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When tapping a blast furnace, a break-through of gas through the taphole at the end of a cast needs to be prevented, both to preserve a healthy state of the taphole and to prevent gas and dust from escaping into the environment. In this paper, two acoustic techniques are presented that can be used to prevent gas emissions from the taphole at the end of a cast. The first approach is by using the spectral analysis of the recorded data, the second approach uses a neural network to recognize the popping sounds that announce the slag–gas interface is approaching the taphole level. It was found that with both methods the end of the cast is detected about 4–2 min before the cast is ended.

This is the second in a series of papers which intend to explore conceptual ways of distinguishing between families in the q-Askey scheme and uniform ways of parametrizing the families. For a system of polynomials pn (x)in the -Askey scheme satisfying Lpn = hnpn with a second order q-difference operator the q-Zhedanov algebra is the algebra generated by operators L and X (multiplication by x). It has two relations in which essentially five coefficients occur. Vanishing of one or more of the coefficients corresponds to a subfamily or limit family of the Askey–Wilson polynomials. An arrow from one family to another means that in the latter family one more coefficient vanishes. This yields the q-Zhedanov scheme given in this paper. The q-hypergeometric expression of pn (x) can be interpreted as an expansion of pn (x) in terms of certain Newton polynomials. In our previous paper (Contemporary Math. 780) we used Verde-Star’s clean parametrization of such expansions and we obtained a q-Verde-Star scheme, where vanishing of one or more of these parameters corresponds to a subfamily or limit family. The actions of the operators L and X on the Newton polynomials can be expressed in terms of the Verde-Star parameters, and thus the coefficients for the q-Zhedanov algebra can be expressed in terms of these parameters. There are interesting differences between the q-Verde-Star scheme and the q-Zhedanov scheme, which are discussed in the paper.

In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker–Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations.

This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a regularized least squares formulation in a continuous infinite-dimensional setting that is based on stable variational time-space formulations of the parabolic partial differential equation. This allows us to derive a priori as well as a posteriori error bounds for the recovered states with respect to a certain reference solution. In these bounds the regularization parameter is disentangled from the underlying discretization. An important ingredient for the derivation of a posteriori bounds is the construction of suitable Fortin operators which allow us to control oscillation errors stemming from the discretization of dual norms. Moreover, the variational framework allows us to contrive preconditioners for the discrete problems whose application can be performed in linear time, and for which the condition numbers of the preconditioned systems are uniformly proportional to that of the regularized continuous problem. In particular, we provide suitable stopping criteria for the iterative solvers based on the a posteriori error bounds. The presented numerical experiments quantify the theoretical findings and demonstrate the performance of the numerical scheme in relation with the underlying discretization and regularization.

The refractory thickness of the blast furnace hearth is a critical parameter for the safe operation of a blast furnace. A replacement of the refractory comes with high costs and, therefore, unnecessary repairs should be avoided. A promising method to estimate the refractory thickness is the impact-echo method. For the interpretation of impact-echo, often, the peaks in the spectrum of the measured signal are associated with the echo from an interface. In this paper, it is shown that this method should be used with care when applied to multi-layered structures. An alternative approach is presented for the calculation of the thickness of multi-layered structures. This method solves the wave equation based on transmission line theory. The formula is checked against 2D finite element analysis and for simple layered structures, good agreement is found. Three cases are simulated using finite element models. Both the impact-echo method and the proposed method overestimate the refractory thickness.

Nonsymmetric interpolation Laurent polynomials in $n$ variables are introduced, with the interpolation points depending on $q$ and on a $n$-tuple of parameters $\tau=(\tau_1,\ldots,\tau_n)$. When $\tau_i=st^{n-i}$ Okounkov's $3$-parameter $BC_n$-type interpolation Macdonald polynomials are recovered from the nonsymmetric interpolation Laurent polynomials through Hecke algebra symmetrisation with respect to a type $C_n$ Hecke algebra action. In the appendix we give some conjectures about extra vanishing, based on Mathematica computations in rank two., Comment: 32 pages, 9 figures; v5: thoroughly revised version, including a new introduction. To appear in Transformation Groups

In atmospheric modeling, superparameterization (SP) has gained popularity as a technique to improve cloud and convection representations in large-scale models by coupling them locally to cloud-resolving models. We show how the different representations of cloud water in the local and the global models in SP lead to a suppression of cloud advection and ultimately to a systematic underrepresentation of the cloud amount in the large-scale model. We demonstrate this phenomenon in a regional SP experiment with the global model OpenIFS coupled to the local model Dutch Atmospheric Large Eddy Simulation, as well as in an idealized setup, where the large-scale model is replaced by a simple advection scheme. As a starting point for mitigating the problem of suppressed cloud advection, we propose a scheme where the spatial variability of the local model's total water content is enhanced in order to match the global model's cloud condensate amount. The proposed scheme enhances the cloud condensate amount in the test cases, however a large discrepancy remains, caused by rapid dissipation of the clouds added by the proposed scheme.

