1,721,031 research outputs found
The cone of completely positive matrices
No full textWe study the cone of completely positive (cp) matrices for the first
interesting case . This is a semialgebraic set, which means that the
polynomial equalities and inequlities that define its boundary can be derived.
We characterize the different loci of this boundary and we examine the two open
sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and
able to compute the cp-factorization even for matrices in the boundary. With
our results, many new example cases can be produced and several insightful
numerical experiments are performed that illustrate the difficulty of the
cp-factorization problem
The Cone of Completely Positive Matrices
No full textAbstract
We study the cone of completely positive (cp) matrices for the first interesting case
n
=
5
. This is a semialgebraic set for which the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem.Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659Georg-August-Universität Göttingen http://dx.doi.org/10.13039/50110000338
Tensor methods for the numerical solution of high-dimensional parametric partial differential equations
No full textThis thesis deals with tensor methods for the numerical solution of parametric partial differential equations (PDEs). Because of their parametric dependence, these differential equations exhibit a high dimensionality. Numerical methods like the Galerkin method quickly reach their limits, since the resulting linear system grows exponentially with the number of parameters. This is known as the curse of dimensionality.
The work can be roughly divided into three parts. The first part covers the theory of parametric PDEs using the leading example of a diffusion equation with parametric coefficients. We consider both a so called affine coefficient as well as a log-normal one. The necessary theoretical foundations for the numerical treatment of these problems are laid out.
The second part is a review of the state of the art of tensor methods. These have gained a lot of recognition in recent years. We introduce a number of tensor decomposition formats and discuss their advantages and disadvantages. In particular, we devise a general tensor representation and unite some concepts from mathematics and quantum physics where these formats have been known for a while. We highlight especially the methods for low rank approximation with tensors, because these will be used for the numerical solution of the parametric diffusion equation.
In the last part, we present our own results. We developed a number of numerical methods in order to facilitate the use of tensor methods especially in the log-normal case. Additionally, we present error estimators, both for the affine and the log-normal case, that allow for the comparison of the errors resulting from the different discretisations. These enable us to formulate an adaptive algorithm for the numerical solution of these equations. Finally, we illustrate the applicability of our methods with some numerical experiments
The cone of completely positive matrices
No full textWe study the cone of completely positive (cp) matrices for the first
interesting case . This is a semialgebraic set, which means that the
polynomial equalities and inequlities that define its boundary can be derived.
We characterize the different loci of this boundary and we examine the two open
sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and
able to compute the cp-factorization even for matrices in the boundary. With
our results, many new example cases can be produced and several insightful
numerical experiments are performed that illustrate the difficulty of the
cp-factorization problem
Adaptive stochastic Galerkin FEM with hierarchical tensor representations
Full text linkThe solution of PDE with stochastic data commonly leads to very high-dimensional algebraic problems, e.g. when multiplicative noise is present. The Stochastic Galerkin FEM considered in this paper then suffers from the curse of dimensionality. This is directly related to the number of random variables required for an adequate representation of the random fields included in the PDE. With the presented new approach, we circumvent this major complexity obstacle by combining two highly efficient model reduction strategies, namely a modern low-rank tensor representation in the tensor train format of the problem and a refinement algorithm on the basis of a posteriori error estimates to adaptively adjust the different employed discretizations. The adaptive adjustment includes the refinement of the FE mesh based on a residual estimator, the problem-adapted stochastic discretization in anisotropic Legendre Wiener chaos and the successive increase of the tensor rank. Computable a posteriori error estimators are derived for all error terms emanating from the discretizations and the iterative solution with a preconditioned ALS scheme of the problem. Strikingly, it is possible to exploit the tensor structure of the problem to evaluate all error terms very efficiently. A set of benchmark problems illustrates the performance of the adaptive algorithm with higher-order FE. Moreover, the influence of the tensor rank on the approximation quality is investigated
Efficient training of Gaussian processes with tensor product structure
No full textAbstract To determine the optimal set of hyperparameters of a Gaussian process based on a large number of training data, both a linear system and a trace estimation problem must be solved. In this paper, we focus on establishing numerical methods for the case where the covariance matrix is given as the sum of possibly multiple Kronecker products, i.e., can be identified as a tensor. As such, we will represent this operator and the training data in the tensor train format. Based on the AME n method and Krylov subspace methods, we derive an efficient scheme for computing the matrix functions required for evaluating the gradient and the objective function in hyperparameter optimization
Riemannian thresholding methods for row-sparse and low-rank matrix recovery
No full textIn this paper, we present modifications of the iterative hard thresholding
(IHT) method for recovery of jointly row-sparse and low-rank matrices. In
particular a Riemannian version of IHT is considered which significantly
reduces computational cost of the gradient projection in the case of rank-one
measurement operators, which have concrete applications in blind deconvolution.
Experimental results are reported that show near-optimal recovery for Gaussian
and rank-one measurements, and that adaptive stepsizes give crucial
improvement. A Riemannian proximal gradient method is derived for the special
case of unknown sparsity
Riemannian thresholding methods for row-sparse and low-rank matrix recovery
No full textIn this paper, we present modifications of the iterative hard thresholding
(IHT) method for recovery of jointly row-sparse and low-rank matrices. In
particular a Riemannian version of IHT is considered which significantly
reduces computational cost of the gradient projection in the case of rank-one
measurement operators, which have concrete applications in blind deconvolution.
Experimental results are reported that show near-optimal recovery for Gaussian
and rank-one measurements, and that adaptive stepsizes give crucial
improvement. A Riemannian proximal gradient method is derived for the special
case of unknown sparsity
tPARAFAC2: tracking evolving patterns in (incomplete) temporal data
No full textAbstract Tensor factorizations have been widely used for the task of uncovering patterns in various domains. Often, the input is time-evolving, shifting the goal to tracking the evolution of the underlying patterns instead. To adapt to this more complex setting, existing methods incorporate temporal regularization but they either have overly constrained structural requirements or lack uniqueness which is crucial for interpretation. In this paper, in order to capture the underlying evolving patterns, we introduce t(emporal)PARAFAC2, which utilizes temporal smoothness regularization on the evolving factors. Previously, Alternating Optimization and Alternating Direction Method of Multipliers-based algorithmic approach has been introduced to fit the PARAFAC2 model to fully observed data. In this paper, we extend this algorithmic framework to the case of partially observed data and use it to fit the tPARAFAC2 model to complete and incomplete datasets with the goal of revealing evolving patterns. Our numerical experiments on simulated datasets demonstrate that tPARAFAC2 can extract the underlying evolving patterns more accurately compared to the state-of-the-art in the presence of high amounts of noise and missing data. Using two real datasets, we also demonstrate the effectiveness of the algorithmic approach in terms of handling missing data and tPARAFAC2 model in terms of revealing evolving patterns. The paper provides an extensive comparison of different approaches for handling missing data within the proposed framework, and discusses both the advantages and limitations of tPARAFAC2 model
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