A literature survey of low-rank tensor approximation techniques

During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors

Preconditioned Low-Rank Methods for High-Dimensional Elliptic PDE Eigenvalue Problems

We consider elliptic PDE eigenvalue problems on a tensorized domain, discretized such that the resulting matrix eigenvalue problem Ax=λx exhibits Kronecker product structure. In particular, we are concerned with the case of high dimensions, where standard approaches to the solution of matrix eigenvalue problems fail due to the exponentially growing degrees of freedom. Recent work shows that this curse of dimensionality can in many cases be addressed by approximating the desired solution vector x in a low-rank tensor format. In this paper, we use the hierarchical Tucker decomposition to develop a low-rank variant of LOBPCG, a classical preconditioned eigenvalue solver. We also show how the ALS and MALS (DMRG) methods known from computational quantum physics can be adapted to the hierarchical Tucker decomposition. Finally, a combination of ALS and MALS with LOBPCG and with our low-rank variant is proposed. A number of numerical experiments indicate that such combinations represent the methods of choic

An Error Analysis Of Galerkin Projection Methods For Linear Systems With Tensor Product Structure

Recent results on the convergence of a Galerkin projection method for the Sylvester equation are extended to more general linear systems with tensor product structure. In the Hermitian positive definite case, explicit convergence bounds are derived for Galerkin projection based on tensor products of rational Krylov subspaces. The results can be used to optimize the choice of shifts for these methods. Numerical experiments demonstrate that the convergence rates predicted by our bounds appear to be sharp

MATHICSE Technical Report : Low-rank tensor approximation for high-order correlation functions of Gaussian random fields

Gaussian random fields are widely used as building blocks for modeling stochastic processes. This paper is concerned with the efficient representation of d-point correlations for such fields, which in turn enables the representation of more general stochastic processes that can be expressed as a function of one (or several) Gaussian random fields. Our representation consists of two ingredients. In the first step, we replace the random field by a truncated Karhunen-Loève expansion and analyze the resulting error. The parameters describing the d-point correlation can be arranged in a tensor, but its storage grows exponentially in d. To avoid this, the second step consists of approximating the tensor in a low-rank tensor format, the so called Tensor Train decomposition. By exploiting the particular structure of the tensor, an approximation algorithm is derived that does not need to form this tensor explicitly and allows to process correlations of order as high as d = 20. The resulting representation is very compact and its use is illustrated for elliptic partial differential equations with random Gaussian forcing terms

Calprotectin (S100A8/S100A9) and Myeloperoxidase: Co-Regulators of Formation of Reactive Oxygen Species

Inflammatory mediators trigger polymorphonuclear neutrophils (PMN) to produce reactive oxygen species (ROS: O2-, H2O2, ∙OH). Mediated by myeloperoxidase in PMN, HOCl is formed, detectable in a chemiluminescence (CL) assay. We have shown that the abundant cytosolic PMN protein calprotectin (S100A8/A9) similarly elicits CL in response to H2O2 in a cell-free system. Myeloperoxidase and calprotectin worked synergistically. Calprotectin-induced CL increased, whereas myeloperoxidase-triggered CL decreased with pH > 7.5. Myeloperoxidase needed NaCl for CL, calprotectin did not. 4-hydroxybenzoic acid, binding ∙OH, almost abrogated calprotectin CL, but moderately increased myeloperoxidase activity. The combination of native calprotectin, or recombinant S100A8/A9 proteins, with NaOCl markedly enhanced CL. NaOCl may be the synergistic link between myeloperoxidase and calprotectin. Surprisingly- and unexplained- at higher concentration of S100A9 the stimulation vanished, suggesting a switch from pro-oxidant to anti-oxidant function. We propose that the ∙OH is predominant in ROS production by calprotectin, a function not described before

Risks to carbon storage from land-use change revealed by peat thickness maps of Peru

This work was funded by NERC (grant ref. NE/R000751/1) to I.T.L., A.H., K.H.R., E.T.A.M., C.M.A., T.R.B., G.D. and E.C.D.G.; Leverhulme Trust (grant ref. RPG-2018-306) to K.H.R., L.E.S.C. and C.E.W.; Gordon and Betty Moore Foundation (grant no. 5439, MonANPeru network) to T.R.B., E.N.H.C. and G.F.; Wildlife Conservation Society to E.N.H.C.; Concytec/British Council/Embajada Británica Lima/Newton Fund (grant ref. 220–2018) to E.N.H.C. and J.D.; Concytec/NERC/Embajada Británica Lima/Newton Fund (grant ref. 001–2019) to E.N.H.C. and N.D.; the governments of the United States (grant no. MTO-069018) and Norway (grant agreement no. QZA-12/0882) to K.H.; and NERC Knowledge Exchange Fellowship (grant ref no. NE/V018760/1) to E.N.H.C.Tropical peatlands are among the most carbon-dense ecosystems but land-use change has led to the loss of large peatland areas, associated with substantial greenhouse gas emissions. To design effective conservation and restoration policies, maps of the location and carbon storage of tropical peatlands are vital. This is especially so in countries such as Peru where the distribution of its large, hydrologically intact peatlands is poorly known. Here field and remote sensing data support the model development of peatland extent and thickness for lowland Peruvian Amazonia. We estimate a peatland area of 62,714 km2 (5th and 95th confidence interval percentiles of 58,325 and 67,102 km2, respectively) and carbon stock of 5.4 (2.6–10.6) PgC, a value approaching the entire above-ground carbon stock of Peru but contained within just 5% of its land area. Combining the map of peatland extent with national land-cover data we reveal small but growing areas of deforestation and associated CO2 emissions from peat decomposition due to conversion to mining, urban areas and agriculture. The emissions from peatland areas classified as forest in 2000 represent 1–4% of Peruvian CO2 forest emissions between 2000 and 2016. We suggest that bespoke monitoring, protection and sustainable management of tropical peatlands are required to avoid further degradation and CO2 emissions.PostprintPeer reviewe

Your Resting Brain CAREs about Your Risky Behavior

Research on the neural correlates of risk-related behaviors and personality traits has provided insight into mechanisms underlying both normal and pathological decision-making. Task-based neuroimaging studies implicate a distributed network of brain regions in risky decision-making. What remains to be understood are the interactions between these regions and their relation to individual differences in personality variables associated with real-world risk-taking.We employed resting state functional magnetic resonance imaging (R-fMRI) and resting state functional connectivity (RSFC) methods to investigate differences in the brain's intrinsic functional architecture associated with beliefs about the consequences of risky behavior. We obtained an individual measure of expected benefit from engaging in risky behavior, indicating a risk seeking or risk-averse personality, for each of 21 participants from whom we also collected a series of R-fMRI scans. The expected benefit scores were entered in statistical models assessing the RSFC of brain regions consistently implicated in both the evaluation of risk and reward, and cognitive control (i.e., orbitofrontal cortex, nucleus accumbens, lateral prefrontal cortex, dorsal anterior cingulate). We specifically focused on significant brain-behavior relationships that were stable across R-fMRI scans collected one year apart. Two stable expected benefit-RSFC relationships were observed: decreased expected benefit (increased risk-aversion) was associated with 1) stronger positive functional connectivity between right inferior frontal gyrus (IFG) and right insula, and 2) weaker negative functional connectivity between left nucleus accumbens and right parieto-occipital cortex.Task-based activation in the IFG and insula has been associated with risk-aversion, while activation in the nucleus accumbens and parietal cortex has been associated with both risk seeking and risk-averse tendencies. Our results suggest that individual differences in attitudes toward risk-taking are reflected in the brain's functional architecture and may have implications for engaging in real-world risky behaviors