Background: Anterior and posterior pelvic tilt appears to play a role in total hip arthroplasty (THA) stability. When changing from the standing to the sitting position, the pelvis typically rotates posteriorly while the hips flex and this affects the femoro-acetabular positions. This case-control study compares changes in 3-D acetabular cup orientation during functional pelvic tilt between posterior THA dislocations vs stable THAs. Methods: Standing and sitting 3-D cup orientation was compared between fifteen posterior dislocations vs 233 prospectively followed stable THAs. 3-D cup orientation was calculated using previously validated trigonometric algorithms on biplanar radiographs. Those algorithms combine the angles in the three anatomical planes (coronal inclination, transverse version, and sagittal ante-inclination) in the standing position with the change in sagittal pelvic tilt from standing to sitting to calculate the 3-D orientation in the sitting position. Results: The standing cup orientation of the dislocated THAs was only characterized by a lower coronal inclination (P =.039). Compared with the controls, from standing to sitting, they showed less posterior pelvic tilt (P

We prove that all entire transcendental entire functions have infinite topological entropy., Comment: 13 pages

In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by Führer and Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven. In the present work, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated. The proof of the latter easily extends to a large class of least-squares formulations.

Background: Sagittal pelvic dynamics mainly consist of the pelvis rotating anteriorly or posteriorly while the hips flexes, and this affects the femoroacetabular or THA configuration. Thus far, it is unknown how the acetabular cup of the THA in the individual patient reorients with changing sagittal pelvic dynamics.Questions/purposes: The aim of this study was to validate a method that establishes the three-dimensional (3-D) acetabular cup orientation with changing sagittal pelvic dynamics and describe these changes during functional pelvic dynamics.Methods: A novel trigonometric mathematical model, which was incorporated into an easy-to-use tool, was tested. The model connected sagittal tilt, transverse version, and coronal inclination of the acetabular cup during sagittal pelvic tilt. Furthermore, the effect of sagittal pelvic tilt on the 3-D reorientation of acetabular cups was simulated for cups with different initial positions. Twelve pelvic CT images of patients who underwent THA were taken and rotated around the hip axis to different degrees of anterior and posterior sagittal pelvic tilt (± 30°) to simulate functional pelvic tilt in various body positions. For each simulated pelvic tilt, the transverse version and coronal inclination of the cup were manually measured and compared with those measured in a mathematical model in which the 3-D cup positions were calculated. Next, this model was applied to different acetabular cup positions to simulate the effect of sagittal pelvic dynamics on the 3-D orientation of the acetabular cup in the coronal and transverse plane. After pelvic tilt was applied, the intraclass correlation coefficients of 108 measured and calculated coronal and transverse cup orientation angles were 0.963 and 0.990, respectively, validating the clinical use of the mathematical model.Results: The changes in 3-D acetabular cup orientation by functional pelvic tilt differed substantially between cups with different initial positions; the change in transverse version was much more pronounced in cups with low coronal inclination (from 50° to -29°) during functional pelvic tilt than in cups with a normal coronal inclination (from 39° to -11°) or high coronal inclination (from 31° to 2°). However, changes in coronal inclination were more pronounced in acetabular cups with high transverse version.Conclusion: Using a simple algorithm to determine the dynamic 3-D reorientation of the acetabular cup during functional sagittal pelvic tilt, we demonstrated that the 3-D effect of functional pelvic tilt is specific to the initial acetabular cup orientation and thus per THA patient.Clinical Relevance: Future studies concerning THA (in)stability should not only include the initial acetabular cup orientation, but also they need to incorporate the effect of sagittal pelvic dynamics on the individual 3-D acetabular cup orientation. Clinicians can also use the developed tool, www.3d-hip.com, to calculate the acetabular cup's orientation in other instances, such as for patients with spinopelvic imbalance.

The study of the dynamics of an holomorphic map near a fixed point is a central topic in complex dynamical systems. In this paper we will consider the corresponding random setting: given a probability measure $\nu$ with compact support on the space of germs of holomorphic maps fixing the origin, we study the compositions $f_n\circ\cdots\circ f_1$, where each $f_i$ is chosen independently with probability $\nu$. As in the deterministic case, the stability of the family of the random iterates is mostly determined by the linear part of the germs in the support of the measure. A particularly interesting case occurs when all Lyapunov indices vanish, in which case stability implies simultaneous linearizability of all germs in $supp(\nu)$., Comment: 30 pages

Uniform preconditioners for operators of negative order discretized by (dis)continuous piecewise polynomials of any order are constructed from a boundedly invertible operator of opposite order discretized by continuous piecewise linears. Besides the cost of the application of the latter discretized operator, the other cost of the preconditioner scales linearly with the number of mesh cells. Compared to earlier proposals, the preconditioner has the following advantages: It does not require the inverse of a non-diagonal matrix; it applies without any mildly grading assumption on the mesh; and it does not require a barycentric refinement of the mesh underlying the trial space.

