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In a 3 D bounded and C 1 , α -smooth domain Ω, α ∈ (0 , 1) , we distribute a cluster of nanoparticles enjoying moderately contrasting relative permittivity and permeability which can be anisotropic. We show that the effective permittivity and permeability generated by such cluster is explicitly characterized by the corresponding electric and magnetic polarization tensors of the fixed shape. The error of the approximation of the scattered fields corresponding to the cluster and the effective medium is inversely proportional to the dilution parameter c r : = δ a , where a is the maximum diameter of the nanoparticles and δ is the minimum distance between them. The constant of the proportionality is given in terms of a-priori bounds on the cluster of nanoparticles (i.e. upper and lower bounds on their permittivity and permeability parameters, upper bound on the dilution parameter c r , the used incident frequency and the domain Ω). A key point of the analysis is to show that the Foldy-Lax field appearing in the meso-scale approximation, derived in [17] , is a discrete form of a (continuous) system of Lippmann-Schwinger equations with a related effective permittivity and permeability contrasts. To derive this, we prove that the Lippmann-Schwinger operator, for the Maxwell system, is invertible in the Hölder spaces. As a by-product, this shows a Hölder regularity property of the electromagnetic fields up to the boundary of the inhomogeneity. [ABSTRACT FROM AUTHOR]

This paper studies the approximation capacity of ReLU neural networks with norm constraint on the weights. We prove upper and lower bounds on the approximation error of these networks for smooth function classes. The lower bound is derived through the Rademacher complexity of neural networks, which may be of independent interest. We apply these approximation bounds to analyze the convergences of regression using norm constrained neural networks and distribution estimation by GANs. In particular, we obtain convergence rates for over-parameterized neural networks. It is also shown that GANs can achieve optimal rate of learning probability distributions, when the discriminator is a properly chosen norm constrained neural network. [ABSTRACT FROM AUTHOR]

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root and we compute its multiplicity structure. More precisely, given a polynomial system f = (f 1 , ... , f N) ∈ C [ x 1 , ... , x n ] N , we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so-called inverse system that describes the multiplicity structure at the root. We use α -theory test to certify the quadratic convergence, and to give bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria. [ABSTRACT FROM AUTHOR]

Statistical descriptions of particulate phases and granular media based on the notion of a property distribution are particularly advantageous for dense, many-particle dispersions since they support the direct evaluation of Eulerian property statistics, permit a kinematically simple treatment of particle collisions and feature a model complexity that is independent of the number of physical particles. On the minus side, the property distribution is governed by a high-dimensional population balance equation (PBE) whose moment reduction is accompanied by a challenging closure problem and engenders intricate dynamical nonlinearities. Circumventing both the curse of dimensionality and the moment closure problem, we present a Eulerian Monte Carlo (EMC) solution method that harnesses a representation of the property distribution in terms of an ensemble of property samples and is based on a statistical enforcement of the spatially and temporally discretized PBE. After application of a kinetic finite volume scheme, the PBE is subjected to a fractional steps decomposition that separates spatial transport from transport in property space. Conceptually, spatial transport is emulated by assembling the ensemble in one spatial grid cell from samples drawn from the ensembles in neighbouring cells at the previous time point. In this regard, we propose a two-level sampling scheme that is sensitive to large-scale features of the marginal velocity distribution and effectively reduces the instantaneous sampling error. The property transport step, by contrast, is based on the method of characteristics and involves the propagation of individual samples as property characteristics in time. While the sampling error afflicting the property statistics needs to be controlled using a top-level time or ensemble averaging, the EMC method remains well-conditioned for an arbitrary number of samples, is easy to implement and involves a computational expense that scales only linearly with the number of particle properties. Based on two benchmark examples, we analyze the spatial and statistical convergence properties of the EMC method and demonstrate its capacity to capture multiple crossings of particle trajectories. A third example involves the two-way coupled transport of size-polydispersed inertial particles in an inhomogeneous flow field. • We present a Eulerian Monte Carlo (EMC) method for solving multivariate population balances. • The particle property distribution is represented in terms of the total number density and an ensemble of property samples. • In a spatial grid cell, the samples evolve as jump-drift processes with neighbour and mean-field interactions. • While closure problems do not arise, a statistical approximation error occurs. • The EMC method is well-suited for capturing crossing trajectories in inertial particulate flows. [ABSTRACT FROM AUTHOR]

We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results. [ABSTRACT FROM AUTHOR]

Consider the classical supervised learning problem: we are given data (y i , x i) , i ≤ n , with y i a response and x i ∈ X a covariates vector, and try to learn a model f ˆ : X → R to predict future responses. Random feature methods map the covariates vector x i to a point ϕ (x i) in a higher dimensional space R N , via a random featurization map ϕ . We study the use of random feature methods in conjunction with ridge regression in the feature space R N. This can be viewed as a finite-dimensional approximation of kernel ridge regression (KRR), or as a stylized model for neural networks in the so called lazy training regime. We define a class of problems satisfying certain spectral conditions on the underlying kernels, and a hypercontractivity assumption on the associated eigenfunctions. These conditions are verified by classical high-dimensional examples. Under these conditions, we prove a sharp characterization of the error of random feature ridge regression. In particular, we address two fundamental questions: (1) What is the generalization error of KRR? (2) How big N should be for the random feature approximation to achieve the same error as KRR? In this setting, we prove that KRR is well approximated by a projection onto the top ℓ eigenfunctions of the kernel, where ℓ depends on the sample size n. We show that the test error of random feature ridge regression is dominated by its approximation error and is larger than the error of KRR as long as N ≤ n 1 − δ for some δ > 0. We characterize this gap. For N ≥ n 1 + δ , random features achieve the same error as the corresponding KRR, and further increasing N does not lead to a significant change in test error. [ABSTRACT FROM AUTHOR]

