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In this article, we propose a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. To this end, we derive a posteriori error estimates for the smallest eigenvalue which is assumed to be simple and the associated left and right eigenvectors. The practical computation of these estimators requires the estimation of a constant called prefactor, which we can express as the spectral norm of some operator. We provide some elements of theoretical analysis which illustrate the link between the expression of the prefactor we obtain here and its well-known expression in the case of symmetric eigenvalue problems, either using the notion of numerical range of the operator, or via a perturbative analysis. Lastly, we propose a practical method in order to estimate this prefactor which yields interesting numerical results on actual test cases. We provide detailed numerical simulations on two-dimensional examples including a multigroup neutron diffusion equation. [ABSTRACT FROM AUTHOR]

The concept of the Shortest Vector Problem (SVP) has surprisingly been used widely in many applications of lattice-based cryptography, notably in public-key cryptanalysis. One of the applications is to develop a well-known algorithm of lattice reduction, namely the LLL (Lenstra- Lenstra-Lovasz) algorithm. The LLL algorithm is known to be able to reduce the basis of a lattice to a minimum set of vectors, which is called the LLL-reduced basis. In this paper, we investigate the properties of the LLL-reduced basis for some different factor δ values. By changing and adjusting the value of factor δ, the proposed value of factor δ in the LLL-reduced basis produces some interesting properties. We also looked into the relationship between the initial vector and the factor δ in the LLL-reduced basis and developed some related properties. [ABSTRACT FROM AUTHOR]

In this paper, a nonsmooth semilinear parabolic partial differential equation (PDE) is considered. For a reduced basis (RB) approach, a space-time formulation is used to develop a certified a-posteriori error estimator. This error estimator is adopted to the presence of the discrete empirical interpolation method (DEIM) as approximation technique for the nonsmoothness. The separability of the estimated error into an RB and a DEIM part then guides the development of an adaptive RB-DEIM algorithm, combining both offline phases into one. Numerical experiments show the capabilities of this novel approach in comparison with classical RB and RB-DEIM approaches. [ABSTRACT FROM AUTHOR]

Abstract The simulation of magnetic bearings involves highly non-linear physics, with high dependency on the input variation. Moreover, such a simulation is time consuming and can’t run, within realistic computation time for control purposes, when using classical computation methods. On the other hand, classical model reduction techniques fail to achieve the required precision within the allowed computation window. To address this complexity, this work proposes a combination of physics-based computing methods, model reduction techniques and machine learning algorithms, to tackle the requirements. The physical model used to represent the magnetic bearing is the classical Cauer Ladder Network method, while the model reduction technique is applied on the error of the physical model’s solution. Later on, in the latent space a machine learning algorithm is used to predict the evolution of the correction in the latent space. The results show an improvement of the solution without scarifying the computation time. The solution is computed in almost real-time (few milliseconds), and compared to the finite element reference solution.

Summary: This paper addresses the overwhelming computational resources needed with standard numerical approaches to simulate architected materials. Those multiscale heterogeneous lattice structures gain intensive interest in conjunction with the improvement of additive manufacturing as they offer, among many others, excellent stiffness‐to‐weight ratios. We develop here a dedicated HPC solver that benefits from the specific nature of the underlying problem in order to drastically reduce the computational costs (memory and time) for the full fine‐scale analysis of lattice structures. Our purpose is to take advantage of the natural domain decomposition into cells and, even more importantly, of the geometrical and mechanical similarities among cells. Our solver consists in a so‐called inexact FETI‐DP method where the local, cell‐wise operators and solutions are approximated with reduced order modeling techniques. Instead of considering independently every cell, we end up with only few principal local problems to solve and make use of the corresponding principal cell‐wise operators to approximate all the others. It results in a scalable algorithm that saves numerous local factorizations. Our solver is applied for the isogeometric analysis of lattices built by spline composition, which offers the opportunity to compute the reduced basis with macro‐scale data, thereby making our method also multiscale and matrix‐free. The solver is tested against various 2D and 3D analyses. It shows major gains compared to black‐box solvers; in particular, problems of several millions of degrees of freedom can be solved with a simple computer within few minutes. [ABSTRACT FROM AUTHOR]

