Institute of Numerical Mathematics

The authors systematically investigate two easily computed measures of the effective number of spatial degrees of freedom (ESDOF), or number of independently varying spatial patterns, of a time-varying field of data. The first measure is based on matching the mean and variance of the time series of the spatially integrated squared anomaly of the field to a chi-squared distribution. The second measure, which is equivalent to the first for a long time sample of normally distributed field values, is based on the partitioning of variance between the EOFs. Although these measures were proposed almost 30 years ago, this paper aims to provide a comprehensive discussion of them that may help promote their more widespread use.

Abstract Demands are growing rapidly in the operational prediction and applications communities for forecasts that fill the gap between medium-range weather and long-range or seasonal forecasts. Based on the potential for improved forecast skill at the subseasonal to seasonal time range, the Subseasonal to Seasonal (S2S) Prediction research project has been established by the World Weather Research Programme/World Climate Research Programme. A main deliverable of this project is the establishment of an extensive database containing subseasonal (up to 60 days) forecasts, 3 weeks behind real time, and reforecasts from 11 operational centers, modeled in part on the The Observing System Research and Predictability Experiment (THORPEX) Interactive Grand Global Ensemble (TIGGE) database for medium-range forecasts (up to 15 days). The S2S database, available to the research community since May 2015, represents an important tool to advance our understanding of the subseasonal to seasonal time range that has been considered for a long time as a “desert of predictability.” In particular, this database will help identify common successes and shortcomings in the model simulation and prediction of sources of subseasonal to seasonal predictability. For instance, a preliminary study suggests that the S2S models significantly underestimate the amplitude of the Madden–Julian oscillation (MJO) teleconnections over the Euro-Atlantic sector. The S2S database also represents an important tool for case studies of extreme events. For instance, a multimodel combination of S2S models displays higher probability of a landfall over the islands of Vanuatu 2–3 weeks before Tropical Cyclone Pam devastated the islands in March 2015.

We show that the time-dependent variational principle provides a unifying framework for time-evolution methods and optimization methods in the context of matrix product states. In particular, we introduce a new integration scheme for studying time evolution, which can cope with arbitrary Hamiltonians, including those with long-range interactions. Rather than a Suzuki-Trotter splitting of the Hamiltonian, which is the idea behind the adaptive time-dependent density matrix renormalization group method or time-evolving block decimation, our method is based on splitting the projector onto the matrix product state tangent space as it appears in the Dirac-Frenkel time-dependent variational principle. We discuss how the resulting algorithm resembles the density matrix renormalization group (DMRG) algorithm for finding ground states so closely that it can be implemented by changing just a few lines of code and it inherits the same stability and efficiency. In particular, our method is compatible with any Hamiltonian for which ground-state DMRG can be implemented efficiently. In fact, DMRG is obtained as a special case of our scheme for imaginary time evolution with infinite time step.