It is shown that weighted linear combinations of the generalised- α method and certain related higher order schemes allow for the formulation of unconditionally stable single step time integration methods with improved second order accuracy and more targeted high-frequency damping. It is also shown that, if the user controlled high frequency damping parameter ρ ∞ is set to zero, the new schemes can be expressed as linear multistep backward difference formulae and, in a particular case, recover Park's method. The performance of the proposed methods is illustrated in terms of mathematical analysis and a number of linear and nonlinear numerical examples including finite element based solutions of the incompressible Navier-Stokes equations. • New single step, second order, unconditionally stable implicit time integration schemes with controlled numerical damping. • New second order backward difference formulae. • Smaller approximation errors than generalised- α method and standard second order backward difference formula. • Trivial computer implementation and negligible additional computational cost. [ABSTRACT FROM AUTHOR]

In this paper, we propose an augmented subspace based adaptive proper orthogonal decomposition (POD) method for solving the time dependent partial differential equations. We use the difference between the approximation obtained in the augmented subspace and that obtained in the original POD subspace to construct an error indicator, by which we obtain a general framework for augmented subspace based adaptive POD method. We then provide two strategies to construct the augmented subspaces, the residual type augmented subspace and the coarse-grid approximation type augmented subspace. We apply our new methods to two typical 3D advection-diffusion equations with the advection being the Kolmogorov flow and the ABC flow. Numerical results show that both the residual type augmented subspace based adaptive POD method and the coarse-grid approximation type augmented subspace based adaptive POD method are more efficient than the existing adaptive POD methods, especially for the advection dominated models. • Propose an augmented subspace approximation based error indicator for the POD approximation of the time dependent PDEs. • Obtain a general framework of augmented subspace based adaptive POD method for solving the time dependent PDEs. • Provide two strategies to obtain the augmented subspaces, the residual type and the coarse-grid approximation type. • Numerical experiments show that the new method is more efficient than the existing adaptive POD methods. [ABSTRACT FROM AUTHOR]

Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen the emergence of neural PDE solvers, leveraging deep networks to effectively tackle high dimensional PDE problems. This study introduces Inf-SupNet, a model-based unsupervised learning approach designed to acquire solutions for a specific category of elliptic PDEs. The fundamental concept behind Inf-SupNet involves incorporating the inf-sup formulation of the underlying PDE into the loss function. The analysis reveals that the global solution error can be bounded by the sum of three distinct errors: the numerical integration error, the duality gap of the loss function (training error), and the neural network approximation error for functions within Sobolev spaces. To validate the efficacy of the proposed method, numerical experiments conducted in high dimensions demonstrate its stability and accuracy across various boundary conditions, as well as for both semi-linear and nonlinear PDEs. • Introduce a novel InfSupNet for learning PDE solutions. • Establish the equivalence between the PDE and the inf-sup formulation. • Decompose the total error and estimate the error bounds. • Demonstrate the effectiveness of our method in comprehensive experiments. [ABSTRACT FROM AUTHOR]

It is increasingly realized that taking stochastic effects into account is important in order to study biological cells. However, the corresponding mathematical formulation, the chemical master equation (CME), suffers from the curse of dimensionality and thus solving it directly is not feasible for most realistic problems. In this paper we propose a dynamical low-rank algorithm for the CME that reduces the dimensionality of the problem by dividing the reaction network into partitions. Only reactions that cross partitions are subject to an approximation error (everything else is computed exactly). This approach, compared to the commonly used stochastic simulation algorithm (SSA, a Monte Carlo method), has the advantage that it is completely noise-free. This is particularly important if one is interested in resolving the tails of the probability distribution. We show that in some cases (e.g. for the lambda phage) the proposed method can drastically reduce memory consumption and run time and provide better accuracy than SSA. • We propose a low-rank algorithm that does not rely on single orbital basis functions. • The proposed approach is better suited for biological/chemical reaction networks. • We demonstrate better accuracy and improved efficiency compared to SSA for some problems. • The proposed complexity reduction algorithm is noise free (in contrast to SSA). • We demonstrate that our algorithm accurately resolves the tails of the distribution (in contrast to SSA). [ABSTRACT FROM AUTHOR]

This work provides new results for the analysis of random sequences in terms of ℓ p -compressibility. The results characterize the degree in which a random sequence can be approximated by its best k -sparse version under different rates of significant coefficients (compressibility analysis). In particular, the notion of strong ℓ p -characterization is introduced to denote a random sequence that has a well-defined asymptotic limit (sample-wise) of its best k -term approximation error when a fixed rate of significant coefficients is considered (fixed-rate analysis). The main theorem of this work shows that the rich family of asymptotically mean stationary (AMS) processes has a strong ℓ p -characterization. Furthermore, we present results that characterize and analyze the ℓ p -approximation error function for this family of processes. Adding ergodicity in the analysis of AMS processes, we introduce a theorem demonstrating that the approximation error function is constant and determined in closed-form by the stationary mean of the process. Our results and analyses contribute to the theory and understanding of discrete-time sparse processes and, on the technical side, confirm how instrumental the point-wise ergodic theorem is to determine the compressibility expression of discrete-time processes even when stationarity and ergodicity assumptions are relaxed. [ABSTRACT FROM AUTHOR]

We study convergence analysis of the first-order stochastic consensus model with random batch interactions. The proposed model can be obtained via random batch method (RBM) from the first-order nonlinear consensus model. This model has two competing mechanisms, namely intrinsic free flow and nonlinear consensus interaction terms. From the competition between the two mechanisms, the original (full batch) model can admit relative equilibria and relaxation of the dynamics to the relative equilibrium in a large coupling regime. In authors' earlier work, we have studied the RBM approximation and its uniform error analysis. In this paper, we present two convergence analysis of RBM solutions toward the relative equilibrium. More precisely, we show that the variances of displacement processes between the full batch and random batch solutions tend to zero exponentially fast, as time goes to infinity. Second, we also show that, almost surely, the diameter process of displacement tends to zero exponentially. [ABSTRACT FROM AUTHOR]

In this paper, we propose a unified theoretical and practical spherical approximation framework for functional inverse problems on the hypersphere S d − 1. More specifically, we consider recovering spherical fields directly in the continuous domain using functional penalised basis pursuit problems with generalised total variation (gTV) regularisation terms. Our framework is compatible with various measurement types as well as non-differentiable convex cost functionals. Via a novel representer theorem, we characterise their solution sets in terms of spherical splines with sparse innovations. We use this result to derive an approximate canonical spline-based discretisation scheme, with vanishing approximation error. To solve the resulting finite-dimensional optimisation problem, we propose an efficient and provably convergent primal-dual splitting algorithm. We illustrate the versatility of our framework on real-life examples from the field of environmental sciences. [ABSTRACT FROM AUTHOR]