We prove a-posteriori error-estimates for reduced-order modeling of quasilinear parabolic PDEs with non-monotone nonlinearity. We consider the solution of a semi-discrete in space equation as reference, and therefore incorporate reduced basis-, empirical interpolation-, and time-discretization-errors in our consideration. Numerical experiments illustrate our results. [ABSTRACT FROM AUTHOR]

Finding the shortest vector in a lattice is a NP-hard problem. The best known approximation algorithm for this problem is LLL algorithm with the approximation factor of $\alpha ^{\frac {n-1}{2}}$ , $\alpha \geq \frac {4}{3}$ , which is not a good approximation factor. This work proposes a new polynomial time approximation algorithm for the shortest lattice vector problem. The proposed method makes use of only integer arithmetic and does not require Gram-Schmidt orthogonal basis for generating reduced basis. The proposed method is able to obtain an approximation factor of $\frac {1}{(1-\delta)}$ , where $0 \leq \delta \lt 1$ .

In a previous work, we introduced, in the context of gravitational wave science, an initial study on an automated domain-decomposition approach for a reduced basis through hp-greedy refinement. The approach constructs local reduced bases of lower dimensionality than global ones, with the same or higher accuracy. These "light" local bases should imply both faster evaluations when predicting new waveforms and faster data analysis, particularly faster statistical inference (the forward and inverse problems, respectively). In this approach, however, we have previously found important dependence on several hyperparameters, which do not appear in a global reduced basis. This naturally leads to the problem of hyperparameter optimization (HPO), which is the subject of this paper. Here, we compare the efficiency of the Bayesian approach against grid and random searches, which are two order of magnitude slower. Then, we tackle the problem of HPO through Bayesian optimization.We find that, for the cases studied here of gravitational waves from the collision of two spinning but non-precessing black holes, for the same accuracy, local hp-greedy reduced bases with HPO have a lower dimensionality of up to 4×, depending on the desired accuracy. This factor should directly translate into a parameter estimation speedup in the context of reduced order quadratures, for instance. Such acceleration might help in the near real-time requirements for electromagnetic counterparts of gravitational waves from compact binary coalescences. The code developed for this project is available open source from public repositories. This paper is an invited contribution to the Special Issue "Recent Advances in Gravity: A Themed Issue in Honor of Prof. Jorge Pullin on his 60th Anniversary". [ABSTRACT FROM AUTHOR]

We introduce a novel heterogeneous multi‐scale method for the frequency response analysis of waves propagating in porous domains with a complex micro‐structure. In this, the domain is decomposed onto polygonal coarse elements each clustering its local micro‐structure. The micro‐structure is resolved using the virtual element method hence providing ample flexibility into capturing fine scale heterogeneities of arbitrary polygonal shapes. The upscaling is performed through a set of numerically evaluated multi‐scale basis functions. Rather than evaluating the basis for a broad band set of frequencies, a reduced order modeling approach is opted for. In this, the basis is evaluated for a select subset of frequencies, that is used as a basis for the evaluation of the remaining set through a proper orthogonal decomposition procedure. The solution of the coupled governing equations is then performed at the coarse‐scale at a reduced computational cost. The performance of the method in terms of accuracy and computational efficiency is evaluated through a set of numerical examples for poro‐elastic materials with heterogeneities of various shapes. [ABSTRACT FROM AUTHOR]

This paper develops geographic style maps containing two‐dimensional lattices in all known periodic crystals parameterized by recent complete invariants. Motivated by rigid crystal structures, lattices are considered up to rigid motion and uniform scaling. The resulting space of two‐dimensional lattices is a square with identified edges or a punctured sphere. The new continuous maps show all Bravais classes as low‐dimensional subspaces, visualize hundreds of thousands of lattices of real crystal structures from the Cambridge Structural Database, and motivate the development of continuous and invariant‐based crystallography. [ABSTRACT FROM AUTHOR]