Abstract. The Scenario Model Intercomparison Project (ScenarioMIP) defines and coordinates the main set of future climate projections, based on concentration-driven simulations, within the Coupled Model Intercomparison Project phase 6 (CMIP6). This paper presents a range of its outcomes by synthesizing results from the participating global coupled Earth system models. We limit our scope to the analysis of strictly geophysical outcomes: mainly global averages and spatial patterns of change for surface air temperature and precipitation. We also compare CMIP6 projections to CMIP5 results, especially for those scenarios that were designed to provide continuity across the CMIP phases, at the same time highlighting important differences in forcing composition, as well as in results. The range of future temperature and precipitation changes by the end of the century (2081–2100) encompassing the Tier 1 experiments based on the Shared Socioeconomic Pathway (SSP) scenarios (SSP1-2.6, SSP2-4.5, SSP3-7.0 and SSP5-8.5) and SSP1-1.9 spans a larger range of outcomes compared to CMIP5, due to higher warming (by close to 1.5 ∘C) reached at the upper end of the 5 %–95 % envelope of the highest scenario (SSP5-8.5). This is due to both the wider range of radiative forcing that the new scenarios cover and the higher climate sensitivities in some of the new models compared to their CMIP5 predecessors. Spatial patterns of change for temperature and precipitation averaged over models and scenarios have familiar features, and an analysis of their variations confirms model structural differences to be the dominant source of uncertainty. Models also differ with respect to the size and evolution of internal variability as measured by individual models' initial condition ensemble spreads, according to a set of initial condition ensemble simulations available under SSP3-7.0. These experiments suggest a tendency for internal variability to decrease along the course of the century in this scenario, a result that will benefit from further analysis over a larger set of models. Benefits of mitigation, all else being equal in terms of societal drivers, appear clearly when comparing scenarios developed under the same SSP but to which different degrees of mitigation have been applied. It is also found that a mild overshoot in temperature of a few decades around mid-century, as represented in SSP5-3.4OS, does not affect the end outcome of temperature and precipitation changes by 2100, which return to the same levels as those reached by the gradually increasing SSP4-3.4 (not erasing the possibility, however, that other aspects of the system may not be as easily reversible). Central estimates of the time at which the ensemble means of the different scenarios reach a given warming level might be biased by the inclusion of models that have shown faster warming in the historical period than the observed. Those estimates show all scenarios reaching 1.5 ∘C of warming compared to the 1850–1900 baseline in the second half of the current decade, with the time span between slow and fast warming covering between 20 and 27 years from present. The warming level of 2 ∘C of warming is reached as early as 2039 by the ensemble mean under SSP5-8.5 but as late as the mid-2060s under SSP1-2.6. The highest warming level considered (5 ∘C) is reached by the ensemble mean only under SSP5-8.5 and not until the mid-2090s.

Deep neural networks currently demonstrate state-of-the-art performance in\nseveral domains. At the same time, models of this class are very demanding in\nterms of computational resources. In particular, a large amount of memory is\nrequired by commonly used fully-connected layers, making it hard to use the\nmodels on low-end devices and stopping the further increase of the model size.\nIn this paper we convert the dense weight matrices of the fully-connected\nlayers to the Tensor Train format such that the number of parameters is reduced\nby a huge factor and at the same time the expressive power of the layer is\npreserved. In particular, for the Very Deep VGG networks we report the\ncompression factor of the dense weight matrix of a fully-connected layer up to\n200000 times leading to the compression factor of the whole network up to 7\ntimes.\n

The INMCM3.0 climate model has formed the basis for the development of a new climate-model version: the INMCM4.0. It differs from the previous version in that there is an increase in its spatial resolution and some changes in the formulation of coupled atmosphere-ocean general circulation models. A numerical experiment was conducted on the basis of this new version to simulate the present-day climate. The model data were compared with observational data and the INMCM3.0 model data. It is shown that the new model adequately reproduces the most significant features of the observed atmospheric and oceanic climate. This new model is ready to participate in the Coupled Model Intercomparison Project Phase 5 (CMIP5), the results of which are to be used in preparing the fifth assessment report of the Intergovernmental Panel on Climate Change (IPCC).

Using NASA's A‐Train satellite measurements, we evaluate the accuracy of cloud water content (CWC) and water vapor mixing ratio (H 2 O) outputs from 19 climate models submitted to the Phase 5 of Coupled Model Intercomparison Project (CMIP5), and assess improvements relative to their counterparts for the earlier CMIP3. We find more than half of the models show improvements from CMIP3 to CMIP5 in simulating column‐integrated cloud amount, while changes in water vapor simulation are insignificant. For the 19 CMIP5 models, the model spreads and their differences from the observations are larger in the upper troposphere (UT) than in the lower or middle troposphere (L/MT). The modeled mean CWCs over tropical oceans range from ∼3% to ∼15× of the observations in the UT and 40% to 2× of the observations in the L/MT. For modeled H 2 Os, the mean values over tropical oceans range from ∼1% to 2× of the observations in the UT and within 10% of the observations in the L/MT. The spatial distributions of clouds at 215 hPa are relatively well‐correlated with observations, noticeably better than those for the L/MT clouds. Although both water vapor and clouds are better simulated in the L/MT than in the UT, there is no apparent correlation between the model biases in clouds and water vapor. Numerical scores are used to compare different model performances in regards to spatial mean, variance and distribution of CWC and H 2 O over tropical oceans. Model performances at each pressure level are ranked according to the average of all the relevant scores for that level.