The modularity is the best known and widely used quality function for community detection in graphs. We investigate the approximability of the modularity maximization problem and some related problems. We first design a polynomial-time 0.4209-additive approximation algorithm for the modularity maximization problem, which improves the current best additive approximation error of 0.4672. Our theoretical analysis also demonstrates that the proposed algorithm obtains a nearly-optimal solution for any instance with a high modularity value. We next design a polynomial-time 0.1660-additive approximation algorithm for the maximum modularity cut problem. Finally, we extend our algorithm to some related problems. [ABSTRACT FROM AUTHOR]

As an extension of the standard paradigm in statistical learning theory, we introduce the concept of r -learnability, 0 < r ≤ 1 , which is a notion very closely related to that of nonexact oracle inequalities (see Lecue and Mendelson (2012) [7]). The r -learnability concept can enable so-called fast learning rates (along with corresponding sample complexity-type bounds) to be established at the cost of multiplying the approximation error term by an extra (1 + r) -factor in the learning error estimate. We establish a new, general r -learning bound (nonexact oracle inequality) yielding fast learning rates in probability (up to at most a logarithmic factor) for proper learning in the general setting of an agnostic model, essentially only assuming a uniformly bounded squared loss function and a hypothesis class of finite VC-dimension (that is, finite pseudo-dimension). [ABSTRACT FROM AUTHOR]

We study wave-packets in nonlinear periodic media in arbitrary (d) spatial dimension, modeled by the cubic Gross-Pitaevskii equation. In the asymptotic setting of small and broad wave-packets with N ∈ N carrier Bloch waves the effective equations for the envelopes are first order coupled mode equations (CMEs). We provide a rigorous justification of the effective equations. The estimate of the asymptotic error is carried out in an L 1 -norm in the Bloch variables. This translates to a supremum norm estimate in the physical variables. In order to investigate the existence of gap solitons of the d -dimensional CMEs, we discuss spectral gaps of the CMEs. For N = 4 and d = 2 a family of time harmonic gap solitons is constructed formally asymptotically and numerically. Moving gap solitons have not been found for d > 1 and for the considered values of N due to the absence of a spectral gap in the standard moving frame variables. [ABSTRACT FROM AUTHOR]

In this paper, we present a generalization of the parameterization method, introduced by Cabré, Fontich and De la Llave, to center manifolds associated to non-hyperbolic fixed points of discrete dynamical systems. As a byproduct, we find a new proof for the existence and regularity of center manifolds. However, in contrast to the classical center manifold theorem, our parameterization method will simultaneously obtain the center manifold and its conjugate center dynamical system. Furthermore, we will provide bounds on the error between approximations of the center manifold and the actual center manifold, as well as bounds for the error in the conjugate dynamical system. [ABSTRACT FROM AUTHOR]

In this paper we obtain a new constant in the Pólya-Vinogradov inequality. Our argument follows previously established techniques which use the Fourier expansion of an interval to reduce to Gauss sums. Our improvement comes from approximating an interval by a function with slower decay on the edges and this allows for a better estimate of the ℓ 1 norm of the Fourier transform. This approximation induces an error for our original sums which we deal with by combining some ideas of Hildebrand with Garaev and Karatsuba concerning long character sums. [ABSTRACT FROM AUTHOR]

A hierarchical solver is proposed for solving sparse ill-conditioned linear systems in parallel. The solver is based on a modification of the LoRaSp method, but employs a deferred-compression technique, which provably reduces the approximation error and significantly improves efficiency. Moreover, the deferred-compression technique introduces minimal overhead and does not affect parallelism. As a result, the new solver achieves linear computational complexity under mild assumptions and excellent parallel scalability. To demonstrate the performance of the new solver, we focus on applying it to solve sparse linear systems arising from ice sheet modeling. The strong anisotropic phenomena associated with the thin structure of ice sheets creates serious challenges for existing solvers. To address the anisotropy, we additionally developed a customized partitioning scheme for the solver, which captures the strong-coupling direction accurately. In general, the partitioning can be computed algebraically with existing software packages, and thus the new solver is generalizable for solving other sparse linear systems. Our results show that ice sheet problems of about 300 million degrees of freedom have been solved in just a few minutes using 1024 processors. • We introduced the deferred-compression technique in hierarchical solvers for solving sparse ill-conditioned linear systems. • We developed a parallel hierarchical solver and compared its performance with other state-of-the-art methods. • We solved a problem from ice sheet modeling with 300 million degrees of freedom in a few minutes on a thousand processors. [ABSTRACT FROM AUTHOR]

Polyhedral-type approximations of convex-like domains in C d have been considered recently by the second author. In particular, the decay rate of the error in optimal volume approximation as a function of the number of facets has been obtained. In this article, we take these studies further by investigating polyhedra constructed using random points (Poisson or binomial process) on the boundary of a strongly C -convex domain. We determine the rate of error in volume approximation of the domain by random polyhedra, and conjecture the precise value of the minimal limiting constant. Analogous to the real case, the exponent appearing in the error rate of random volume approximation coincides with that of optimal volume approximation, and can be interpreted in terms of the Hausdorff dimension of a naturally-occurring metric space. Moreover, the limiting constant is conjectured to depend on the Möbius-Fefferman measure , which is a complex analogue of the Blaschke surface area measure. Finally, we also prove L 1 -convergence, variance bounds, and normal approximation. [ABSTRACT FROM AUTHOR]

In the present paper we establish estimates for the error of approximation (in the L p -norm) achieved by neural network (NN) operators. The above estimates have been given by means of an averaged modulus of smoothness introduced by Sendov and Popov, also known with the name of τ -modulus, in case of bounded and measurable functions on the interval [ − 1 , 1 ]. As a consequence of the above estimates, we can deduce an L p convergence theorem for the above family of NN operators in case of functions which are bounded, measurable, and Riemann integrable on the above interval. In order to reach the above aims, we preliminarily establish a number of results; among them we can mention an estimate for the p -norm of the operators, and an asymptotic type theorem for the NN operators in case of functions belonging to Sobolev spaces. [ABSTRACT FROM AUTHOR]