Summary: Shape optimization is an effective tool to improve the aerodynamic performance of tall buildings by introducing minor modifications to the original project. Nevertheless, economic criteria demand efficient cross sections aiming at maximizing the building's profitability. These two contradictory criteria are commonly handled by adopting multi‐objective optimization approaches seeking the definition of Pareto fronts. However, the aerodynamic nonlinear features of low‐aspect‐ratio cross sections typically adopted in architectural practice can cause wind‐induced acceleration response surfaces over the considered design domain with multiple local minima that eventually lead to discontinuous Pareto fronts with non‐convex regions. This study delves into this problem and proposes a design framework that effectively combines the reduced basis method with multi‐objective optimization techniques to carry out the aerodynamic shape optimization using surrogates trained with CFD simulations. The ability of the optimization strategy to properly define the non‐convex regions of discontinuous Pareto fronts is successfully leveraged by adopting the weighted min–max method. [ABSTRACT FROM AUTHOR]

In this work, a simple, easy to use MATLAB code is presented for the optimal design of materials for 3D domains. For the optimal design of materials, the theoretical framework of topology optimization and that of homogenization were utilized to develop a formulation where the design of the micro-structure of the material is affected among others by the loading and boundary conditions of the 3D macro domain. The final result of the micro-scale can then be converted into an stl file, which can be utilized for 3D printing; however, the continuity of the unit cells when assembled to form the macro structure should be taken into account. The transition of the design of the material problem formulation from 2D to 3D domains generates drastically increased computational needs in order to perform the design procedures, which might narrow its formulation scales and the corresponding sizes of the adopted finite element discretization. Thus, in addition to the optimal design of materials implementation, the utilization of three different model order reduction (MOR) approaches is presented, aiming to assist towards the reduction of the computational cost of the two scales formulation. On-the-fly reduced order model, proper orthogonal decomposition (POD), and approximate reanalysis (AR) following the combined approximations are the three approaches adopted for the purposes of this study, while the code implementation enables the addition of new ones easily. [ABSTRACT FROM AUTHOR]

A model order reduction approach combining reduced basis (RB) projection and sub-structuring by domain decomposition is developed to tackle nonlinear elasto-plasticity in parametrized coupled thermo-hydro-mechanical (THM) systems. The region-specific occurrence of plasticity in the THM process is exploited in domain decomposition to facilitate the simplified construction of localized reduced subspaces in order to perform projection on a multi-domain basis. Traditional RB projection is leveraged in the elastic zone while the computationally expensive nonlinear iterative procedure is confined in the zone where plasticity is assumed to be restricted. The applicability of the approach to parametric problems is investigated through inverse identification of material parameters involving a nonlinear THM model system for nuclear waste repository applications. The model reduction strategy is observed to significantly reduce the computational effort involved with up to 12 times faster simulation compared to finite element (FE)-driven inverse identification procedure. • A model reduction technique is developed combining reduced basis projection and domain sub-structuring. • Region-specific occurrence of plasticity is exploited to construct the localized reduced subspaces. • The methodology is used for the identification of material parameters in a geological storage model problem. • The model reduction strategy is observed to significantly reduce the computational effort. [ABSTRACT FROM AUTHOR]

The main part of the computational cost required for solving the problem of optimal material design with extreme properties using a topology optimization formulation is devoted to solving the equilibrium system of equations derived through the implementation of the finite element method (FEM). To reduce this computational cost, among other methodologies, various model order reduction (MOR) approaches can be utilized. In this work, a simple Matlab code for solving the topology optimization for the design of materials combined with three different model order reduction approaches is presented. The three MOR approaches presented in the code implementation are the proper orthogonal decomposition (POD), the on-the-fly reduced order model construction and the approximate reanalysis (AR) following the combined approximations approach. The complete code, containing all participating functions (including the changes made to the original ones), is provided. [ABSTRACT FROM AUTHOR]

Linear kinetic transport equations play a critical role in optical tomography, radiative transfer and neutron transport. The fundamental difficulty hampering their efficient and accurate numerical resolution lies in the high dimensionality of the physical and velocity/angular variables and the fact that the problem is multiscale in nature. Leveraging the existence of a hidden low-rank structure hinted by the diffusive limit, in this work, we design and test the angular-space reduced order model for the linear radiative transfer equation, the first such effort based on the celebrated reduced basis method (RBM). Our method is built upon a high-fidelity solver employing the discrete ordinates method in the angular space, an asymptotic preserving upwind discontinuous Galerkin method for the physical space, and an efficient synthetic accelerated source iteration for the resulting linear system. Addressing the challenge of the parameter values (or angular directions) being coupled through an integration operator, the first novel ingredient of our method is an iterative procedure where the macroscopic density is constructed from the RBM snapshots, treated explicitly and allowing a transport sweep, and then updated afterwards. A greedy algorithm can then proceed to adaptively select the representative samples in the angular space and form a surrogate solution space. The second novelty is a least squares density reconstruction strategy, at each of the relevant physical locations, enabling the robust and accurate integration over an arbitrarily unstructured set of angular samples toward the macroscopic density. Numerical experiments indicate that our method is effective for computational cost reduction in a variety of regimes. [ABSTRACT FROM AUTHOR]