We prove that entire transcendental holomorphic functions with an omitted value have infinite entropy. A proof for general transcendental entire functions will be given in an upcoming paper., Comment: 4 pages

We analyze Galerkin discretizations of a new well-posed mixed space–time variational formulation of parabolic partial differential equations. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space–time discretization methods introduced by Andreev (2013, Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal., 33, 242–260) and by Steinbach (2015, Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math., 15, 551–566).

For the Poisson problem in two dimensions, posed on a domain partitioned into axis-aligned rectangles with up to one hanging node per edge, we envision an efficient error reduction step in an instance-optimal $hp$-adaptive finite element method. Central to this is the problem: Which increase in local polynomial degree ensures $p$-robust contraction of the error in energy norm? We reduce this problem to a small number of saturation problems on the reference square, and provide strong numerical evidence for their solution., 18 pages, with one table and 4 figures

The large-scale ocean circulation is strongly influenced by mesoscale turbulent eddies. Mesoscale ocean eddies exist on spatial scales in an approximate range of 10-100 km. Therefore, global climate models need grid resolutions smaller than 10 km in their ocean component in order to directly resolve these turbulent motions. Due to computational limitations, such high resolution is still infeasible in current climate models, and the effects of turbulent eddies must be parameterized. In this thesis we propose novel stochastic methodologies for eddy parameterization. A key assumption for the developed methodologies is the availability of reference data, albeit in small amounts relative to the problem dimensionality. This reference data can come from such places as real-life observations or highly resolving, but restricted, simulations. This enables us to propose data-driven stochastic parameterizations that are informed by empirical references in addition to physical insight. We use the stochastic parameterizations to drive reduced models that, otherwise, would omit most mesoscale features. We show that these stochastic models accurately reproduce both the key statistical and physical criteria of the references. Additionally, this thesis presents solutions to define our methodologies in a memory-efficient way for global ocean model applications.

In the present work, we consider weakly-singular integral equations arising from linear second-order elliptic PDE systems with constant coefficients, including, e.g. linear elasticity. We introduce a general framework for optimal convergence of adaptive Galerkin BEM. We identify certain abstract conditions for the underlying meshes, the corresponding mesh-refinement strategy, and the ansatz spaces that guarantee that the weighted-residual error estimator is reliable and converges at optimal algebraic rate if used within an adaptive algorithm. These conditions are satisfied, e.g. for discontinuous piecewise polynomials on simplicial meshes as well as certain ansatz spaces used for isogeometric analysis. Technical contributions include the localization of (non-local) fractional Sobolev norms and local inverse estimates for the (non-local) boundary integral operators associated to the PDE system.

We study the computational complexity of approximating the partition function of the ferromagnetic Ising model with the external field parameter $\lambda$ on the unit circle in the complex plane. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics, but have also been key in the recent deterministic approximation scheme for all $|\lambda|\neq 1$ by Liu, Sinclair, and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle, and on the tantalising question of what happens around $\lambda=1$, where on one hand the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability, and on the other hand the presence of Lee-Yang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point $\lambda=1$, and more generally on the entire unit circle. For an integer $\Delta\geq 3$ and edge interaction parameter $b\in (0,1)$ we show #P-hardness for approximating the partition function on graphs of maximum degree $\Delta$ on the arc of the unit circle where the Lee-Yang zeros are dense. This result contrasts with known approximation algorithms when $|\lambda|\neq 1$ or when $\lambda$ is in the complementary arc around $1$ of the unit circle. Our work thus gives a direct connection between the presence/absence of Lee-Yang zeros and the tractability of efficiently approximating the partition function on bounded-degree graphs., Comment: 40 pages, 1 figure. We have included a new result for the case $b\in [1-2/\Delta,1)$. This essentially gives a complete picture of the complexity of the problem. An extended abstract has been presented at SODA 2021

International audience; This study considers an efficient method for the estimation of quantiles associated to very small levels of probability (up to O(10−9)), where the scalar performance function J is complex (eg, output of an expensive‐to‐run finite element model), under a probability measure that can be recast as a multivariate standard Gaussian law using an isoprobabilistic transformation. A surrogate‐based approach (Gaussian Processes) combined with adaptive experimental designs allows to iteratively increase the accuracy of the surrogate while keeping the overall number of J evaluations low. Direct use of Monte‐Carlo simulation even on the surrogate model being too expensive, the key idea consists in using an importance sampling method based on an isotropic‐centered Gaussian with large standard deviation permitting a cheap estimation of small quantiles based on the surrogate model. Similar to AK‐MCS as presented in the work of Schöbi et al., (2016), the surrogate is adaptively refined using a parallel infill criterion of an algorithm suitable for very small failure probability estimation. Additionally, a multi‐quantile selection approach is developed, allowing to further exploit high‐performance computing architectures. We illustrate the performances of the proposed method on several two to eight‐dimensional cases. Accurate results are obtained with less than 100 evaluations of J on the considered benchmark cases.