Abstract. Soil is currently thought to be a sink for carbon; however, the response of this sink to increasing levels of atmospheric carbon dioxide and climate change is uncertain. In this study, we analyzed soil organic carbon (SOC) changes from 11 Earth system models (ESMs) contributing simulations to the Coupled Model Intercomparison Project Phase 5 (CMIP5). We used a reduced complexity model based on temperature and moisture sensitivities to analyze the drivers of SOC change for the historical and high radiative forcing (RCP 8.5) scenarios between 1850 and 2100. ESM estimates of SOC changed over the 21st century (2090–2099 minus 1997–2006) ranging from a loss of 72 Pg C to a gain of 253 Pg C with a multi-model mean gain of 65 Pg C. Many ESMs simulated large changes in high-latitude SOC that ranged from losses of 37 Pg C to gains of 146 Pg C with a multi-model mean gain of 39 Pg C across tundra and boreal biomes. All ESMs showed cumulative increases in global NPP (11 to 59%) and decreases in SOC turnover times (15 to 28%) over the 21st century. Most of the model-to-model variation in SOC change was explained by initial SOC stocks combined with the relative changes in soil inputs and decomposition rates (R2 = 0.89, p < 0.01). Between models, increases in decomposition rate were well explained by a combination of initial decomposition rate, ESM-specific Q10-factors, and changes in soil temperature (R2 = 0.80, p < 0.01). All SOC changes depended on sustained increases in NPP with global change (primarily driven by increasing CO2). Many ESMs simulated large accumulations of SOC in high-latitude biomes that are not consistent with empirical studies. Most ESMs poorly represented permafrost dynamics and omitted potential constraints on SOC storage, such as priming effects, nutrient availability, mineral surface stabilization, and aggregate formation. Future models that represent these constraints are likely to estimate smaller increases in SOC storage over the 21st century.

Abstract. The Last Glacial Maximum (LGM, ∼ 21 000 years ago) has been a major focus for evaluating how well state-of-the-art climate models simulate climate changes as large as those expected in the future using paleoclimate reconstructions. A new generation of climate models has been used to generate LGM simulations as part of the Paleoclimate Modelling Intercomparison Project (PMIP) contribution to the Coupled Model Intercomparison Project (CMIP). Here, we provide a preliminary analysis and evaluation of the results of these LGM experiments (PMIP4, most of which are PMIP4-CMIP6) and compare them with the previous generation of simulations (PMIP3, most of which are PMIP3-CMIP5). We show that the global averages of the PMIP4 simulations span a larger range in terms of mean annual surface air temperature and mean annual precipitation compared to the PMIP3-CMIP5 simulations, with some PMIP4 simulations reaching a globally colder and drier state. However, the multi-model global cooling average is similar for the PMIP4 and PMIP3 ensembles, while the multi-model PMIP4 mean annual precipitation average is drier than the PMIP3 one. There are important differences in both atmospheric and oceanic circulations between the two sets of experiments, with the northern and southern jet streams being more poleward and the changes in the Atlantic Meridional Overturning Circulation being less pronounced in the PMIP4-CMIP6 simulations than in the PMIP3-CMIP5 simulations. Changes in simulated precipitation patterns are influenced by both temperature and circulation changes. Differences in simulated climate between individual models remain large. Therefore, although there are differences in the average behaviour across the two ensembles, the new simulation results are not fundamentally different from the PMIP3-CMIP5 results. Evaluation of large-scale climate features, such as land–sea contrast and polar amplification, confirms that the models capture these well and within the uncertainty of the paleoclimate reconstructions. Nevertheless, regional climate changes are less well simulated: the models underestimate extratropical cooling, particularly in winter, and precipitation changes. These results point to the utility of using paleoclimate simulations to understand the mechanisms of climate change and evaluate model performance.

