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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]

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]

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]

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]

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 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]

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]

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]

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]

A mode of analysis of the errors in a mathematical table is provided by examining the residuals, the differences between the entries that appear in the table and the appropriate computed values. For the most part, the residuals used here are integer-valued, related to the precision of the table entries. These residuals can provide information about typographical errors, round-off errors, calculation errors and errors due to approximation. Residuals may also be used to make inferences about unknown (to us today) parameters that were used in the construction of a table. The methodology is illustrated by examining four different historical tables. [ABSTRACT FROM AUTHOR]

We consider a generalised Webster's equation for describing wave propagation in curved tubular structures such as variable diameter acoustic wave guides. Webster's equation in generalised form has been rigorously derived in a previous article starting from the wave equation, and it approximates cross-sectional averages of the propagating wave. Here, the approximation error is estimated by an a posteriori technique. [ABSTRACT FROM AUTHOR]

A persistent item in a stream is one that occurs regularly in the stream without necessarily contributing significantly to the volume of the stream. Persistent items are often associated with anomalies in network streams, such as botnet traffic and click fraud. While it is important to track persistent items in an online manner, it is challenging to zero-in on such items in a massive distributed stream. We present the first communication-efficient distributed algorithms for tracking persistent items in a data stream whose elements are partitioned across many different sites. We consider both infinite window and sliding window settings, and present algorithms that can track persistent items approximately with a probabilistic guarantee on the approximation error. Our algorithms have a provably low communication cost, and a low rate of false positives and false negatives, with a high probability. We present detailed results from an experimental evaluation that show the communication cost is small, and that the false positive and false negative rates are typically much lower than theoretical guarantees. [ABSTRACT FROM AUTHOR]

We propose a new proof of the Laplace approximation for multiple integrals and consequently find new bounds for the approximation error. The main advantages of our approach are its simplicity, an explicit form of the bounds and small coefficient in the main error term, which depend on the smallest eigenvalue of the Hessian matrix, only. On the other hand, the method seems indicating much larger values of the parameter then really necessary. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the approximation of a function by its interpolating multilinear spline and the approximation of its derivatives by the derivatives of the corresponding spline. We obtain the exact uniform approximation error on classes of functions with moduli of continuity bounded above by certain majorants. [Copyright &y& Elsevier]

Abstract: A compressed sensing method consists of a rectangular measurement matrix, with , together with an associated recovery algorithm, . Compressed sensing methods aim to construct a high quality approximation to any given input vector using only as input. In particular, we focus herein on instance optimal nonlinear approximation error bounds for and of the form for , where is the best possible -term approximation to . In this paper we develop a compressed sensing method whose associated recovery algorithm, , runs in -time, matching a lower bound up to a factor. This runtime is obtained by using a new class of sparse binary compressed sensing matrices of near optimal size in combination with sublinear-time recovery techniques motivated by sketching algorithms for high-volume data streams. The new class of matrices is constructed by randomly subsampling rows from well-chosen incoherent matrix constructions which already have a sub-linear number of rows. As a consequence, fewer random bits than previously required are needed in order to select the rows utilized by the fast reconstruction algorithms considered herein. [Copyright &y& Elsevier]

Abstract: Subscription covering is an optimization that reduces the number of subscriptions propagated, and hence, the size of routing tables, in a content-based publish-subscribe system. However, detecting covering relationships among subscriptions can be a costly computational task; this cost can potentially reduce the utility of covering as an optimization. We introduce an alternate approach called approximate subscription covering, that can provide much of the benefits of subscription covering at a fraction of the cost. By forgoing an exhaustive search for covering subscriptions in favor of an approximate search, it is shown that the time complexity of covering detection can be dramatically reduced. The tradeoff between efficiency of covering detection and the approximation error is demonstrated through the analysis of indexes for multi-attribute subscriptions based on space filling curves. [Copyright &y& Elsevier]

Abstract: An approximation scheme is a family of homogeneous subsets of a quasi-Banach space , such that , , and . Continuing the line of research originating at the classical paper by Bernstein, we give several characterizations of the approximation schemes with the property that, for every sequence , there exists such that (in this case we say that satisfies Shapiro’s Theorem). If is a Banach space, as above exists if and only if, for every sequence , there exists such that . We give numerous examples of approximation schemes satisfying Shapiro’s Theorem. [Copyright &y& Elsevier]

