process therefore is much closer to the analysis of ill-posed problems than standard linear algebra-based solution methods.
In this work we analyze an optimized artificial fixed-stress iteration scheme for the numerical approximation of the Biot system modelling fluid flow in deformable porous media. The iteration is based on a prescribed constant artificial volumetric mean total stress in the first half step. The optimization comes through the adaptation of a numerical stabilization or tuning parameter and aims at an acceleration of the iterations. The separated subproblems of fluid flow, written as a mixed first order in space system, and mechanical deformation are discretized by space-time finite element methods of arbitrary order. Continuous and discontinuous discretizations of the time variable are encountered. The convergence of the iteration schemes is proved for the continuous and fully discrete case. The choice of the optimization parameter is identified in the proofs of convergence of the iterations. The analyses are illustrated and confirmed by numerical experiments.
We propose a new cross-conv algorithm for approximate computation of convolution in different low-rank tensor formats (tensor train, Tucker, Hierarchical Tucker). It has better complexity with respect to the tensor rank than previous approaches. The new algorithm has a high potential impact in different applications. The key idea is based on applying cross approximation in the frequency domain, where convolution becomes a simple elementwise product. We illustrate efficiency of our algorithm by computing the three-dimensional Newton potential and by presenting preliminary results for solution of the Hartree-Fock equation on tensor-product grids.
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