Complexity Analysis of a Fast Directional Matrix-Vector Multiplication

We consider a fast, data-sparse directional method to realize matrix-vector products related to point evaluations of the Helmholtz kernel. The method is based on a hierarchical partitioning of the point sets and the matrix. The considered directional multi-level approximation of the Helmholtz kernel can be applied even on high-frequency levels efficiently. We provide a detailed analysis of the almost linear asymptotic complexity of the presented method. Our numerical experiments are in good agreement with the provided theory.Comment: 20 pages, 2 figures, 1 tabl

External validation of a model to predict the survival of patients presenting with a spinal epidural metastasis

The surgical treatment of spinal metastases is evolving. The major problem is the selection of patients who may benefit from surgical treatment. One of the criteria is an expected survival of at least 3 months. A prediction model has been previously developed. The present study has been performed in order to validate externally the model and to demonstrate that this model can be generalized to other institutions and other countries than the Netherlands. Data of 356 patients from five centers in Germany, Spain, Sweden, and the Netherlands who were treated for metastatic epidural spinal cord compression were collected. Hazard ratios in the test population corresponded with those of the developmental population. However, the observed and the expected survival were different. Analysis revealed that the baseline hazard function was significantly different. This tempted us to combine the data and develop a new prediction model. Estimating iteratively, a baseline hazard was composed. An adapted prediction model is presented. External validation of a prediction model revealed a difference in expected survival, although the relative contribution of the specific hazard ratios was the same as in the developmental population. This study emphasized the need to check the baseline hazard function in external validation. A new model has been developed using an estimated baseline hazar

Application of hierarchical matrices for computing the Karhunen-Loève expansion

Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen-Lo`eve expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of sparse hierarchical matrix techniques for this. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield an efficient and fast discretisation of the random fields presented

Comparison of some Reduced Representation Approximations

In the field of numerical approximation, specialists considering highly complex problems have recently proposed various ways to simplify their underlying problems. In this field, depending on the problem they were tackling and the community that are at work, different approaches have been developed with some success and have even gained some maturity, the applications can now be applied to information analysis or for numerical simulation of PDE's. At this point, a crossed analysis and effort for understanding the similarities and the differences between these approaches that found their starting points in different backgrounds is of interest. It is the purpose of this paper to contribute to this effort by comparing some constructive reduced representations of complex functions. We present here in full details the Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM) together with other approaches that enter in the same category

Approximation of integral operators using product-convolution expansions

International audienceWe consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on functions is a computationally intensive problem necessary for many practical problems. We analyze a technique called product-convolution expansion: the operator is locally approximated by a convolution, allowing to design fast numerical algorithms based on the fast Fourier transform. We design various types of expansions, provide their explicit rates of approximation and their complexity depending on the time varying impulse response smoothness. This analysis suggests novel wavelet based implementations of the method with numerous assets such as optimal approximation rates, low complexity and storage requirements as well as adaptivity to the kernels regularity. The proposed methods are an alternative to more standard procedures such as panel clustering, cross approximations, wavelet expansions or hierarchical matrices