Learning can be seen as approximating an unknown function by interpolating the training data. Although Kriging offers a solution to this problem, it requires the prior specification of a kernel and it is not scalable to large datasets. We explore a numerical approximation approach to kernel selection/construction based on the simple premise that a kernel must be good if the number of interpolation points can be halved without significant loss in accuracy (measured using the intrinsic RKHS norm ‖ ⋅ ‖ associated with the kernel). We first test and motivate this idea on a simple problem of recovering the Green's function of an elliptic PDE (with inhomogeneous coefficients) from the sparse observation of one of its solutions. Next we consider the problem of learning non-parametric families of deep kernels of the form K 1 (F n (x) , F n (x ′)) with F n + 1 = (I d + ϵ G n + 1) ∘ F n and G n + 1 ∈ span { K 1 (F n (x i) , ⋅) }. With the proposed approach constructing the kernel becomes equivalent to integrating a stochastic data driven dynamical system, which allows for the training of very deep (bottomless) networks and the exploration of their properties. These networks learn by constructing flow maps in the kernel and input spaces via incremental data-dependent deformations/perturbations (appearing as the cooperative counterpart of adversarial examples) and, at profound depths, they (1) can achieve accurate classification from only one data point per class (2) appear to learn archetypes of each class (3) expand distances between points that are in different classes and contract distances between points in the same class. For kernels parameterized by the weights of Convolutional Neural Networks, minimizing approximation errors incurred by halving random subsets of interpolation points, appears to outperform training (the same CNN architecture) with relative entropy and dropout. • The relative error made by interpolating a subset of the data and half of that subset can be used as an ordering principle for learning kernels with good generalization properties. • The resulting algorithm (Kernel Flows, KF) learns kernels by simulating a data driven dynamical system and enables scalability of Kernel Regression. • As a regression tool KF leads to improved accuracy and can recover the rough coefficients of elliptic PDEs from partial measurements. • KF learns a kernel capable of generalization from one sample per class and learns archetypes of each class. KF unrolls the classical Swiss Roll Cheesecake classification data set. • KF applied to Convolutional Neural Network architecture can outperform training with relative entropy and dropout. [ABSTRACT FROM AUTHOR]

We present an approximation scheme for functions in three dimensions, that requires only their samples on the Cartesian grid, under the assumption that the functions are sufficiently concentrated in both space and frequency. The scheme is based on expanding the given function in the basis of generalized prolate spheroidal wavefunctions, with the expansion coefficients given by weighted dot products between the samples of the function and the samples of the basis functions. As numerical implementations require all expansions to be finite, we present a truncation rule for the expansions. Finally, we derive a bound on the overall approximation error in terms of the assumed space/frequency concentration. [ABSTRACT FROM AUTHOR]

Abstract Let E n (f) α , β , γ denote the error of best approximation by polynomials of degree at most n in the space L 2 (ϖ α , β , γ) on the triangle { (x , y) : x , y ≥ 0 , x + y ≤ 1 } , where ϖ α , β , γ (x , y) ≔ x α y β (1 − x − y) γ for α , β , γ > − 1. Our main result gives a sharp estimate of E n (f) α , β , γ in terms of the error of best approximation for higher order derivatives of f in appropriate Sobolev spaces. The result also leads to a characterization of E n (f) α , β , γ by a weighted K -functional. [ABSTRACT FROM AUTHOR]

Abstract Electroencephalography (EEG) source imaging is an ill-posed inverse problem that requires accurate conductivity modelling of the head tissues, especially the skull. Unfortunately, the conductivity values are difficult to determine in vivo. In this paper, we show that the exact knowledge of the skull conductivity is not always necessary when the Bayesian approximation error (BAE) approach is exploited. In BAE, we first postulate a probability distribution for the skull conductivity that describes our (lack of) knowledge on its value, and model the effects of this uncertainty on EEG recordings with the help of an additive error term in the observation model. Before the Bayesian inference, the likelihood is marginalized over this error term. Thus, in the inversion we estimate only our primary unknown, the source distribution. We quantified the improvements in the source localization when the proposed Bayesian modelling was used in the presence of different skull conductivity errors and levels of measurement noise. Based on the results, BAE was able to improve the source localization accuracy, particularly when the unknown (true) skull conductivity was much lower than the expected standard conductivity value. The source locations that gained the highest improvements were shallow and originally exhibited the largest localization errors. In our case study, the benefits of BAE became negligible when the signal-to-noise ratio dropped to 20 dB. Highlights • Bayesian uncertainty modelling was used for the unknown skull conductivity in EEG. • Modelling error statistics were pre-computed before the source analysis. • Source localization accuracy improved particularly for the sources close to the skull. • Computational complexity was essentially the same as in the plain source analysis. • Bayesian uncertainty modelling is easily adjustable to include other modelling errors. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate optimal linear approximations (n -approximation numbers) of the embeddings from the Sobolev spaces H r (r > 0) for various equivalent norms and the Gevrey type spaces G α , β (α , β > 0) on the sphere S d and on the ball B d , where the approximation error is measured in the L 2 -norm. We obtain preasymptotics, asymptotics, and strong equivalences of the above approximation numbers as a function in n and the dimension d. We emphasize that all equivalence constants in the above preasymptotics and asymptotics are independent of the dimension d and n. As a consequence we obtain that for the absolute error criterion the approximation problems I d : H r → L 2 are weakly tractable if and only if r > 1 , not uniformly weakly tractable, and do not suffer from the curse of dimensionality. We also prove that for any α , β > 0 , the approximation problems I d : G α , β → L 2 are uniformly weakly tractable, not polynomially tractable, and quasi-polynomially tractable if and only if α ≥ 1. [ABSTRACT FROM AUTHOR]