Linear finite element harmonic vibration analysis is an important tool when investigating structural performance. In order to construct the requested frequency response functions by sweeping through the frequencies, the full finite element model is typically reduced by a Galerkin projection, using either the solution of a free vibration problem, or with load‐dependent Ritz vectors. Here we construct an approximation of the model reduction matrix using an intermediate reduced‐model stage based on the Coherent Nodal Clusters technique. Thus we arrive at a two‐stage model‐reduction procedure. Although there is an additional approximation the reduced models are obtained at a much lower cost than the one‐stage models. In order to ameliorate the loss of accuracy involved in the additional approximation, we introduce enhancement of the reduced bases by inexpensive residual‐minimizing iterations. [ABSTRACT FROM AUTHOR]

In this article, a greedy reduced basis algorithm is proposed for the solution of structural acoustic systems with parameter and implicit frequency dependence. The underlying equations of linear time‐harmonic elastodynamics and acoustics are discretized using the finite element and boundary element method, respectively. The solution within the parameter domain is determined by a linear combination of reduced basis vectors. This basis is generated iteratively and given by the responses of the structural acoustic system at certain parameter samples. A greedy approach is followed by evaluating the next basis vector at the parameter sample which is currently approximated worst. The algorithm runs on a small training set which bounds the memory requirements and allows applications to large‐scale problems with high‐dimensional parameter domains. The computational efficiency of the proposed scheme is illustrated based on two numerical examples: a point‐excited spherical shell submerged in water and a satellite structure subject to a diffuse sound pressure field excitation. [ABSTRACT FROM AUTHOR]

This work presents a local reduced-order method for computing bifurcation diagrams in 2D Rayleigh–Bénard convection problems. The proposed method is based on Proper Orthogonal Decomposition, and employs a reduced-order study of the regularity of solutions to detect new solution branches within the bifurcation diagram. The locality of the method is achieved through k-means clustering, and the selection of local problems in the online stage is done in the solution space. All hyperparameters of the reduced method, such as the number of clusters or the number of POD modes, are estimated by deterministic criteria. Furthermore, the offline–online splitting strategy for online calculations is explicitly outlined. The method is applied to a single-parameter problem and a two-parameter problem, showing its ability to rapidly compute bifurcation diagrams with small errors. Compared to a standard approach that samples each branch separately, the local approach produces more accurate results in less computational time. [ABSTRACT FROM AUTHOR]

In this work, a simple, easy to use MATLAB code is presented for the optimal design of materials for 3D domains. For the optimal design of materials, the theoretical framework of topology optimization and that of homogenization were utilized to develop a formulation where the design of the micro-structure of the material is affected among others by the loading and boundary conditions of the 3D macro domain. The final result of the micro-scale can then be converted into an stl file, which can be utilized for 3D printing; however, the continuity of the unit cells when assembled to form the macro structure should be taken into account. The transition of the design of the material problem formulation from 2D to 3D domains generates drastically increased computational needs in order to perform the design procedures, which might narrow its formulation scales and the corresponding sizes of the adopted finite element discretization. Thus, in addition to the optimal design of materials implementation, the utilization of three different model order reduction (MOR) approaches is presented, aiming to assist towards the reduction of the computational cost of the two scales formulation. On-the-fly reduced order model, proper orthogonal decomposition (POD), and approximate reanalysis (AR) following the combined approximations are the three approaches adopted for the purposes of this study, while the code implementation enables the addition of new ones easily.