The radiative effects from increased concentrations of well‐mixed greenhouse gases (WMGHGs) represent the most significant and best understood anthropogenic forcing of the climate system. The most comprehensive tools for simulating past and future climates influenced by WMGHGs are fully coupled atmosphere‐ocean general circulation models (AOGCMs). Because of the importance of WMGHGs as forcing agents it is essential that AOGCMs compute the radiative forcing by these gases as accurately as possible. We present the results of a radiative transfer model intercomparison between the forcings computed by the radiative parameterizations of AOGCMs and by benchmark line‐by‐line (LBL) codes. The comparison is focused on forcing by CO 2 , CH 4 , N 2 O, CFC‐11, CFC‐12, and the increased H 2 O expected in warmer climates. The models included in the intercomparison include several LBL codes and most of the global models submitted to the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4). In general, the LBL models are in excellent agreement with each other. However, in many cases, there are substantial discrepancies among the AOGCMs and between the AOGCMs and LBL codes. In some cases this is because the AOGCMs neglect particular absorbers, in particular the near‐infrared effects of CH 4 and N 2 O, while in others it is due to the methods for modeling the radiative processes. The biases in the AOGCM forcings are generally largest at the surface level. We quantify these differences and discuss the implications for interpreting variations in forcing and response across the multimodel ensemble of AOGCM simulations assembled for the IPCC AR4.

Abstract We consider simulation of the present day climate with the use of the climate model INM-CM48 in comparison with the result of the previous model INMCM4.0 which used different parameterizations of many physical processes and also in comparison with the model INM-CM5 which uses the same parameterizations, but with better spatial resolution. It is shown that the model INM-CM48 reproduces the modern climate better than the model INMCM4.0 in most indicators.

Abstract. The Last Glacial Maximum (LGM, 21 000 years ago) is one of the suite of paleoclimate simulations included in the current phase of the Coupled Model Intercomparison Project (CMIP6). It is an interval when insolation was similar to the present, but global ice volume was at a maximum, eustatic sea level was at or close to a minimum, greenhouse gas concentrations were lower, atmospheric aerosol loadings were higher than today, and vegetation and land-surface characteristics were different from today. The LGM has been a focus for the Paleoclimate Modelling Intercomparison Project (PMIP) since its inception, and thus many of the problems that might be associated with simulating such a radically different climate are well documented. The LGM state provides an ideal case study for evaluating climate model performance because the changes in forcing and temperature between the LGM and pre-industrial are of the same order of magnitude as those projected for the end of the 21st century. Thus, the CMIP6 LGM experiment could provide additional information that can be used to constrain estimates of climate sensitivity. The design of the Tier 1 LGM experiment (lgm) includes an assessment of uncertainties in boundary conditions, in particular through the use of different reconstructions of the ice sheets and of the change in dust forcing. Additional (Tier 2) sensitivity experiments have been designed to quantify feedbacks associated with land-surface changes and aerosol loadings, and to isolate the role of individual forcings. Model analysis and evaluation will capitalize on the relative abundance of paleoenvironmental observations and quantitative climate reconstructions already available for the LGM.