Physics-informed Neural Network (PINN) is a promising tool that has been applied in a variety of physical phenomena described by partial differential equations (PDE). However, it has been observed that PINNs are difficult to train in certain "stiff" problems, which include various nonlinear hyperbolic PDEs that display shocks in their solutions. Recent studies added a diffusion term to the PDE, and an artificial viscosity (AV) value was manually tuned to allow PINNs to solve these problems. In this paper, we propose three approaches to address this problem, none of which rely on an a priori definition of the artificial viscosity value. The first method learns a global AV value, whereas the other two learn localized AV values around the shocks, by means of a parametrized AV map or a residual-based AV map. We applied the proposed methods to the inviscid Burgers equation and the Buckley-Leverett equation, the latter being a classical problem in Petroleum Engineering. The results show that the proposed methods are able to learn both a small AV value and the accurate shock location and improve the approximation error over a nonadaptive global AV alternative method. • Methods to add adaptive artificial viscosity to train PINNs to solve hyperbolic PDEs with shocks. • Adaptive methods accurately learn both the value and location of the artificial viscosity. • Successfully solved the Inviscid Burgers and Buckley-Leverett equations. • Additional computation cost over the baseline PINN is small. [ABSTRACT FROM AUTHOR]

Polynomial chaos expansion (PCE) is a versatile tool widely used in uncertainty quantification and machine learning, but its successful application depends strongly on the accuracy and reliability of the resulting PCE-based response surface. High accuracy typically requires high polynomial degrees, demanding many training points especially in high-dimensional problems through the curse of dimensionality. So-called sparse PCE concepts work with a much smaller selection of basis polynomials compared to conventional PCE approaches and can overcome the curse of dimensionality very efficiently, but have to pay specific attention to their strategies of choosing training points. Furthermore, the approximation error resembles an uncertainty that most existing PCE-based methods do not estimate. In this study, we develop and evaluate a fully Bayesian approach to establish the PCE representation via joint shrinkage priors and Markov chain Monte Carlo. The suggested Bayesian PCE model directly aims to solve the two challenges named above: achieving a sparse PCE representation and estimating uncertainty of the PCE itself. The embedded Bayesian regularizing via the joint shrinkage prior allows using higher polynomial degrees for given training points due to its ability to handle underdetermined situations, where the number of considered PCE coefficients could be much larger than the number of available training points. We also explore multiple variable selection methods to construct sparse PCE expansions based on the established Bayesian representations, while globally selecting the most meaningful orthonormal polynomials given the available training data. We demonstrate the advantages of our Bayesian PCE and the corresponding sparsity-inducing methods on several benchmarks. • Polynomial Chaos expansions are surrogates for computationally expensive models. • A key problem is the number of polynomial expansion terms in high dimensions. • We regularize the polynomial coefficients with a Bayesian joint shrinkage prior. • This is combined with statistical methods for Bayesian variable selection. • Our method outperforms existing methods in terms of accuracy and predictive performance. [ABSTRACT FROM AUTHOR]

We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parametric partial differential equations (PDEs). Neural operators have gained significant attention in recent years for their ability to approximate the parameter-to-solution maps defined by PDEs using as training data solutions of PDEs at a limited number of parameter samples. The computational cost of BIPs can be drastically reduced if the large number of PDE solves required for posterior characterization are replaced with evaluations of trained neural operators. However, reducing error in the resulting BIP solutions via reducing the approximation error of the neural operators in training can be challenging and unreliable. We provide an a priori error bound result that implies certain BIPs can be ill-conditioned to the approximation error of neural operators, thus leading to inaccessible accuracy requirements in training. To reliably deploy neural operators in BIPs, we consider a strategy for enhancing the performance of neural operators: correcting the prediction of a trained neural operator by solving a linear variational problem based on the PDE residual. We show that a trained neural operator with error correction can achieve a quadratic reduction of its approximation error, all while retaining substantial computational speedups of posterior sampling when models are governed by highly nonlinear PDEs. The strategy is applied to two numerical examples of BIPs based on a nonlinear reaction–diffusion problem and deformation of hyperelastic materials. We demonstrate that posterior representations of the two BIPs produced using trained neural operators are greatly and consistently enhanced by error correction. • Neural operators with error-correction as surrogate models for nonlinear PDEs. • The error in Bayesian inversion is controlled by the neural operator approximation error. • Correction via solving a linear variational problem defined by the PDE residual. • Correction leads to quadratic error reduction of the neural operator approximation error. • Significantly improved inference results from error-corrected neural operators. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximation of f α (x) = | x | α , α > 0 in L ∞ [ − 1 , 1 ] by its Fourier–Legendre partial sum S n (α) (x). We derive the upper and lower bounds of the approximation error in the L ∞ -norm that are valid uniformly for all n ≥ n 0 for some n 0 ≥ 1. Such an optimal L ∞ -estimate requires a judicious summation rule that can recover the lost half order if one uses a naive summation. Consequently, we can obtain the explicit Bernstein-type constant B ∞ (α) ≔ lim n → ∞ n α ‖ f α − S n (α) ‖ L ∞ = 2 Γ (α) π | sin α π 2 |. Interestingly, using a similar argument, we can show that the Fourier–Chebyshev sum has the same Bernstein-type constant B ∞ (α) as the Legendre case. [ABSTRACT FROM AUTHOR]

Large-margin classifiers are powerful techniques for classification problems. Although binary large-margin classifiers are heavily studied, multicategory problems are more complicated and challenging. A common approach is to construct k different decision functions for a k -class problem with a sum-to-zero constraint. However, such a constraint can be inefficient. Moreover, many large-margin classifiers can be sensitive to outliers in the training sample. In this article, we use the angle-based classification framework to avoid the explicit sum-to-zero constraint, and we propose two adaptively weighted large-margin classification techniques. Our new methods are Fisher consistent and more robust against outliers under suitable conditions. Numerical experiments further indicate that our methods give competitive and stable performance when compared with existing approaches. [ABSTRACT FROM AUTHOR]

Consider an abstract operator L which acts on monomials x n according to L x n = λ n x n + ν n x n − 2 for λ n and ν n some coefficients. Let P n ( x ) be eigenpolynomials of degree n of L : L P n ( x ) = λ n P n ( x ) . A classification of all the cases for which the polynomials P n ( x ) are orthogonal is provided. A general derivation of the algebras explaining the bispectrality of the polynomials is given. The resulting algebras prove to be central extensions of the Askey–Wilson algebra and its degenerate cases. [ABSTRACT FROM AUTHOR]