The seminal Lee-Yang theorem states that for any graph the zeros of the partition function of the ferromagnetic Ising model lie on the unit circle in $\mathbb C$. In fact the union of the zeros of all graphs is dense on the unit circle. In this paper we study the location of the zeros for the class of graphs of bounded maximum degree $d\geq 3$, both in the ferromagnetic and the anti-ferromagnetic case. We determine the location exactly as a function of the inverse temperature and the degree $d$. An important step in our approach is to translate to the setting of complex dynamics and analyze a dynamical system that is naturally associated to the partition function., 24 pages, 3 figures. Made a number of small clarifications, corrections and changes in notation. Results remain unchanged. To appear in the Journal of the London Mathematical Society

Diffusion models with constant boundaries and constant drift function have been successfully applied to model phenomena in a wide range of areas in psychology. In recent years, more complex models with time-dependent boundaries and space-time-dependent drift functions have gained popularity. One obstacle to the empirical and theoretical evaluation of these models is the lack of simple and efficient numerical algorithms for computing their first-passage time distributions. In the present work we use a known series expansion for the first-passage time distribution for models with constant drift function and constant boundaries to simplify the Kolmogorov backward equation for models with time-dependent boundaries and space-time-dependent drift functions. We show how a simple Crank–Nicolson scheme can be used to efficiently solve the simplified equation.

Many countries are currently dealing with the COVID-19 epidemic and are searching for an exit strategy such that life in society can return to normal. To support this search, computational models are used to predict the spread of the virus and to assess the efficacy of policy measures before actual implementation. The model output has to be interpreted carefully though, as computational models are subject to uncertainties. These can stem from, e.g., limited knowledge about input parameters values or from the intrinsic stochastic nature of some computational models. They lead to uncertainties in the model predictions, raising the question what distribution of values the model produces for key indicators of the severity of the epidemic. Here we show how to tackle this question using techniques for uncertainty quantification and sensitivity analysis. We assess the uncertainties and sensitivities of four exit strategies implemented in an agent-based transmission model with geographical stratification. The exit strategies are termed Flattening the Curve, Contact Tracing, Intermittent Lockdown and Phased Opening. We consider two key indicators of the ability of exit strategies to avoid catastrophic health care overload: the maximum number of prevalent cases in intensive care (IC), and the total number of IC patient-days in excess of IC bed capacity. Our results show that uncertainties not directly related to the exit strategies are secondary, although they should still be considered in comprehensive analysis intended to inform policy makers. The sensitivity analysis discloses the crucial role of the intervention uptake by the population and of the capability to trace infected individuals. Finally, we explore the existence of a safe operating space. For Intermittent Lockdown we find only a small region in the model parameter space where the key indicators of the model stay within safe bounds, whereas this region is larger for the other exit strategies., Author summary Many countries are currently dealing with the COVID-19 epidemic and are looking for an exit strategy such that life in society can return to normal. For that purpose computational models are used to predict the spread of the virus and to assess the efficacy of policy measures before putting them into practice. These models are subject to uncertainties (due to, for instance, limited knowledge of the parameter values), which can lead to a large variability in model predictions. It is therefore fundamental to assess which range of values a model produces for key indicators of the severity of the epidemic. We present here the results of the uncertainty and sensitivity analysis of four exit strategies simulated with an individual-based model of the COVID-19 transmission. As key indicators of the severity of the pandemic we consider the maximum number of cases in intensive care and the total number of intensive care patient-days in excess. Our results show the crucial role of the intervention uptake by the population, of the reduction in the level of transmission by intervention and of the capability to trace infected individuals.

In simulations of multiscale dynamical systems, not all relevant processes can be resolved explicitly. Taking the effect of the unresolved processes into account is important, which introduces the need for paramerizations. We present a machine-learning method, used for the conditional resampling of observations or reference data from a fully resolved simulation. It is based on the probabilistic classiffcation of subsets of reference data, conditioned on macroscopic variables. This method is used to formulate a parameterization that is stochastic, taking the uncertainty of the unresolved scales into account. We validate our approach on the Lorenz 96 system, using two different parameter settings which are challenging for parameterization methods., 27 pages, 17 figures. Submitted

Copyright © 2021 The Author(s). The VECMA toolkit enables automated Verification, Validation and Uncertainty Quantification (VVUQ) for complex applications that can be deployed on emerging exascale platforms and provides support for software applications for any domain of interest. The toolkit has four main components including EasyVVUQ for VVUQ workflows, FabSim3 for automation and tool integration, MUSCLE3 for coupling multiscale models and QCG tools to execute application workflows on high performance computing (HPC). A more recent addition to the VECMAtk is EasySurrogate for various types of surrogate methods. In this paper, we present five tutorials from different application domains that apply these VECMAtk components to perform uncertainty quantification analysis, use surrogate models, couple multiscale models and execute sensitivity analysis on HPC. This paper aims to provide hands-on experience for practitioners aiming to test and contrast with their own applications. This work was supported by the VECMA project, which has received funding from the European Union Horizon 2020 research and innovation programme under grant agreement No. 800925. The development of MUSCLE3 and its respective description was supported by the Netherlands eScience Center and NWO under the e-MUSC project. The development of ISR3D was supported by the InSilc project and the In Silico World (ISW) project (European Union Horizon 2020 research and innovation programme grant agreements #777119 and #101016503 respectively). The calculations were performed in the Poznan Supercomputing and Networking Center.