Abstract: In the previous paper (Almira and Oikhberg, 2010 ), the authors investigated the existence of an element x of a quasi-Banach space X whose errors of best approximation by a given approximation scheme (defined by ) decay arbitrarily slowly. In this work, we consider the question of whether x witnessing the slowness rate of approximation can be selected in a prescribed subspace of X. In many particular cases, the answer turns out to be positive. [Copyright &y& Elsevier]

Abstract: We describe how to use Schoenberg’s theorem for a radial kernel combined with existing bounds on the approximation error functions for Gaussian kernels to obtain a bound on the approximation error function for the radial kernel. The result is applied to the exponential kernel and Student’s kernel. To establish these results we develop a general theory regarding mixtures of kernels. We analyze the reproducing kernel Hilbert space (RKHS) of the mixture in terms of the RKHS’s of the mixture components and prove a type of Jensen inequality between the approximation error function for the mixture and the approximation error functions of the mixture components. [ABSTRACT FROM AUTHOR]

Abstract: We propose an early stopping algorithm for learning gradients. The motivation is to choose “useful” or “relevant” variables by a ranking method according to norms of partial derivatives in some function spaces. In the algorithm, we used an early stopping technique, instead of the classical Tikhonov regularization, to avoid over-fitting. After stating dimension-dependent learning rates valid for any dimension of the input space, we present a novel error bound when the dimension is large. Our novelty is the independence of power index of the learning rates on the dimension of the input space. [Copyright &y& Elsevier]

Abstract: Moving least-square (MLS) is an approximation method for data interpolation, numerical analysis and statistics. In this paper we consider the MLS method in learning theory for the regression problem. Essential differences between MLS and other common learning algorithms are pointed out: lack of a natural uniform bound for estimators and the pointwise definition. The sample error is estimated in terms of the weight function and the finite dimensional hypothesis space. The approximation error is dealt with for two special cases for which convergence rates for the total error measuring the global approximation on the whole domain are provided. [Copyright &y& Elsevier]

Abstract: Learning gradients is one approach for variable selection and feature covariation estimation when dealing with large data of many variables or coordinates. In a classification setting involving a convex loss function, a possible algorithm for gradient learning is implemented by solving convex quadratic programming optimization problems induced by regularization schemes in reproducing kernel Hilbert spaces. The complexity for such an algorithm might be very high when the number of variables or samples is huge. We introduce a gradient descent algorithm for gradient learning in classification. The implementation of this algorithm is simple and its convergence is elegantly studied. Explicit learning rates are presented in terms of the regularization parameter and the step size. Deep analysis for approximation by reproducing kernel Hilbert spaces under some mild conditions on the probability measure for sampling allows us to deal with a general class of convex loss functions. [Copyright &y& Elsevier]

We formalize the notion and initiate the investigation of approximate testing for arbitrary forms of the error term. Until now only the case of absolute error had been addressed ignoring the fact that often only the most significant figures of a numerical calculation are valid. This work considers approximation errors whose magnitude grows with the size of the input to the program. We demonstrate the viability of this new concept by addressing the basic and benchmark problem of self-testing for the class of linear and polynomial functions. We obtain stronger versions of results of Ergu¨n et al. (Proceedings of the 37th FOCS, 1996, pp. 592–601) by exploiting elegant techniques from Hyers–Ulam stability theory. [Copyright &y& Elsevier]

Deep neural networks have successfully been trained in various application areas with stochastic gradient descent. However, there exists no rigorous mathematical explanation why this works so well. The training of neural networks with stochastic gradient descent has four different discretization parameters: (i) the network architecture; (ii) the amount of training data; (iii) the number of gradient steps; and (iv) the number of randomly initialized gradient trajectories. While it can be shown that the approximation error converges to zero if all four parameters are sent to infinity in the right order, we demonstrate in this paper that stochastic gradient descent fails to converge for ReLU networks if their depth is much larger than their width and the number of random initializations does not increase to infinity fast enough. [ABSTRACT FROM AUTHOR]

We continue the research on the asymptotic and preasymptotic decay of singular numbers for tensor product Hilbert–Sobolev type embeddings in high dimensions with special emphasis on the influence of the underlying dimension d. The main focus in this paper lies on tensor products involving univariate Sobolev type spaces with different smoothness. We study the embeddings into L 2 and H 1. In other words, we investigate the worst-case approximation error measured in L 2 and H 1 when only n linear measurements of the function are available. Recent progress in the field shows that accurate bounds on the singular numbers are essential for recovery bounds using only function values. The asymptotic bounds in our setting are known for a long time. In this paper we contribute the correct asymptotic constant and explicit bounds in the preasymptotic range for n. We complement and improve on several results in the literature. In addition, we refine the error bounds coming from the setting where the smoothness vector is moderately increasing, which has been already studied by Papageorgiou and Woźniakowski. [ABSTRACT FROM AUTHOR]