For the 2D/3D time-dependent magnetohydrodynamics (MHD) system, we propose a new second order backward difference formula Newton scheme (SBDFN) which is a combination of second order backward difference approximation of the time derivative terms and Newton treatment of the nonlinear terms. Meanwhile, the finite element method is applied as spatial discretization. Firstly, the optimal convergence of spatially semi-discrete form is deduced. Secondly, the stability and well-posedness of SBDFN scheme are provided under τ < C , where C is independent of h . Based on these results, optimal error estimates of the scheme in time are proved by negative norm technique. Finally, numerical experiments are carried out to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

We introduce an approximation scheme for almost bandlimited functions which are sufficiently concentrated in a disk, based on their equally spaced samples on a Cartesian grid. The scheme is based on expanding the function into a series of two-dimensional prolate spheroidal wavefunctions, and estimating the expansion coefficients using the available samples. We prove that the approximate expansion coefficients have particularly simple formulas, in the form of a dot product of the available samples with samples of the basis functions. We also derive error bounds for the error incurred by approximating the expansion coefficients as well as by truncating the expansion. In particular, we derive a bound on the approximation error in terms of the assumed space/frequency concentration, and provide a simple truncation rule to control the length of the expansion and the resulting approximation error. [ABSTRACT FROM AUTHOR]

Recently, it has been shown in Hairer et al. (2015) that there exists a system of stochastic differential equations (SDE) on the time interval [ 0 , T ] with infinitely often differentiable and bounded coefficients such that the Euler scheme with equidistant time steps converges to the solution of this SDE at the final time in the strong sense but with no polynomial rate. Even worse, in Jentzen (2016) it has been shown that for any sequence ( a n ) n ∈ N ⊂ ( 0 , ∞ ) , which may converge to zero arbitrarily slowly, there exists an SDE on [ 0 , T ] with infinitely often differentiable and bounded coefficients such that no approximation of the solution of this SDE at the final time based on n evaluations of the driving Brownian motion at fixed time points can achieve a smaller absolute mean error than the given number a n . In the present article we generalize the latter result to the case when the approximations may choose the location as well as the number of the evaluation sites of the driving Brownian motion in an adaptive way dependent on the values of the Brownian motion observed so far. [ABSTRACT FROM AUTHOR]

Pore-scale models are useful for understanding and upscaling the mechanical deformation of fractured porous media. The highest fidelity solutions are obtained by direct numerical simulation (DNS), in which the governing equations are discretized and solved with a fine-grid solver (e.g., finite elements) over a 3D image (e.g., X-ray) of a porous sample. However, the downside of DNS is its high computational cost. We present a pore-level multiscale method (PLMM) that approximates DNS efficiently and with controllable accuracy in modeling the linear elastic response of porous solids with arbitrary microstructure and crack pattern. PLMM decomposes the solid into subdomains, over which local basis and correction functions are built. These functions are then coupled with a global problem to yield an approximate solution, whose errors can be iteratively corrected. A key feature of PLMM is that the decomposition need not conform to the cracks, unlike a previous formulation by the authors. This paves the way towards solving crack nucleation and growth problems in the future without the need to dynamically update the decomposition. We represent cracks diffusely using a phase-field variable and explore three strategies for capturing them through either the basis or correction functions. The implications of each strategy on the convergence rate and computational cost are analyzed. • Pore-level multiscale method developed for deformation of fractured porous media. • The method decomposes a porous solid into subdomains to decouple computations. • The decomposition need not conform to cracks, removing a key previous barrier. • Implications for fixed-crack versus evolving-crack problems are analyzed. • Approximation errors can be estimated and controlled iteratively. [ABSTRACT FROM AUTHOR]

Uncertainty parameters are ubiquitous in engineering and have a significant influence on the performance of mechanical systems. When evaluating the effect of uncertainty parameters on time-dependent response, a previous method is based on building surrogate models at each time instant for the amplitude, phase, and trend of the response, which can be obtained through signal decomposition techniques such as Hilbert–Huang transform. However, for a nonlinear oscillatory response, not only the response itself but also its amplitude might contain vibration components. Therefore, when modeling nonlinear responses, the previous method encounters an approximation error due to the increasingly complicated relationship between uncertainty parameters and the amplitude over time. To reduce this error, this paper proposes an amplitude-nested surrogate model using double-layer Hilbert–Huang transform through two steps. First, the response and its amplitude are successively decomposed through Hilbert–Huang transform to obtain the first- and second-layer decomposition results, respectively. Then, the polynomial surrogate models are built for the amplitude, phase, and trend in the first- and second-layer results at each time instant. The proposed method can improve the accuracy of the previous single-layer surrogate model without increasing the number of samples. Numerical examples have verified the effectiveness of the proposed method. This method can expand uncertainty analysis methods of dynamics. [Display omitted] [ABSTRACT FROM AUTHOR]

We present an Equation/Variable free machine learning (EVFML) framework for the control of the collective dynamics of complex/multiscale systems modeled via microscopic/agent-based simulators. The approach obviates the need for construction of surrogate, reduced-order models. The proposed implementation consists of three steps: (A) from high-dimensional agent-based simulations, machine learning, in particular, non-linear manifold learning (Diffusion Maps (DMs)) identifies a set of coarse-grained variables that parametrize the low-dimensional manifold on which the emergent/collective dynamics evolve. The out-of-sample extension and pre-image problems, i.e. the construction of non-linear mappings from the high-dimensional input space to the low-dimensional manifold and back, are solved by coupling DMs with the Nyström extension and Geometric Harmonics, respectively; (B) having identified the manifold and its coordinates, we exploit the Equation-free approach to perform numerical bifurcation analysis of the emergent dynamics; then (C) based on the previous steps, we design data-driven embedded wash-out controllers that drive the agent-based simulators to their intrinsic, imprecisely known, emergent open-loop unstable steady-states, thus demonstrating that the scheme is robust against numerical approximation errors and modeling uncertainty. The efficiency of the framework is illustrated by controlling emergent unstable (i) traveling waves of a deterministic agent-based model of traffic dynamics, and (ii) equilibria of a stochastic financial market model with mimesis. • We address a numerical methodology for the data-driven control of complex systems. • It bypasses the need to construct surrogate, reduced-order machine-learning models. • It is robust against modeling uncertainties and numerical approximation errors. • We used agent-based models of traffic dynamics, and a financial market. [ABSTRACT FROM AUTHOR]

