In this work, we consider the problem of estimating the probability distribution, the quantile or the conditional expectation above the quantile, the so called conditional-value-at-risk, of output quantities of complex random differential models by the MLMC method. We follow the approach of (reference), which recasts the estimation of the above quantities to the computation of suitable parametric expectations. In this work, we present novel computable error estimators for the estimation of such quantities, which are then used to optimally tune the MLMC hierarchy in a continuation type adaptive algorithm. We demonstrate the efficiency and robustness of our adaptive continuation-MLMC in an array of numerical test cases.

This report presents the latest methods of optimisation under uncertainties investigated in the ExaQUte project, and their applications to problems related to civil and wind engineering. The measure of risk throughout the report is the conditional value at risk. First, the reference method is presented: the derivation of sensitivities of the risk measure; their accurate computation; and lastly, a practical optimisation algorithm with adaptive statistical estimation. Second, this method is directly applied to a nonlinear relaxation oscillator (FitzHugh–Nagumo model) with numerical experiments to demonstrate its performance. Third, the optimisation method is adapted to the shape optimisation of an airfoil and illustrated by a large-scale experiment on a computing cluster. Finally, the benchmark of the shape optimisation of a tall building under a turbulent flow is presented, followed by an adaptation of the optimisation method. All numerical experiments showcase the open-source software stack of the ExaQUte project for large-scale computing in a distributed environment.

We study the use of multi-level Monte Carlo methods for wind engineering. This report brings together methodological research on uncertainty quantification and work on target applications of the ExaQUte project in wind and civil engineering. First, a multi-level Monte Carlo for the estimation of the conditional value at risk and an adaptive algorithm are presented. Their reliability and performance are shown on the time-average of a non-linear oscillator and on the lift coefficient of an airfoil, with both preset and adaptively refined meshes. Then, we propose an adaptive multi-fidelity Monte Carlo algorithm for turbulent fluid flows where multilevel Monte Carlo methods were found to be inefficient. Its efficiency is studied and demonstrated on the benchmark problem of quantifying the uncertainty on the drag force of a tall building under random turbulent wind conditions. All numerical experiments showcase the open-source software stack of the ExaQUte project for large-scale computing in a distributed environment.

In this report, we study the use of Multi-Level Monte Carlo (MLMC) methods for time dependent problems. It was found that the usability of MLMC methods depends strongly on whether or not the underlying time dependent problem is chaotic in nature. Numerical experiments are conducted on both simple problems, as well as fluid flow problems of practical interest to the ExaQUte project, to demonstrate this. For the non-chaotic cases, the hypotheses that enable the use of MLMC methods were found to be satisfied. For the chaotic cases, especially the case of high Reynolds’ number fluid flow, the hypotheses were not satisfied. However, it was found that correlations between the different levels were high enough to merit the use of multi-fidelity or control-variate approaches. It was also noted that MLMC methods could work for chaotic problems if the time window of analysis were chosen to be small enough. Future studies are proposed to examine this possibility.

This report brings together methodological research on stochastic optimisation and work on benchmark and target applications of the ExaQute project, with a focus on unsteady problems. A practical, general method for the optimisation of the conditional value at risk is proposed. Three different optimisation problems are described: an oscillator problem selected as a suitable trial and illustration case; the shape optimisation of an airfoil, chosen as a benchmark application in the project; the shape optimisation of a tall building, which is the challenging target application set for ExaQUte. For each problem, the current developments and results are presented, the application of the proposed method is discussed, and the work to be done until the end of the project is laid out.

In the following sections, the formulation of a possible stochastic optimisation problem relevant to the ExaQUte project is first presented

This deliverable presents the final software release of Kratos Multiphysics, together with the XMC library, Hyperloom and PyCOMPSs API definitions [13]. This release also contains the latest developements on MPI parallel remeshing in ParMmg. This report is meant to serve as a supplement to the public release of the software. Kratos is “a framework for building parallel, multi-disciplinary simulation software, aiming at modularity, extensibility, and high performance. Kratos is written in C++, and counts with an extensive Python interface”. XMC is “a Python library for parallel, adaptive, hierarchical Monte Carlo algorithms, aiming at reliability, modularity, extensibility and high performance“. Hyperloom and PyCOMPSs are environments for enabling parallel and distributed computation. ParMmg is an open source software which offers the parallel mesh adaptation of three dimensional volume meshes.