Abstract. The mid-Holocene (6000 years ago) is a standard time period for the evaluation of the simulated response of global climate models using palaeoclimate reconstructions. The latest mid-Holocene simulations are a palaeoclimate entry card for the Palaeoclimate Model Intercomparison Project (PMIP4) component of the current phase of the Coupled Model Intercomparison Project (CMIP6) – hereafter referred to as PMIP4-CMIP6. Here we provide an initial analysis and evaluation of the results of the experiment for the mid-Holocene. We show that state-of-the-art models produce climate changes that are broadly consistent with theory and observations, including increased summer warming of the Northern Hemisphere and associated shifts in tropical rainfall. Many features of the PMIP4-CMIP6 simulations were present in the previous generation (PMIP3-CMIP5) of simulations. The PMIP4-CMIP6 ensemble for the mid-Holocene has a global mean temperature change of −0.3 K, which is −0.2 K cooler than the PMIP3-CMIP5 simulations predominantly as a result of the prescription of realistic greenhouse gas concentrations in PMIP4-CMIP6. Biases in the magnitude and the sign of regional responses identified in PMIP3-CMIP5, such as the amplification of the northern African monsoon, precipitation changes over Europe, and simulated aridity in mid-Eurasia, are still present in the PMIP4-CMIP6 simulations. Despite these issues, PMIP4-CMIP6 and the mid-Holocene provide an opportunity both for quantitative evaluation and derivation of emergent constraints on the hydrological cycle, feedback strength, and potentially climate sensitivity.

This review is devoted to recent developments in blood flow modelling. It begins with the discussion of blood rheology and its non-Newtonian properties. After that we will present some modelling methods where blood is considered as a heterogeneous fluid composed of plasma and blood cells. Namely, we will describe the method of Dissipative Particle Dynamics and will present some results of blood flow modelling. The last part of this paper deals with one-dimensional global models of blood circulation. We will explain the main ideas of this approach and will present some examples of its application.

Pseudoskeleton approximation and some other problems require the knowledge of sufficiently well-conditioned submatrix in a large-scale matrix. The quality of a submatrix can be measured by modulus of its determinant, also known as volume. In this paper we discuss a search algorithm for the maximum-volume submatrix which already proved to be useful in several matrix and tensor approximation algorithms. We investigate the behavior of this algorithm on random matrices and present some its applications, including maximization of a bivariate functional. 1

Tensors arise naturally in high-dimensional problems in chemistry, financial mathematics, and many other areas. The numerical treatment of such problems is difficult due to the curse of dimensionality: the number of unknowns and the computational complexity grow exponentially with the dimension of the problem. To break the curse of dimensionality, low-parametric representations, or formats, have to be used. In this paper we make use of the TT-format (tensor-train format) which is one of the most effective stable representations of high-dimensional tensors. Basic linear algebra operations in the TT-format are now well developed. Our goal is to provide a "black-box" type of solver for linear systems where both the matrix and the right-hand side are in the TT-format. An efficient DMRG (density matrix renormalization group) method is proposed, and several tricks are employed to make it work. The numerical experiments confirm the effectiveness of our approach.

Snow feedback is expected to amplify global warming caused by increasing concentrations of atmospheric greenhouse gases. The conventional explanation is that a warmer Earth will have less snow cover, resulting in a darker planet that absorbs more solar radiation. An intercomparison of 17 general circulation models, for which perturbations of sea surface temperature were used as a surrogate climate change, suggests that this explanation is overly simplistic. The results instead indicate that additional amplification or moderation may be caused both by cloud interactions and longwave radiation. One measure of this net effect of snow feedback was found to differ markedly among the 17 climate models, ranging from weak negative feedback in some models to strong positive feedback in others.