We consider a minimal residual discretization of a simultaneous space-time variational formulation of parabolic evolution equations. Under the usual ‘LBB’ stability condition on pairs of trial- and test spaces we show quasi-optimality of the numerical approximations without assuming symmetry of the spatial part of the differential operator. Under a stronger LBB condition we show error estimates in an energy-norm that are independent of this spatial differential operator.

Op 11 mei 2020 overleed in Amsterdam op 84-jarige leeftijd Aïda Beatrijs Paalman-de Miranda, emeritus hoogleraar zuivere wiskunde aan de Universiteit van Amsterdam. Zij kwam als 17-jarige uit Paramaribo naar Nederland om in Amsterdam wiskunde te studeren en ontwikkelde zich tot prominent lid van het Mathematisch Instituut van de Universiteit van Amsterdam, de voorloper van het huidige Korteweg-de Vries Instituut voor Wiskunde. Haar oud-collega’s Peter van Emde Boas, Jan van Mill en Jan Wiegerinck herdenken haar

We present the VECMA toolkit (VECMAtk), a flexible software environment for single and multiscale simulations that introduces directly applicable and reusable procedures for verification, validation (V&V), sensitivity analysis (SA) and uncertainty quantification (UQ). It enables users to verify key aspects of their applications, systematically compare and validate the simulation outputs against observational or benchmark data, and run simulations conveniently on any platform from the desktop to current multi-petascale computers. In this sequel to our paper on VECMAtk which we presented last year, we focus on a range of functional and performance improvements that we have introduced, cover newly introduced components, and applications examples from seven different domains such as conflict modelling and environmental sciences. We also present several implemented patterns for UQ/SA and V&V, and guide the reader through one example concerning COVID-19 modelling in detail., 23 pages, 10 figures

As a computationally attractive alternative for global large eddy simulations (LESs), we investigate the possibility of using comprehensive three‐dimensional LESs as a superparameterization that can replace all traditional parameterizations of atmospheric processes that are currently used in global models. We present the technical design for a replacement of the parameterization for clouds, convection, and turbulence of the global atmospheric model of the European Centre for Medium‐Range Weather Forecasts by the Dutch Atmospheric Large Eddy Simulation model. The model coupling consists of bidirectional data exchange between the global model and the high‐resolution LES models embedded within the columns of the global model. Our setup allows for selective superparameterization, that is, for applying superparameterization in local regions selected by the user, while keeping the standard parameterization of the global model intact outside this region. Computationally, this setup can result in major geographic load imbalance, because of the large difference in computational load between superparameterized and nonsuperparameterized model columns. To resolve this issue, we use a modular design where the local and global models are kept as distinct model codes and organize the model coupling such that all the local models run in parallel, separate from the global model. First simulation results, employing this design, demonstrate the potential of our approach.

The reliability of the transmission grid is challenged by the integration of intermittent renewable energy sources into the grid. For model-based reliability studies, it is important to have suitable models available of renewable energy sources like wind and solar power. In this study, we investigate to what extent the power output of wind farms can be modeled with discrete Hidden Markov Models (HMMs). The parameters of the HMMs are inferred from measurement data from multiple turbines in a wind farm. We use these models both for individual turbine output and for total aggregated power output of multiple turbines. When modeling individual turbine output, the hidden process in the HMM is instrumental in capturing the dependencies between the output of the different turbines. It is important to account for these dependencies in order to correctly capture the upper quantiles (90%, 95%, 99%) of the distribution of the wind farm aggregated power output. We show that despite their simple structure, HMMs are able to reproduce important features of the power output of wind farms. This opens up possibilities to model and analyze these features with methods and techniques stemming from the field of Markov models and stochastic processes.

A conjecture of Sokal (2001) regarding the domain of non-vanishing for independence polynomials of graphs, states that given any natural number $\Delta \ge 3$, there exists a neighborhood in $\mathbb C$ of the interval $[0, \frac{(\Delta-1)^{\Delta-1}}{(\Delta-2)^{\Delta}})$ on which the independence polynomial of any graph with maximum degree at most $\Delta$ does not vanish. We show here that Sokal's Conjecture holds, as well as a multivariate version, and prove optimality for the domain of non-vanishing. An important step is to translate the setting to the language of complex dynamical systems., Comment: We have updated the file partly based on some comments from a referee. The file is now 20 pages and contains one figure. Accepted in Michigan Mathematical Journal

A new approach to parameterizing subgrid-scale processes is proposed: The impact of the unresolved dynamics on the resolved dynamics (i.e., the eddy forcing) is represented by a series expansion in dynamical spatial modes that stem from the energy budget of the resolved dynamics. It is demonstrated that the convergence in these so-called energy modes is faster by orders of magnitude than the convergence in Fourier-type modes. Moreover, a novel way to test parameterizations in models is explored. The resolved dynamics and the corresponding instantaneous eddy forcing are defined via spatial filtering that accounts for the representation error of the equations of motion on the low-resolution model grid. In this way, closures can be tested within the high-resolution model, and the effects of different parameterizations related to different energy pathways can be isolated. In this study, the focus is on parameterizations of the baroclinic energy pathway. The corresponding standard closure in ocean models, the Gent–McWilliams (GM) parameterization, is also tested, and it is found that the GM field acts like a stabilizing direction in phase space. The GM field does not project well on the eddy forcing and hence fails to excite the model’s intrinsic low-frequency variability, but it is able to stabilize the model.