The ExaQUte project participates in the Pilot on Open Research Data launched by the European Commission (EC) along with the H2020 program. This pilot is part of the Open Access to Scientific Publications and Research Data program in H2020. The goal of the program is to foster access to research data generated in H2020 projects. The use of a Data anagement Plan (DMP) is required for all projects participating in the Open Research Data Pilot, in which they will specify what data will be kept for the longer term. The underpinning idea is that Horizon 2020 beneficiaries have to make their research data findable, accessible, interoperable and re-usable (FAIR), to ensure it is soundly managed.

This deliverable presents the software release of Kratos Multiphysics, together with the XMC library, Hyperloom and PyCOMPSs API definition [8]. This report is meant to serve as a supplement to the public release of the software. Kratos is “a framework for building parallel, multi-disciplinary simulation software, aiming at modularity, extensibility, and high performance. Kratos is written in C++, and counts with an extensive Python interface”. XMC is a python library for hierarchical Monte Carlo algorithms. Hyperloom and PyCOMPSs are environments for enabling parallel and distributed computation.

ExaQUte Data This repository contains the source files, data and metadata of the different simulations and analyses used to produce the deliverables from the ExaQUte project: https://www.scipedia.com/sj/exaqute-deliverables. This release contains data from: D3.3 Report of ensemble based parallelism for turbulent flows and release of solvers. Section 4.1: Rectangle body. Section 4.2: CAARC building. D3.4 Report on the calibration of parallel methods for transient problems in wind engineering. Section 4.1: Rectangle obstacle with stochastic inlet profile. Section 4.2: High-rise building with stochastic mean profile. Section 4.3: High-rise building with stochastic mean profile and fluctuations. D4.3 Benchmarking report as tested on the available infrastructure. Section 3: Infrastructures benchmark. D4.5 Framework development and release. Section 3: Evaluation (of the framework). D5.5 Report on the application of MLMC to wind engineering applications. Section 2.2.1: Numerical experiments: Van der Pol Oscillator. Section 2.2.2: Numerical experiments: Potential flow over an airfoil. Section 3.2: Application to a turbulent fluid problem. D6.5 Report on stochastic optimization for wind engineering. Section 3: Optimisation of the conditional value-at-risk of an oscillator. Section 4: Stochastic optimisation of an airfoil D7.4 Final report on Stochastic Optimization results. Section 3: Stochastic optimization of twisted tapered tower. For any specific inquiry about the data, please contact the deliverable leaders.

This report addresses the general matter of optimisation under uncertainties, following a previous report on stochastic sensitivities (deliverable 6.2). It describes several theoretical methods, as well their application into implementable algorithms. The specific case of the conditional value at risk chosen as risk measure, with its challenges, is prominently discussed. In particular, the issue of smoothness – or lack thereof – is addressed through several possible approaches. The whole report is written in the context of high-performance computing, with concern for parallelisation and cost-efficiency.

In this deliverable, the ExaQUte xmc library is introduced. This report is meant to serve as a supplement to the publicly release of the library. In the following sections, the ExaQUte xmc library is described along with its current and future capabilities. The structure of the library, along with its dynamic import mechanism, are described using samples of code. The algorithms behind the example files supplied with the public release are explained in detail as well.

This documents describes several studies undertaken to assess the applicability of MultiLevel Monte Carlo (MLMC) methods to problems of interest; namely in turbulent fluid flow over civil engineering structures. Several numerical experiments are presented wherein the convergence of quantities of interest with mesh parameters are studied at different Reynolds’ numbers and geometries. It was found that MLMC methods could be used successfully for low Reynolds’ number flows when combined with appropriate Adaptive Mesh Refinement (AMR) strategies. However, the hypotheses for optimal MLMC performance were found to not be satisfied at higher turbulent Reynolds’ numbers despite the use of AMR strategies. Recommendations are made for future research directions based on these studies. A tentative outline for an MLMC algorithm with adapted meshes is made, as well as recommendations for alternatives to MLMC methods for cases where the underlying assumptions for optimal MLMC performance are not satisfied.

This paper proposes to address the issue of complexity reduction for the numerical simulation of multiscale media in a quasi-periodic setting. We consider a stationary elliptic diffusion equation defined on a domain D such that D̅ is the union of cells { D̅ i } i ∈ I and we introduce a two-scale representation by identifying any function v ( x ) defined on D with a bi-variate function v ( i , y ), where i ∈ I relates to the index of the cell containing the point x and y ∈ Y relates to a local coordinate in a reference cell Y . We introduce a weak formulation of the problem in a broken Sobolev space V ( D ) using a discontinuous Galerkin framework. The problem is then interpreted as a tensor-structured equation by identifying V ( D ) with a tensor product space ℝ I ⊗ V ( Y ) of functions defined over the product set I × Y . Tensor numerical methods are then used in order to exploit approximability properties of quasi-periodic solutions by low-rank tensors.