Abstract. The modeling of paleoclimate, using physically based tools, is increasingly seen as a strong out-of-sample test of the models that are used for the projection of future climate changes. New to the Coupled Model Intercomparison Project (CMIP6) is the Tier 1 Last Interglacial experiment for 127 000 years ago (lig127k), designed to address the climate responses to stronger orbital forcing than the midHolocene experiment, using the same state-of-the-art models as for the future and following a common experimental protocol. Here we present a first analysis of a multi-model ensemble of 17 climate models, all of which have completed the CMIP6 DECK (Diagnostic, Evaluation and Characterization of Klima) experiments. The equilibrium climate sensitivity (ECS) of these models varies from 1.8 to 5.6 ∘C. The seasonal character of the insolation anomalies results in strong summer warming over the Northern Hemisphere continents in the lig127k ensemble as compared to the CMIP6 piControl and much-reduced minimum sea ice in the Arctic. The multi-model results indicate enhanced summer monsoonal precipitation in the Northern Hemisphere and reductions in the Southern Hemisphere. These responses are greater in the lig127k than the CMIP6 midHolocene simulations as expected from the larger insolation anomalies at 127 than 6 ka. New synthesis for surface temperature and precipitation, targeted for 127 ka, have been developed for comparison to the multi-model ensemble. The lig127k model ensemble and data reconstructions are in good agreement for summer temperature anomalies over Canada, Scandinavia, and the North Atlantic and for precipitation over the Northern Hemisphere continents. The model–data comparisons and mismatches point to further study of the sensitivity of the simulations to uncertainties in the boundary conditions and of the uncertainties and sparse coverage in current proxy reconstructions. The CMIP6–Paleoclimate Modeling Intercomparison Project (PMIP4) lig127k simulations, in combination with the proxy record, improve our confidence in future projections of monsoons, surface temperature, and Arctic sea ice, thus providing a key target for model evaluation and optimization.

Abstract. We present results from an ensemble of eight climate models, each of which has carried out simulations of the early Eocene climate optimum (EECO, ∼ 50 million years ago). These simulations have been carried out in the framework of the Deep-Time Model Intercomparison Project (DeepMIP; http://www.deepmip.org, last access: 10 January 2021); thus, all models have been configured with the same paleogeographic and vegetation boundary conditions. The results indicate that these non-CO2 boundary conditions contribute between 3 and 5 ∘C to Eocene warmth. Compared with results from previous studies, the DeepMIP simulations generally show a reduced spread of the global mean surface temperature response across the ensemble for a given atmospheric CO2 concentration as well as an increased climate sensitivity on average. An energy balance analysis of the model ensemble indicates that global mean warming in the Eocene compared with the preindustrial period mostly arises from decreases in emissivity due to the elevated CO2 concentration (and associated water vapour and long-wave cloud feedbacks), whereas the reduction in the Eocene in terms of the meridional temperature gradient is primarily due to emissivity and albedo changes owing to the non-CO2 boundary conditions (i.e. the removal of the Antarctic ice sheet and changes in vegetation). Three of the models (the Community Earth System Model, CESM; the Geophysical Fluid Dynamics Laboratory, GFDL, model; and the Norwegian Earth System Model, NorESM) show results that are consistent with the proxies in terms of the global mean temperature, meridional SST gradient, and CO2, without prescribing changes to model parameters. In addition, many of the models agree well with the first-order spatial patterns in the SST proxies. However, at a more regional scale, the models lack skill. In particular, the modelled anomalies are substantially lower than those indicated by the proxies in the southwest Pacific; here, modelled continental surface air temperature anomalies are more consistent with surface air temperature proxies, implying a possible inconsistency between marine and terrestrial temperatures in either the proxies or models in this region. Our aim is that the documentation of the large-scale features and model–data comparison presented herein will pave the way to further studies that explore aspects of the model simulations in more detail, for example the ocean circulation, hydrological cycle, and modes of variability, and encourage sensitivity studies to aspects such as paleogeography, orbital configuration, and aerosols.

Abstract. A robust and efficient time integrator for dynamical tensor approximation in the tensor train or matrix product state format is presented. The method is based on splitting the projector onto the tangent space of the tensor manifold. The algorithm can be used for updating time-dependent tensors in the given data-sparse tensor train / matrix product state format and for computing an approximate solution to high-dimensional tensor differential equations within this data-sparse format. The formulation, implementation and theoretical properties of the proposed integrator are studied, and numerical experiments with problems from quantum molecular dynamics and with iterative processes in the tensor train format are included. Key words. Tensor train, matrix product state, low-rank approximation, time-varying tensors, tensor differential equations, splitting integrator. AMS subject classifications. 15A18,15A69,65F99,65L05 1. Introduction. In