GAMM Mitteilungen最新文献

Matrix functions are a central topic of linear algebra, and problems requiring their numerical approximation appear increasingly often in scientific computing. We review various limited-memory methods for the approximation of the action of a large-scale matrix function on a vector. Emphasis is put on polynomial methods, whose memory requirements are known or prescribed a priori. Methods based on explicit polynomial approximation or interpolation, as well as restarted Arnoldi methods, are treated in detail. An overview of existing software is also given, as well as a discussion of challenging open problems.

Efficient numerical linear algebra is a core ingredient in many applications across almost all scientific and industrial disciplines. With this survey we want to illustrate that numerical linear algebra has played and is playing a crucial role in enabling and improving data science computations with many new developments being fueled by the availability of data and computing resources. We highlight the role of various different factorizations and the power of changing the representation of the data as well as discussing topics such as randomized algorithms, functions of matrices, and high-dimensional problems. We briefly touch upon the role of techniques from numerical linear algebra used within deep learning.

We review the recent use of functions of matrices in the analysis of graphs and networks, with special focus on centrality and communicability measures and diffusion processes. Both undirected and directed networks are considered, as well as dynamic (temporal) networks. Computational issues are also addressed.

Data assimilation is a method that combines observations (ie, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three-dimensional and four-dimensional variational data assimilation (3D-Var and 4D-Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimization problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.

In the discussion section of Jung and Staat¹, the statement “a factor of 1000 when the ventricular-like” should be corrected to “a factor of 10 if the ventricular-like”.

The online version has been corrected.

Important conditions in structural analysis are the fulfillment of the balance of linear momentum (vanishing resultant forces) and the balance of angular momentum (vanishing resultant moment), which is not a priori satisfied for arbitrary element formulations. In this contribution, we analyze a mixed least-squares (LS) finite element formulation for linear elasticity with explicit consideration of the balance of angular momentum. The considered stress-displacement (σ − u) formulation is based on the squared L²(ℬ)-norm minimization of the residuals of a first-order system of differential equations. The formulation is constructed by means of two residuals, that is, the balance of linear momentum and the constitutive equation. Motivated by the crucial point of weighting factors within LS formulations, a scale independent formulation is constructed. The displacement approximation is performed by standard Lagrange polynomials and the stress approximation with Raviart-Thomas functions. The latter ansatz functions do not a priori fulfill the symmetry of the Cauchy stress tensor. Therefore, a redundant residual, the balance of angular momentum ((x − x₀) × (divσ + f) + axl[σ − σ^T]), is introduced and the results are discussed from the engineering point of view, especially for coarse mesh discretizations. However, this formulation shows an improvement compared to standard LS σ − u formulations, which is considered here in a numerical study.

In this contribution a phase field model for ductile fracture with linear isotropic hardening is presented. An energy functional consisting of an elastic energy, a plastic dissipation potential and a Griffith type fracture energy constitutes the model. The application of an unaltered radial return algorithm on element level is possible due to the choice of an appropriate coupling between the nodal degrees of freedom, namely the displacement and the crack/fracture fields. The degradation function models the mentioned coupling by reducing the stiffness of the material and the plastic contribution of the energy density in broken material. Furthermore, to solve the global system of differential equations comprising the balance of linear momentum and the quasi-static Ginzburg-Landau type evolution equation, the application of a monolithic iterative solution scheme becomes feasible. The compact model is used to perform 3D simulations of fracture in tension. The computed plastic zones are compared to the dog-bone model that is used to derive validity criteria for K_IC measurements.