Error bounds and complexity bounds in numerical analysis and information-based complexity are often proved for functions that are defined on very simple domains, such as a cube, a torus, or a sphere. We study optimal error bounds for the approximation or integration of functions defined on D d ⊂ R d and only assume that D d is a bounded Lipschitz domain. Some results are even more general. We study three different concepts to measure the complexity: order of convergence, asymptotic constant, and explicit uniform bounds, i.e., bounds that hold for all n (number of pieces of information) and all (normalized) domains. It is known for many problems that the order of convergence of optimal algorithms does not depend on the domain D d ⊂ R d . We present examples for which the following statements are true: 1. Also the asymptotic constant does not depend on the shape of D d or the imposed boundary values, it only depends on the volume of the domain. 2. There are explicit and uniform lower (or upper, respectively) bounds for the error that are only slightly smaller (or larger, respectively) than the asymptotic error bound. [ABSTRACT FROM AUTHOR]

It is known that for a ϱ -weighted L q approximation of single variable functions defined on a finite or infinite interval, whose r th derivatives are in a ψ -weighted L p space, the minimal error of approximations that use n samples of f is proportional to ‖ ω 1 ∕ α ‖ L 1 α ‖ f (r) ψ ‖ L p n − r + (1 ∕ p − 1 ∕ q) + , where ω = ϱ ∕ ψ and α = r − 1 ∕ p + 1 ∕ q , provided that ‖ ω 1 ∕ α ‖ L 1 < + ∞. Moreover, the optimal sample points are determined by quantiles of ω 1 ∕ α. In this paper, we show how the error of the best approximation changes when the sample points are determined by a quantizer κ other than ω. Our results can be applied in situations when an alternative quantizer has to be used because ω is not known exactly or is too complicated to handle computationally. The results for q = 1 are also applicable to ϱ -weighted integration over finite and infinite intervals. [ABSTRACT FROM AUTHOR]

The Fourier extension method, also known as the Fourier continuation method, is a method for approximating non-periodic functions on an interval using truncated Fourier series with period larger than the interval on which the function is defined. When the function being approximated is known at only finitely many points, the approximation is constructed as a projection based on this discrete set of points. In this paper we address the issue of estimating the absolute error in the approximation. The error can be expressed in terms of a system of discrete orthogonal polynomials on an arc of the unit circle, and these polynomials are then evaluated asymptotically using Riemann–Hilbert methods. [ABSTRACT FROM AUTHOR]

We investigate dimension-dependent estimates of the approximation error for linear algorithms of sampling recovery on Smolyak grids parametrized by m ∈ N , of periodic d -variate functions from the space with Hölder–Zygmund mixed smoothness α > 0. We consider two subsets of the unit ball in this space. The first one is the subset of functions with homogeneous condition, and the second one is the subset of functions depending on ν active variables (1 ≤ ν ≤ d). For these classes, we prove some upper bounds and lower bounds (for α ≤ 2) of the error of the optimal sampling recovery on Smolyak grids, explicit in d , ν , m when d and m may be large. [ABSTRACT FROM AUTHOR]

In this paper we obtain an approximation for the multivariate Laplace's integral with a large parameter and estimate error term for two cases, when the maximum of the exponent is in the interior of the domain and on the boundary. We are specifically interested in the situation when the function in the exponent depends on the large parameter. As an application we prove weak law of large numbers and central limit theorem. The second result gives different limiting distributions for two cases mentioned above. When the maximum of the exponent is in the interior of the domain it is Normal distribution and if it is on the boundary, it is Exponential in one direction of integration and Normal in other directions. [ABSTRACT FROM AUTHOR]

In this work, we show that uniform integrability is not a necessary condition for central limit theorems (CLT) to hold for normalized multilevel Monte Carlo (MLMC) estimators and we provide near optimal weaker conditions under which the CLT is achieved. In particular, if the variance decay rate dominates the computational cost rate (i.e., β > γ), we prove that the CLT applies to the standard (variance minimizing) MLMC estimator. For other settings where the CLT may not apply to the standard MLMC estimator, we propose an alternative estimator, called the mass-shifted MLMC estimator, to which the CLT always applies. This comes at a small efficiency loss: the computational cost of achieving mean square approximation error O (ϵ 2) is at worst a factor O (log (1 ∕ ϵ)) higher with the mass-shifted estimator than with the standard one. [ABSTRACT FROM AUTHOR]