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A methodology for using random sketching in the context of model order reduction for high-dimensional parameter-dependent systems of equations was introduced in Balabanov and Nouy (Part I; Advances in Computational Mathematics 45:2969–3019, 2019). Following this framework, we here construct a reduced model from a small, efficiently computable random object called a sketch of a reduced model, using minimal residual methods. We introduce a sketched version of the minimal residual based projection as well as a novel nonlinear approximation method, where for each parameter value, the solution is approximated by minimal residual projection onto a subspace spanned by several vectors picked (online) from a dictionary of candidate basis vectors. It is shown that random sketching technique can improve not only efficiency but also numerical stability. A rigorous analysis of the conditions on the random sketch required to obtain a given accuracy is presented. These conditions may be ensured a priori with high probability by considering for the sketching matrix an oblivious embedding of sufficiently large size. Furthermore, a simple and reliable procedure for a posteriori verification of the quality of the sketch is provided. This approach can be used for certification of the approximation as well as for adaptive selection of the size of the random sketching matrix. [ABSTRACT FROM AUTHOR]

We present a reduced basis method for parametrized linear elliptic partial differential equations (PDEs) in a least-squares finite element framework. A rigorous and reliable error estimate is developed, and is shown to bound the error with respect to the exact solution of the PDE, in contrast to estimates that measure error with respect to a finite-dimensional (high-fidelity) approximation. It is shown that the first-order formulation of the least-squares finite element is a key ingredient. The method is demonstrated using numerical examples. [ABSTRACT FROM AUTHOR]

We propose and analyze a Stein variational reduced basis method (SVRB) to solve large-scale PDE-constrained Bayesian inverse problems. To address the computational challenge of drawing numerous samples requiring expensive PDE solves from the posterior distribution, we integrate an adaptive and goal-oriented model reduction technique with an optimization-based Stein variational gradient descent method. We present detailed analyses for the reduced basis approximation errors of the potential and its gradient, the induced errors of the posterior distribution measured by Kullback--Leibler divergence, as well as the induced errors of the Stein variational samples. To demonstrate the computational accuracy and efficiency of SVRB, we report results of numerical experiments on a Bayesian inverse problem governed by a diffusion PDE with random parameters with both uniform and Gaussian prior distributions. Over 100X speedups can be achieved while the accuracy of the approximation of the potential and its gradient is preserved. [ABSTRACT FROM AUTHOR]

Abstract Numerical reduced basis methods are instrumental to solve parameter dependent partial differential equations problems in case of many queries. Bifurcation and instability problems have these characteristics as different solutions emerge by varying a bifurcation parameter. Rayleigh–Bénard convection is an instability problem with multiple steady solutions and bifurcations by varying the Rayleigh number. In this paper the eigenvalue problem of the corresponding linear stability analysis has been solved with this method. The resulting matrices are small, the eigenvalues are easily calculated and the bifurcation points are correctly captured. Nine branches of stable and unstable solutions are obtained with this method in an interval of values of the Rayleigh number. Different basis sets are considered in each branch. The reduced basis method permits one to obtain the bifurcation diagrams with much lower computational cost.

Summary: A method for efficient computational homogenization of hyperelastic materials under finite strains is proposed. Multiple spatial scales are homogenized in a recursive procedure: starting on the smallest scale, few high fidelity FE computations are performed. The resulting fields of deformation gradient fluctuations are processed by a snapshot POD resulting in a reduced basis (RB) model. By means of the computationally efficient RB model, a large set of samples of the homogenized material response is created. This data set serves as the support for the Concentric Interpolation (CI) scheme, interpolating the effective stress and stiffness. Then, the same procedure is invoked on the next larger scale with this CI surrogating the homogenized material law. A three‐scale homogenization process is completed within few hours on a standard workstation. The resulting model is evaluated within minutes on a laptop computer in order to generate fourth‐scale results. Open source code is provided. [ABSTRACT FROM AUTHOR]

Summary: Modal expansion is a workhorse used in many engineering analysis algorithms. One example is the coupled boundary element‐finite element computation of the backscattering target strength of underwater elastic objects. To obtain the modal basis, a free‐vibration (generalized eigenvalue) problem needs to be solved, which tends to be expensive when there are many basis vectors to compute. In the above‐mentioned backscattering example it could be many hundreds or thousands. Excellent algorithms exist to solve the free‐vibration problem, and most use some form of the Rayleigh‐Ritz (RR) procedure. The key to an efficient RR application is a low‐cost transformation into a reduced basis. In this work, a novel, inexpensive a priori transformation is constructed for solid‐mechanics finite element models based on the notion of coherent nodal clusters. The inexpensive RR procedure then leads to significant speedups of the computation of an approximate solution to the free vibration problem. [ABSTRACT FROM AUTHOR]