A 0/1-simplex is the convex hull of n+1 affinely independent vertices of the unit n-cube I^n. It is nonobtuse if none its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0/1-simplices in I^n can be represented by 0/1-matrices P of size n x n whose Gramians have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part D of the transposed inverse of P is doubly stochastic and has the same support as P. The negated negative part C of P^-T is strictly row-substochastic and its support is complementary to that of D, showing that P^-T=D-C has no zero entries and has positive row sums. As a consequence, for each facet F of an acute 0/1-facet S there exists at most one other acute 0/1-simplex T in I^n having F as a facet. We call T the acute neighbor of S at F. If P represents a 0/1-simplex that is merely nonobtuse, P^-T can have entries equal to zero. Its positive part D is still doubly stochastic, but its support may be strictly contained in the support of P. This allows P to be partly decomposable. In theory, this might cause a nonobtuse 0/1-simplex S to have several nonobtuse neighbors at each of its facets. Next, we study nonobtuse 0/1-simplices S having a partly decomposable matrix representation P. We prove that such a simplex also has a block diagonal matrix representation with at least two diagonal blocks, and show that a nonobtuse simplex with partly decomposable matrix representation can be split in mutually orthogonal fully indecomposable simplicial facets whose dimensions add up to n. Using this insight, we are able to extend the one neighbor theorem for acute simplices to a larger class of nonobtuse simplices., 26 pages, 17 figures

Both practice and analysis of p-FEMs and adaptive hp-FEMs raise the question what increment in the current polynomial degree p guarantees a p-independent reduction of the Galerkin error. We answer this question for the p-FEM in the simplified context of homogeneous Dirichlet problems for the Poisson equation in the two dimensional unit square with polynomial data of degree p. We show that an increment proportional to p yields a p-robust error reduction and provide computational evidence that a constant increment does not.

A parabolic cylinder is an invariant, non-recurrent Fatou component Ω of an automorphism F of C2 satisfying: (1) The closure of the ω-limit set of F on Ω contains an isolated fixed point, (2) there exists a univalent map Φ from Ω into C2 conjugating F to the translation (z, w) ↦ (z+ 1 , w) , and (3) every limit map of { F∘ n} on Ω has one-dimensional image. In this paper, we prove the existence of parabolic cylinders for an explicit class of maps, and show that examples in this class can be constructed as compositions of shears and overshears.

In this study, we investigate uncertainties in a large eddy simulation of the atmosphere, employing modern uncertainty quantification methods that have hardly been used yet in this context. When analysing the uncertainty of model results, one can distinguish between uncertainty related to physical parameters whose values are not exactly known, and uncertainty related to modelling choices such as the selection of numerical discretization methods, of the spatial domain size and resolution, and the use of different model formulations. While the former kind is commonly studied e.g. with forward uncertainty propagation, we explore the use of such techniques to also assess the latter kind. From a climate modelling perspective, uncertainties in the convective response and cloud formation are of particular interest, since these affect the cloud-climate feedback, one of the dominant sources of uncertainty in current climate models. Therefore we analyse the DALES model in the RICO case, a well-studied convection benchmark. We use the VECMA toolkit for uncertainty propagation, assessing uncertainties stemming from physical parameters as well as from modelling choices. We find substantial uncertainties due to small random initial state perturbations, and that the choice of advection scheme is the most influential of the modelling choices we assessed. This article is part of the theme issue ‘Reliability and reproducibility in computational science: implementing verification, validation and uncertainty quantification in silico ’.

Using the framework of operator or Calderon preconditioning, uniform preconditioners are constructed for elliptic operators discretized with continuous finite (or boundary) elements. The preconditioners are constructed as the composition of an opposite order operator, discretized on the same ansatz space, and two identical diagonal scaling operators, whose matrix representation is the lumped mass matrix.