In this paper we establish a multiscale approximation for random fields on the sphere using spherical needlets—a class of spherical wavelets. We prove that the semidiscrete needlet decomposition converges in mean and pointwise senses for weakly isotropic random fields on S d , d ≥ 2 . For numerical implementation, we construct a fully discrete needlet approximation of a smooth 2 -weakly isotropic random field on S d and prove that the approximation error for fully discrete needlets has the same convergence order as that for semidiscrete needlets. Numerical examples are carried out for fully discrete needlet approximations of Gaussian random fields and compared to a discrete version of the truncated Fourier expansion. [ABSTRACT FROM AUTHOR]

In this paper we introduce adaptive time step control for simulation of the evolution of ice sheets. The discretization error in the approximations is estimated using “Milne's device” by comparing the result from two different methods in a predictor–corrector pair. Using a predictor–corrector pair the expensive part of the procedure, the solution of the velocity and pressure equations, is performed only once per time step and an estimate of the local error is easily obtained. The stability of the numerical solution is maintained and the accuracy is controlled by keeping the local error below a given threshold using PI-control. Depending on the threshold, the time step Δ t is bound by stability requirements or accuracy requirements. Our method takes a shorter Δ t than an implicit method but with less work in each time step and the solver is simpler. The method is analyzed theoretically with respect to stability and applied to the simulation of a 2D ice slab and a 3D circular ice sheet. The stability bounds in the experiments are explained by and agree well with the theoretical results. [ABSTRACT FROM AUTHOR]

Robust and efficient automatic model order reduction is important in the design process of vibro-acoustic structures. For correct assessment of the reduction error without requiring the solution of the full order model, reliable error estimation techniques are required. Such error estimators can be used in iterative algorithms to automatically determine a reasonable order and expansion point distribution for reduced-order models. An accurate estimation of the approximation error is important in order to design an efficient adaptive model order reduction process requiring as little a priori knowledge of the system response as possible. In this contribution, different techniques for estimating the approximation error of reduced-order models are compared and employed in adaptive algorithms aiming at generating reduced-order models of vibro-acoustic systems. For many applications, vibro-acoustic systems need to be evaluated in a specific frequency range, as certain system characteristics are only observed in this region. Therefore, the employed error estimation methods are assessed regarding their effectiveness estimating the error in a particular frequency region. • We compare methods for estimating the approximation error of reduced-order models. • They are employed in a greedy algorithm to find locations for expansion points. • We show strategies to reduce the number of required error estimator evaluations. • We present an IRKA-like algorithm computing a reasonable size for a reduced model. • Its computational cost is reduced by employing reduced models of intermediate size. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new reduced basis methodology for accelerating the computation of large parameterized systems of high-fidelity integral equations. Core to our methodology is the use of coarse-proxy models (i.e., lower resolution variants of the underlying high-fidelity equations) to identify important samples in the parameter space from which a high quality reduced basis is then constructed. Unlike the more traditional POD or greedy methods for reduced basis construction, our methodology has the benefit of being both easy to implement and embarrassingly parallel. We apply our methodology to the under-served area of integral equations, where the density of the underlying integral operators has traditionally made reduced basis methods difficult to apply. To handle this difficulty, we introduce an operator interpolation technique, based on random sub-sampling, that is aimed specifically at integral operators. To demonstrate the effectiveness of our techniques, we present two numerical case studies, based on the radiative transport equation and a boundary integral formation of the Laplace equation respectively, where our methodology provides a significant improvement in performance over the underlying high-fidelity models for a wide range of error tolerances. Moreover, we demonstrate that for these problems, as the coarse-proxy selection threshold is made more aggressive, the approximation error of our method decreases at an approximately linear rate. Finally, we provide a public repository of our source code with easy instructions for reproducing all results in this paper. • Presents novel model reduction method for many-query integral equation problems. • Uses column-pivoted QR and coarse proxy models to efficiently extract training samples for a reduced basis. • Sparse sampling of integral operators allows for efficient reconstruction of Galerkin projected operators. • Benchmarks on radiative transport and Laplace equation show a 30× speedup with only a 4% sacrifice to accuracy. [ABSTRACT FROM AUTHOR]

We consider the approximation of manifold-valued functions by embedding the manifold into a higher dimensional space, applying a vector-valued approximation operator and projecting the resulting vector back to the manifold. It is well known that the approximation error for manifold-valued functions is close to the approximation error for vector-valued functions. This is not true anymore if we consider the derivatives of such functions. In our paper we give pre-asymptotic error bounds for the approximation of the derivative of manifold-valued function. In particular, we provide explicit constants that depend on the reach of the embedded manifold. [ABSTRACT FROM AUTHOR]

In the present work, a simplified way to apply tri-periodic Boundary Conditions (BCs) in hierarchical algorithms has been devised. The application of these BCs is demonstrated using an in-house algorithm that is used for the calculation of electrostatic interactions in a system of charged particles. The proposed quasi-periodic BCs entail a truncation of the infinite periodic domain with a reasonable cut-off error. For a correct representation of the physics, two properties have to be ensured: the convergence of the electrostatic forces and the isotropy of the electric field. The developed algorithm allows for a rather efficient and precise calculation of the former in a tri-periodic computational domain by separating them in short- and long-range parts, which are calculated exactly and approximately, respectively. The approximation error, computational cost and performance of the proposed algorithm are documented and thoroughly analyzed. Then, an application of the method is presented for dry mono-charged (all particles bear equal charge) particle flows where the fundamentals physics of particle-particle electrostatic interactions are investigated via characteristic length and time scales. It is shown that the underlying mechanism of these interactions is the Coulomb collision, a concept that allows to interpret these interactions in a rather intuitive way. In addition, an attempt is made to provide analytical estimations for several statistical quantities via dimensional analysis based on measured simulation data. Finally, the particle-induced electric field is presented and its characteristics are related to particle motion. • Development of tri-periodic boundary conditions for hierarchical algorithms. • Properties of isotropy and convergence of the particles-induced electric field. • Cut-off, approximation error and cost estimation of the pseudo-particle algorithms. • Application of the numerical method for dry mono-charged particle flows. • Hard collisions-electrostatic interactions analogy via effective Coulomb diameter. [ABSTRACT FROM AUTHOR]