Additive printing allows the "single step" production of virtually any complex mechanical component. However, the manufacturing process involves a layer-by-layer deposition of material, which leads to an anisotropic mechanical behavior of the whole component. This would then entail a very fine 3D model to simulate the mechanical performance accurately. This simulation would also need to be integrated within an iterative design process in order to obtain the most efficient design. Both reasons explain the prohibitive number of calculations needed, which is currently beyond the capacities of existing software and computers. Recent research papers have opened promising pathways for integrating model reduction techniques within the overall topology optimization process. However, these approaches still present challenges such as choosing the minimum number and appropriate selection of the snapshots required to get accurate simulations. In this work, we present a methodology in the combined field of reduced-order modeling and topology optimization. The key idea consists of projecting the higher dimensional system of equations onto a lower dimensional space with the reduced basis vectors constructed using Proper Orthogonal Decomposition (POD). This reduced basis is updated in an incremental "on-the-fly" manner using alternatively costly high-fidelity and more rapid lower fidelity simulation snapshots. The variable-fidelity resolutions of successive approximations to the global system of equations are then integrated into the topology optimization process. The approaches are tested and computational savings and precision are compared, using both minimum compliance and compliant mechanism design benchmark problems. [ABSTRACT FROM AUTHOR]

We deal with the minimization of the H∞-norm of the transfer function of a parameter-dependent descriptor system over the set of admissible parameter values. Subspace frameworks are proposed for such minimization problems where the involved systems are of large order. The proposed algorithms are greedy interpolatary approaches inspired by our recent work [Aliyev et al., SIAM J. Matrix Anal. Appl., 38 (20∞7), pp. 1496-1516] for the computation of the H∞-norm. In this work, we minimize the H∞-norm of a reduced-order parameter-dependent system obtained by two-sided restrictions onto certain subspaces. Then we expand the subspaces so that Hermite interpolation properties hold between the full and reduced-order system at the optimal parameter value for the reduced-order system. We formally establish the superlinear convergence of the subspace frameworks under some smoothness and nondegeneracy assumptions. The fast convergence of the proposed frameworks in practice is illustrated by several large-scale systems. [ABSTRACT FROM AUTHOR]

We present a reduced basis stochastic Galerkin method for partial differential equations with random inputs. In this method, the reduced basis methodology is integrated into the stochastic Galerkin method, resulting in a significant reduction in the cost of solving the Galerkin system. To reduce the main cost of matrix-vector manipulation involved in our reduced basis stochastic Galerkin approach, the secant method is applied to identify the number of reduced basis functions. We present a general mathematical framework of the methodology, validate its accuracy and demonstrate its efficiency with numerical experiments. • Propose a more efficient method compared with the stochastic Galerkin method. • Integrate the reduced basis methodology into the stochastic Galerkin method. • Use the secant method to identify the number of reduced basis needed in the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents a method for the numerical treatment of reaction-convection-diffusion problems with parameter-dependent coefficients that are arbitrary rough and possibly varying at a very fine scale. The presented technique combines the reduced basis (RB) framework with the recently proposed super-localized orthogonal decomposition (SLOD). More specifically, the RB is used for accelerating the typically costly SLOD basis computation, while the SLOD is employed for an efficient compression of the problem's solution operator requiring coarse solves only. The combined advantages of both methods allow one to tackle the challenges arising from parametric heterogeneous coefficients. Given a value of the parameter vector, the method outputs a corresponding compressed solution operator which can be used to efficiently treat multiple, possibly non-affine, right-hand sides at the same time, requiring only one coarse solve per right-hand side. [ABSTRACT FROM AUTHOR]

Summary: The solution of frequency dependent linear systems arising from the discretization of vibro‐acoustic problems requires a significant computational effort in the case of rapidly varying responses. In this paper, we review the use of a greedy reduced basis scheme for the efficient solution in a frequency range. The reduced basis is spanned by responses of the system at certain frequencies that are chosen iteratively based on the response that is currently worst approximated in each step. The approximations at intermediate frequencies as well as the a posteriori estimations of associated errors are computed using a least squares solver. The proposed scheme is applied to the solution of an interior acoustic problem with boundary element method (BEM) and to the solution of coupled structural acoustic problems with finite element method and BEM. The computational times are compared to those of a conventional frequencywise strategy. The results illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

