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In this work, the task to find a simultaneous low rank approximation of several lowest eigenpairs of a matrix-valued symmetric operator is considered. This problem arises, for instance, in the dens...

Calcolo : a quarterly on numerical analysis and theory of computation 59(2), 24 (2022). doi:10.1007/s10092-022-00462-9, Published by Springer Italia, Milano

There is a growing number of multi-domain genomic datasets for human tumors. Multi-domain data are usually interpreted after separately analyzing single-domain data and integrating the results post hoc. Data fusion techniques allow for the real integration of multi-domain data to ideally improve the tumor classification results for the prognosis and prediction of response to therapy. We have previously described the joint singular value decomposition (jSVD) technique as a means of data fusion. Here, we report on the development of these methods in open source code based on R and Python and on the application of these data fusion methods. The Cancer Genome Atlas (TCGA) Skin Cutaneous Melanoma (SKCM) dataset was used as a benchmark to evaluate the potential of the data fusion approaches to improve molecular classification of cancers in a clinically relevant manner. Our data show that the data fusion approach does not generate classification results superior to those obtained using single-domain data. Data from different domains are not entirely independent from each other, and molecular classes are characterized by features that penetrate different domains. Data fusion techniques might be better suited for response prediction, where they could contribute to the identification of predictive features in a domain-independent manner to be used as biomarkers.

Uveal melanoma (UM) is a rare cancer that is well characterized at the molecular level. Two to four classes have been identified by the analyses of gene expression (mRNA, ncRNA), DNA copy number, DNA-methylation and somatic mutations yet no factual integration of these data has been reported. We therefore applied novel algorithms for data fusion, joint Singular Value Decomposition (jSVD) and joint Constrained Matrix Factorization (jCMF), as well as similarity network fusion (SNF), for the integration of gene expression, methylation and copy number data that we applied to the Cancer Genome Atlas (TCGA) UM dataset. Variant features that most strongly impact on definition of classes were extracted for biological interpretation of the classes. Data fusion allows for the identification of the two to four classes previously described. Not all of these classes are evident at all levels indicating that integrative analyses add to genomic discrimination power. The classes are also characterized by different frequencies of somatic mutations in putative driver genes (GNAQ, GNA11, SF3B1, BAP1). Innovative data fusion techniques confirm, as expected, the existence of two main types of uveal melanoma mainly characterized by copy number alterations. Subtypes were also confirmed but are somewhat less defined. Data fusion allows for real integration of multi-domain genomic data.

The 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.

The treatment of high-dimensional problems such as the Schr\"odinger equation can be approached by concepts of tensor product approximation. We present general techniques that can be used for the treatment of high-dimensional optimization tasks and time-dependent equations, and connect them to concepts already used in many-body quantum physics. Based on achievements from the past decade, entanglement-based methods, -- developed from different perspectives for different purposes in distinct communities already matured to provide a variety of tools -- can be combined to attack highly challenging problems in quantum chemistry. The aim of the present paper is to give a pedagogical introduction to the theoretical background of this novel field and demonstrate the underlying benefits through numerical applications on a text book example. Among the various optimization tasks we will discuss only those which are connected to a controlled manipulation of the entanglement which is in fact the key ingredient of the methods considered in the paper. The selected topics will be covered according to a series of lectures given on the topic "New wavefunction methods and entanglement optimizations in quantum chemistry" at the Workshop on Theoretical Chemistry, 18 - 21 February 2014, Mariapfarr, Austria., Comment: Tutorial-review paper, 107 pages, 44 figures, 277 references. Comments are welcome

That farmers rely on the land for their livelihoods is obvious. The converse, that ecosystem services depend on farmers' behaviors, must also be recognized if agricultural productivity is to be improved. In sub Saharan Africa, the 70% of the population employed in the agricultural sector (Sanchez 2002) is engaged in an on-going 'dialogue' with the agricultural natural resource base. Recently, this conversation has not been going well: per capita food production has remained stagnant for the last 40 years so now 180 million on the continent lack adequate food, a number that has increased by 100% since 1970 (Sanchez 2002). To provide adequate diets to the African population, increases in crop yields of 3.0 to 3.5% y-1 are needed (Reardon et al. 2001), but such increases have not been realized as average maize yields have remained static at 1200 kg ha-1.