Lattice-Boltzmann methods are known for their simplicity, efficiency and ease of parallelization, usually relying on uniform Cartesian meshes with a strong bond between spatial and temporal discretization. This fact complicates the crucial issue of reducing the computational cost and the memory impact by automatically coarsening the grid where a fine mesh is unnecessary, still ensuring the overall quality of the numerical solution through error control. This work provides a possible answer to this interesting question, by connecting, for the first time, the field of lattice-Boltzmann Methods (LBM) to the adaptive multiresolution (MR) approach based on wavelets. To this end, we employ a MR multi-scale transform to adapt the mesh as the solution evolves in time according to its local regularity. The collision phase is not affected due to its inherent local nature and because we do not modify the speed of the sound, contrarily to most of the LBM/Adaptive Mesh Refinement (AMR) strategies proposed in the literature, thus preserving the original structure of any LBM scheme. Besides, an original use of the MR allows the scheme to resolve the proper physics by efficiently controlling the accuracy of the transport phase. We carefully test our method to conclude on its adaptability to a wide family of existing lattice Boltzmann schemes, treating both hyperbolic and parabolic systems of equations, thus being less problem-dependent than the AMR approaches, which have a hard time guaranteeing an effective control on the error. The ability of the method to yield a very efficient compression rate and thus a computational cost reduction for solutions involving localized structures with loss of regularity is also shown, while guaranteeing a precise control on the approximation error introduced by the spatial adaptation of the grid. The numerical strategy is implemented on a specific open-source platform called SAMURAI with a dedicated data-structure relying on set algebra. • New strategy for LBM with dynamic mesh adaptation by multiresolution: error-control. • Original LBM scheme without alteration, still reaching a controlled accuracy. Straightforwardly extendable to any LBM scheme. • Very reduced memory footprint preserving a given accuracy prescribed by the user. • Thorough assessment: hyperbolic conservation laws and the incompressible NS. • Algorithms implemented on a dedicated open-source structure called SAMURAI. [ABSTRACT FROM AUTHOR]

The past two decades have borne remarkable progress in the application of the molecular dynamics method in a number of engineering problems. However, the computational efficiency is limited by the massive-atoms system, and the study of rare dynamically-relevant events is challenging at the timescale of molecular dynamics. In this work, a multi-scale molecular simulation algorithm is proposed with a novel toy model that can mimic the state transitions in extensive scenarios. The algorithm consists of two scales, including producing the realistic particle trajectory and probability transition matrix in the molecular dynamics scale and calculating the state distribution and residence time in the Monte Carlo scale. A new state definition is proposed to take the velocity direction into consideration, and different coarsening models are established in the spatial and time scales. The accuracy, efficiency, and robustness of our proposed multi-scale method have been validated, and the general applicability is also demonstrated by explaining two practical applications in the shale gas adsorption and protein folding problems respectively. • A multi-scale molecular simulation algorithm is developed in this paper in both molecular dynamics (MD) scale and Monte Carlo (MC) scale, and is proved to be accurate, robust and efficient. • A novel toy model is proposed for the first time to study typical state transition problems that can be applied in various engineering scenarios, which only needs one atom to generate the chaotic trajectory. • The sensitivities of the toy model are tested, including the effect of external force and channel size, in order to benefit further applications. • A new state definition is proposed to take the velocity direction into consideration, and different coarsening models are developed in the spatial and time scales. • The general applicability is demonstrated by explaining two practical applications in the shale gas adsorption and protein folding problems respectively. [ABSTRACT FROM AUTHOR]

Many particulate systems occurring in nature and technology are adequately described by a number density function (NDF). The numerical solution of the corresponding population balance equation (PBE) is typically accompanied by high computational costs. Quadrature-based moment methods are an approach to reduce the computational complexity by solving only for a set of moments associated with the NDF employing Gaussian quadrature rules to close the moment equations derived from the PBE. The evolution of a population of inertial particles dispersed in a turbulent fluid is governed by a PBE with a phase-space diffusion term as a result of random microscale fluctuations. Considerations on the microscopic behavior concerning the momentum exchange between fluid and dispersed particles suggest that this diffusion term is nonlinear and nonsmooth. The resulting integral terms in the derived moment equations entail large approximation errors when using Gaussian quadrature rules for closure. In this work, we propose a modification of the quadrature method of moments (QMOM), namely the Gauss/anti-Gauss-QMOM (GaG-QMOM), making use of anti-Gaussian quadrature formulae to reduce the large errors due to this particular form of diffusion. This new method is investigated in a series of simple one-dimensional test cases with analytical reference solutions. Moreover, we propose a realizability-preserving variation of the strong-stability preserving Runge-Kutta (RKSSP) schemes that is suited to problems involving phase-space diffusion. Besides the observation that the modified second-order RKSSP scheme can serve as a realizability-preserving alternative to the standard RKSSP scheme of the same order, the numerical results reveal that, compared to the standard QMOM, the GaG-QMOM can reduce the large errors by one to two orders of magnitude. • Modeling turbulence in generalized PBEs may involve nonlinear phase-space diffusion. • Nonlinear/nonsmooth terms in moment equations are particularly challenging for QBMMs. • Gauss/anti-Gauss QMOM can significantly improve approximation of nonsmooth terms. • The suggested modified RKSSP scheme can ensure moment realizability unconditionally. [ABSTRACT FROM AUTHOR]

We present a lower error bound for approximating linear multivariate operators defined over Hilbert spaces in terms of the error bounds for appropriately constructed linear functionals as long as algorithms use function values. Furthermore, some of these linear functionals have the same norm as the linear operators. We then apply this error bound for linear (unweighted) tensor products. In this way we use negative tractability results known for linear functionals to conclude the same negative results for linear operators. In particular, we prove that L 2 -multivariate approximation defined for standard Sobolev space suffers the curse of dimensionality if function values are used although the curse is not present if linear functionals are allowed. [ABSTRACT FROM AUTHOR]