We propose a probabilistic way for reducing the cost of classical projection-based model order reduction methods for parameter-dependent linear equations. A reduced order model is here approximated from its random sketch, which is a set of low-dimensional random projections of the reduced approximation space and the spaces of associated residuals. This approach exploits the fact that the residuals associated with approximations in low-dimensional spaces are also contained in low-dimensional spaces. We provide conditions on the dimension of the random sketch for the resulting reduced order model to be quasi-optimal with high probability. Our approach can be used for reducing both complexity and memory requirements. The provided algorithms are well suited for any modern computational environment. Major operations, except solving linear systems of equations, are embarrassingly parallel. Our version of proper orthogonal decomposition can be computed on multiple workstations with a communication cost independent of the dimension of the full order model. The reduced order model can even be constructed in a so-called streaming environment, i.e., under extreme memory constraints. In addition, we provide an efficient way for estimating the error of the reduced order model, which is not only more efficient than the classical approach but is also less sensitive to round-off errors. Finally, the methodology is validated on benchmark problems. [ABSTRACT FROM AUTHOR]

Summary: Monte Carlo methods are well suited to characterize events of which associated probabilities are not too low with respect to the simulation budget. For very seldom observed events, these approaches do not lead to accurate results. Indeed, the number of samples is often insufficient to estimate such low probabilities (at least 10n+2 samples are needed to estimate a probability of 10−n with 10% relative deviation of the Monte Carlo estimator). Even within the framework of reduced order methods, such as a reduced basis approach, it seems difficult to predict accurately low‐probability events. In this paper, we propose to combine a cross‐entropy method with a reduced basis algorithm to compute rare‐event (failure) probabilities. [ABSTRACT FROM AUTHOR]

It is shown that the computational efficiency of the discrete least-squares (DLS) approximation of solutions of stochastic elliptic PDEs is improved by incorporating a reduced-basis method into the DLS framework. In particular, we consider stochastic elliptic PDEs with an affine dependence on the random variable. The goal is to recover the entire solution map from the parameter space to the finite element space. To this end, first, a reduced-basis solution using a weak greedy algorithm is constructed, then a DLS approximation is determined by evaluating the reduced-basis approximation instead of the full finite element approximation. The main advantage of the new approach is that one only need apply the DLS operator to the coefficients of the reduced-basis expansion, resulting in huge savings in both the storage of the DLS coefficients and the online cost of evaluating the DLS approximation. In addition, the recently developed quasi-optimal polynomial space is also adopted in the new approach, resulting in superior convergence rates for a wider class of problems than previous analyzed. Numerical experiments are provided that illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract We propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to deal with complex parametrized domains in an efficient and straightforward way. The impact of the proposed approach is threefold. First, problems involving parametrizations of complex geometrical shapes and/or large domain deformations can be efficiently solved at full-order by means of the SBM. This unfitted boundary method permits to avoid remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary. Second, the computational effort is reduced by the development of a Reduced Order Model (ROM) technique based on a POD-Galerkin approach. Third, the SBM provides a smooth mapping from the true to the surrogate domain, and for this reason, the stability and performance of the reduced order basis are enhanced. This feature is the net result of the combination of the proposed ROM approach and the SBM. Similarly, the combination of the SBM with a projection-based ROM gives the great advantage of an easy and fast to implement algorithm considering geometrical parametrization with large deformations. The transformation of each geometry to a reference geometry (morphing) is in fact not required. These combined advantages will allow the solution of PDE problems more efficiently. We illustrate the performance of this approach on a number of two-dimensional Stokes flow problems. Highlights • A POD-Galerkin approach to handle problems involving parametrizations of complex geometrical shapes and/or large domain deformations has been developed. • On the contrary to traditional RB-Galerkin approaches, the proposed methodology do not require the transformation of each geometry to a reference one. • The tedious handling of cut cells is avoided at the full order level by the SBM formulation, keeping the good properties that other related methods like CutFEM have demonstrated. • The SBM provides a smooth mapping from the true to the surrogate domain, and for this reason,the stability and performance of the reduced basis are enhanced. • The efficiency and accuracy of the proposed reduced order model have been numerically verified against the full order model. [ABSTRACT FROM AUTHOR]