Data availability Figure 1a,b displays publicly available cumulative death count data for the UK, which were obtained from ref. 22. Source Data are available with this paper. Furthermore, the parameter list—with all input parameters, a description, their default values and reasons for inclusion or exclusion from the Imperial College London CovidSim team—is available as Supplementary Data. Supplementary information: https://static-content.springer.com/esm/art%3A10.1038%2Fs43588-021-00028-9/MediaObjects/43588_2021_28_MOESM1_ESM.pdf Supplementary discussion, Figs. 1–7 and Table 1. Supplementary Data 1: https://static-content.springer.com/esm/art%3A10.1038%2Fs43588-021-00028-9/MediaObjects/43588_2021_28_MOESM2_ESM.xlsx All input parameters, a description, their default values and reasons for inclusion or exclusion from the Imperial College London CovidSim team. Source data Source Data Fig. 1: https://static-content.springer.com/esm/art%3A10.1038%2Fs43588-021-00028-9/MediaObjects/43588_2021_28_MOESM3_ESM.xlsx The computed cumulative death results for Fig. 1 and the observed death count data. Source Data Fig. 2: https://static-content.springer.com/esm/art%3A10.1038%2Fs43588-021-00028-9/MediaObjects/43588_2021_28_MOESM4_ESM.xlsx The Sobol indices of Fig. 2. Copyright © 2022 The Autor(s). Epidemiological modelling has assisted in identifying interventions that reduce the impact of COVID-19. The UK government relied, in part, on the CovidSim model to guide its policy to contain the rapid spread of the COVID-19 pandemic during March and April 2020; however, CovidSim contains several sources of uncertainty that affect the quality of its predictions: parametric uncertainty, model structure uncertainty and scenario uncertainty. Here we report on parametric sensitivity analysis and uncertainty quantification of the code. From the 940 parameters used as input into CovidSim, we find a subset of 19 to which the code output is most sensitive—imperfect knowledge of these inputs is magnified in the outputs by up to 300%. The model displays substantial bias with respect to observed data, failing to describe validation data well. Quantifying parametric input uncertainty is therefore not sufficient: the effect of model structure and scenario uncertainty must also be properly understood. Royal Society’s RAMP initiative: European Union Horizon 2020 research and innovation programme under grant agreement nos. 800925 (VECMA project; www.vecma.eu) and 823712 (CompBioMed2 Centre of Excellence, www.compbiomed.eu); UK EPSRC for the UK High-End Computing Consortium (grant no. EP/R029598/1).

We present an algorithm for the solution of a simultaneous space-time discretization of linear parabolic evolution equations with a symmetric differential operator in space. Building on earlier work, we recast this discretization into a Schur complement equation whose solution is a quasi-optimal approximation to the weak solution of the equation at hand. Choosing a tensor-product discretization, we arrive at a remarkably simple linear system. Using wavelets in time and standard finite elements in space, we solve the resulting system in linear complexity on a single processor, and in polylogarithmic complexity when parallelized in both space and time. We complement these theoretical findings with large-scale parallel computations showing the effectiveness of the method.

Many processes in nature and engineering are governed by partial differential equations (PDEs). We focus on parabolic PDEs, that describe time-dependent phenomena like heat conduction, chemical concentration, and fluid flow. Even if we know that a unique solution exists, we can express it in closed form only under very strict circumstances. So, to understand what it looks like, we turn to numerical approximation. Historically, parabolic PDEs are solved using time-stepping. One first discretizes the PDE in space as to obtain a system of coupled ordinary differential equations in time. This system is then solved using the vast theory for ODEs. While efficient in terms of memory and computational cost, time-stepping schemes take global time steps, which are independent of spatial position. As a result, these methods cannot efficiently resolve details in localized regions of space and time. Moreover, being inherently sequential, they have limited possibilities for parallel computation. In this thesis, we take a different approach and reformulate the parabolic evolution equation as an equation posed in space and time simultaneously. Space-time methods mitigate the aforementioned issues, and moreover produce approximations to the unknown solution that are uniformly quasi-optimal. The focal point of this thesis is the space-time minimal residual (MR) method introduced by R. Andreev, that finds the approximation that minimizes both PDE- and initial error. We discuss its theoretical properties, provide numerical algorithms for its computation, and discuss its applicability in data assimilation (the problem of fusing measured data to its underlying PDE).

Reduced basis approximations of Optimal Control Problems (OCPs) governed by steady partial differential equations (PDEs) with random parametric inputs are analyzed and constructed. Such approximations are based on a Reduced Order Model, which in this work is constructed using the method of weighted Proper Orthogonal Decomposition. This Reduced Order Model then is used to efficiently compute the reduced basis approximation for any outcome of the random parameter. We demonstrate that such OCPs are well-posed by applying the adjoint approach, which also works in the presence of admissibility constraints and in the case of non linear-quadratic OCPs, and thus is more general than the conventional Lagrangian approach. We also show that a step in the construction of these Reduced Order Models, known as the aggregation step, is not fundamental and can in principle be skipped for noncoercive problems, leading to a cheaper online phase. Numerical applications in three scenarios from environmental science are considered, in which the governing PDE is steady and the control is distributed. Various parameter distributions are taken, and several implementations of the weighted Proper Orthogonal Decomposition are compared by choosing different quadrature rules.