A methodology for using random sketching in the context of model order reduction for high-dimensional parameter-dependent systems of equations was introduced in Balabanov and Nouy (Part I; Advances in Computational Mathematics 45:2969–3019, 2019). Following this framework, we here construct a reduced model from a small, efficiently computable random object called a sketch of a reduced model, using minimal residual methods. We introduce a sketched version of the minimal residual based projection as well as a novel nonlinear approximation method, where for each parameter value, the solution is approximated by minimal residual projection onto a subspace spanned by several vectors picked (online) from a dictionary of candidate basis vectors. It is shown that random sketching technique can improve not only efficiency but also numerical stability. A rigorous analysis of the conditions on the random sketch required to obtain a given accuracy is presented. These conditions may be ensured a priori with high probability by considering for the sketching matrix an oblivious embedding of sufficiently large size. Furthermore, a simple and reliable procedure for a posteriori verification of the quality of the sketch is provided. This approach can be used for certification of the approximation as well as for adaptive selection of the size of the random sketching matrix. [ABSTRACT FROM AUTHOR]

The reduced basis method is a suitable technique for finding numerical solutions to partial differential equations that must be obtained for many values of parameters. This method is suitable when researching bifurcations and instabilities of stationary solutions for partial differential equations. It is necessary to solve numerically the partial differential equations along with the corresponding eigenvalue problems of the linear stability analysis of stationary solutions for a large number of bifurcation parameter values. In this paper, the reduced basis method has been used to solve eigenvalue problems derived from the linear stability analysis of stationary solutions in a two-dimensional Rayleigh–Bénard convection problem. The bifurcation parameter is the Rayleigh number, which measures buoyancy. The reduced basis considered belongs to the eigenfunction spaces derived from the eigenvalue problems for different types of solutions in the bifurcation diagram depending on the Rayleigh number. The eigenvalue with the largest real part and its corresponding eigenfunction are easily calculated and the bifurcation points are correctly captured. The resulting matrices are small, which enables a drastic reduction in the computational cost of solving the eigenvalue problems. [ABSTRACT FROM AUTHOR]

Abstract Many physical systems are modeled using partial differential equations (PDEs) with uncertain or random inputs. For such systems, naively propagating a fixed number of samples of the input probability law (or an approximation thereof) through the PDE is often inadequate to accurately quantify the "risk" associated with critical system responses. In this paper, we develop a goal-oriented, adaptive sampling and local reduced basis approximation for PDEs with random inputs. Our method determines a set of samples and an associated (implicit) Voronoi partition of the parameter domain on which we build local reduced basis approximations of the PDE solution. The samples are selected in an adaptive manner using an a posteriori error indicator. A notable advantage of the proposed approach is that the computational cost of the approximation during the adaptive process remains constant. We provide theoretical error bounds for our approximation and numerically demonstrate the performance of our method when compared to widely used adaptive sparse grid techniques. In addition, we tailor our approach to accurately quantify the risk of quantities of interest that depend on the PDE solution. We demonstrate our method on an advection–diffusion example and a Helmholtz example. Highlights • We develop a methodology for adaptively constructing local reduced bases. • We provide rigorous derivations of a posteriori errors for risk measures as QoI. • We introduce an implicit Voronoi-partitioning that underpins the concept of local bases. • We demonstrate the performance of the method using a damped Helmholtz problem and an advection–diffusion problem. • We compare the efficiency and accuracy of our method to local and global sparse grid algorithms. [ABSTRACT FROM AUTHOR]

We propose a forecasting method for predicting epidemiological health series on a two-week horizon at regional and interregional resolution. The approach is based on the model order reduction of parametric compartmental models and is designed to accommodate small amounts of sanitary data. The efficiency of the method is shown in the case of the prediction of the number of infected people and people removed from the collected data, either due to death or recovery, during the two pandemic waves of COVID-19 in France, which took place approximately between February and November 2020. Numerical results illustrate the promising potential of the approach.