In order to eliminate climate uncertainty w.r.t. cloud and convection parametrizations, superpramaterization (SP) [1] has emerged as one of the possible ways forward. We have implemented (regional) superparametrization of the ECMWF weather model OpenIFS [2] by cloud-resolving, three-dimensional large-eddy simulations. This setup, described in [3], contains a two-way coupling between a global meteorological model that resolves large-scale dynamics, with many local instances of the Dutch Atmospheric Large Eddy Simulation (DALES) [4], resolving cloud and boundary layer physics. The model is currently prohibitively expensive to run over climate or even seasonal time scales, and a global SP requires the allocation of millions of cores. In this paper, we study the performance and scaling behavior of the LES models and the coupling code and present our implemented optimizations. We mimic the observed load imbalance with a simple performance model and present strategies to improve hardware utilization in order to assess the feasibility of a world-covering superparametrization. We conclude that (quasi-)dynamical load-balancing can significantly reduce the runtime for such large-scale systems with wide variability in LES time-stepping speeds.

Quantum-access security, where an attacker is granted superposition access to secret-keyed functionalities, is a fundamental security model and its study has inspired results in post-quantum security. We revisit, and fill a gap in, the quantum-access security analysis of the Lamport one-time signature scheme (OTS) in the quantum random oracle model (QROM) by Alagic et al. (Eurocrypt 2020). We then go on to generalize the technique to the Winternitz OTS. Along the way, we develop a tool for the analysis of hash chains in the QROM based on the superposition oracle technique by Zhandry (Crypto 2019) which might be of independent interest., LIPIcs, Vol. 199, 2nd Conference on Information-Theoretic Cryptography (ITC 2021), pages 21:1-21:22

We describe tdULL, an algorithm for computing treedepth decompositions of minimal depth. An implementation was submitted to the exact track of PACE 2020. tdULL is a branch and bound algorithm branching on inclusion-minimal separators., LIPIcs, Vol. 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020), pages 29:1-29:4

This chapter gives an introduction to the book. It surveys multivariable hypergeometric series and integrals, with comparison of the classical, basic and elliptic case, and multivariable (bi)orthogonal polynomials and functions, where root system generalizations of classical orthogonal polynomials get special emphasis. The chapter also provides a global description of the other chapters.

Span programs are an important model of quantum computation due to their tight correspondence with quantum query complexity. For any decision problem $f$, the minimum complexity of a span program for $f$ is equal, up to a constant factor, to the quantum query complexity of $f$. Moreover, this correspondence is constructive. A span program for $f$ with complexity $C$ can be compiled into a bounded error quantum algorithm for $f$ with query complexity $O(C)$, and vice versa. In this work, we prove an analogous connection for quantum time complexity. In particular, we show how to convert a quantum algorithm for $f$ with time complexity $T$ into a span program for $f$ such that it compiles back into a quantum algorithm for $f$ with time complexity $\widetilde{O}(T)$. While the query complexity of quantum algorithms obtained from span programs is well-understood, it is not generally clear how to implement certain query-independent operations in a time-efficient manner. We show that for span programs derived from algorithms with a time-efficient implementation, we can preserve the time efficiency when implementing the span program. This means in particular that span programs not only fully capture quantum query complexity, but also quantum time complexity. One practical advantage of being able to convert quantum algorithms to span programs in a way that preserves time complexity is that span programs compose very nicely. We demonstrate this by improving Ambainis's variable-time quantum search result using our construction through a span program composition for the OR function., 54 pages, 2 figures

© 2020 The Authors. Validation, verification, and uncertainty quantification (VVUQ) of simulation workflows are essential for building trust in simulation results, and their increased use in decision‐making processes. The EasyVVUQ Python library is designed to facilitate implementation of advanced VVUQ techniques in new or existing workflows, with a particular focus on high‐performance computing, middleware agnosticism, and multiscale modeling. Here, the application of EasyVVUQ to five very diverse application areas is demonstrated: materials properties, ocean circulation modeling, fusion reactors, forced human migration, and urban air quality prediction. Horizon 2020 Framework Programme. Grant Number: 800925; Engineering and Physical Sciences Research Council

We consider a class of skew product maps of interval diffeomorphisms over the doubling map. The interval maps fix the end points of the interval. It is assumed that the system has zero fiber Lyapunov exponent at one endpoint and zero or positive fiber Lyapunov exponent at the other endpoint. We prove the appearance of on-off intermittency. This is done using the equivalent description of chaotic walks: random walks driven by the doubling map. The analysis further relies on approximating the chaotic walks by Markov random walks, that are constructed using Markov partitions for the doubling map., 34 pages

Recent years have seen a growing interest in using data-driven (machine-learning) techniques for the construction of cheap surrogate models of turbulent subgrid scale stresses. These stresses display complex spatio-temporal structures, and constitute a difficult surrogate target. In this paper we propose a data-preprocessing step, in which we derive alternative subgrid scale models which are virtually exact for a user-specified set of spatially integrated quantities of interest. The unclosed component of these new subgrid scale models is of the same size as this set of integrated quantities of interest. As a result, the corresponding training data is massively reduced in size, decreasing the complexity of the subsequent surrogate construction.