information theoretic determination of minimax rates of convergence we present some general results determining minimax bounds on statistical risk for density estimation based on certain information theoretic considerations these bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence

on the fundamental limits of adaptive sensing suppose we can sequentially acquire arbitrary linear measurements of an n dimensional vector x resulting in the linear model y a x z where z represents measurement noise if the signal is known to be sparse one would expect the following folk theorem to be true choosing an adaptive strategy which cleverly selects the next row of a based on what has been previously observed should do far better than a nonadaptive strategy which sets the rows of a ahead of time thus not trying to learn anything about the signal in between observations this paper shows that the folk theorem is false we prove that the advantages offered by clever adaptive strategies and sophisticated estimation procedures no matter how intractable over classical compressed acquisition recovery schemes are in general minimal

stochastic first and zeroth order methods for nonconvex stochastic programming in this paper we introduce a new stochastic approximation type algorithm namely the randomized stochastic gradient rsg method for solving an important class of nonlinear possibly nonconvex stochastic programming problems we establish the complexity of this method for computing an approximate stationary point of a nonlinear programming problem we also show that this method possesses a nearly optimal rate of convergence if the problem is convex we discuss a variant of the algorithm which consists of applying a postoptimization phase to evaluate a short list of solutions generated by several independent runs of the rsg method and we show that such modification allows us to improve significantly the large deviation properties of the algorithm these methods are then specialized for solving a class of simulation based optimization problems in which only stochastic zeroth order information is available

sample complexity of solving non cooperative games this paper studies the complexity of solving two classes of non cooperative games in a distributed manner in which the players communicate with a set of system nodes over noisy communication channels the complexity of solving each game class is defined as the minimum number of iterations required to find a nash equilibrium ne of any game in that class with epsilon accuracy first we consider the class mathcal g of all n player non cooperative games with a continuous action space that admit at least one ne using information theoretic inequalities a lower bound on the complexity of solving mathcal g is derived which depends on the kolmogorov 2 epsilon capacity of the constraint set and the total capacity of the communication channels our results indicate that the game class mathcal g can be solved at most exponentially fast we next consider the class of all n player non cooperative games with at least one ne such that the players utility functions satisfy a certain differential constraint we derive lower bounds on the complexity of solving this game class under both gaussian and non gaussian noise models finally we derive upper and lower bounds on the sample complexity of a class of quadratic games it is shown that the complexity of solving this game class scales according to theta left frac 1 epsilon 2 right where epsilon is the accuracy parameter

fourier spectral exponential time differencing methods for multi dimensional space fractional reaction diffusion equations abstract in this paper we propose two numerical methods for solving the space fractional reaction diffusion equations which are based on a fourier spectral approach in space and exponential time differencing schemes in time the advantages of the approaches are that they attain spectral convergence produce a full diagonal representation of the fractional operator and the extension to multiple spatial dimensions is the same as the one dimensional space that can overcome the constraints associated with many of the numerical schemes for these equations such as the computational efficiency caused by the non locality of the fractional operator which results in full dense matrices moreover the proposed schemes are second order convergent unconditionally stable and highly efficient due to the predictor corrector feature when comparing them with the existing method it is observed that the scheme based on using pade 1 1 approximations to the matrix exponential function introduces oscillations with non smooth initial data for some time steps due to high frequency components present in the solution which diminish as the fractional order decreases however the scheme based on pade 0 2 approximations to the matrix exponential function is oscillation free for any time step numerical experiments for well known models from the literature are performed to show the reliability and effectiveness of the proposed methods

subgrid upscaling and mixed multiscale finite elements second order elliptic problems in divergence form with a highly varying leading order coefficient on the scale epsilon can be approximated on coarse meshes of spacing h gg epsilon only if one uses special techniques the mixed variational multiscale method also called subgrid upscaling can be used and this method is extended to allow oversampling of the local subgrid problems the method is shown to be equivalent to the multiscale finite element method when one uses the lowest order raviart thomas spaces and provided that there are no fine scale components in the source function f in the periodic setting a multiscale error analysis based on homogenization theory of the more general subgrid upscaling method shows that the error is o epsilon hm sqrt epsilon h where m 1 moreover m 2 if one uses the second order brezzi douglas marini or brezzi douglas duran fortin spaces and no oversampling the error bounding constant depends only on the hm 1 norm of f and so is independent of small scales when m 1 when oversampling is not used a superconvergence result for the pressure approximation is shown

zeroth order method for distributed optimization with approximate projections this paper studies the problem of minimizing a sum of possible nonsmooth convex functions that are corresponding to multiple interacting nodes subject to a convex state constraint set time varying directed network is considered here two types of computational constraints are investigated in this paper one where the information of gradients is not available and the other where the projection steps can only be calculated approximately we devise a distributed zeroth order method the implementation of which needs only functional evaluations and approximate projection in particular we show that the proposed method generates expected function value sequences that converge to the optimal value provided that the projection errors decrease at appropriate rates

information theoretic lower bounds on the oracle complexity of stochastic convex optimization relative to the large literature on upper bounds on complexity of convex optimization lesser attention has been paid to the fundamental hardn4516420ess of these problems given the extensive use of convex optimization in machine learning and statistics gaining an understanding of these complexity theoretic issues is important in this paper we study the complexity of stochastic convex optimization in an oracle model of computation we introduce a new notion of discrepancy between functions and use it to reduce problems of stochastic convex optimization to statistical parameter estimation which can be lower bounded using information theoretic methods using this approach we improve upon known results and obtain tight minimax complexity estimates for various function classes

an adaptive gmsfem for high contrast flow problems in this paper we derive an a posteriori error indicator for the generalized multiscale finite element method gmsfem framework this error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high contrast coefficients the gmsfem which has recently been introduced in 13 allows solving multiscale parameter dependent problems at a reduced computational cost by constructing a reduced order representation of the solution on a coarse grid the main idea of the method consists of 1 the construction of snapshot space 2 the construction of the offline space and 3 the construction of the online space the latter for parameter dependent problems in 13 it was shown that the gmsfem provides a flexible tool to solve multiscale problems with a complex input space by generating appropriate snapshot offline and online spaces in this paper we study an adaptive enrichment procedure and derive an a posteriori error indicator which gives an estimate of the local error over coarse grid regions we consider two kinds of error indicators where one is based on the l 2 norm of the local residual and the other is based on the weighted h 1 norm of the local residual where the weight is related to the coefficient of the elliptic equation we show that the use of weighted h 1 norm residual gives a more robust error indicator which works well for cases with high contrast media the convergence analysis of the method is given in our analysis we do not consider the error due to the fine grid discretization of local problems and only study the errors due to the enrichment numerical results are presented that demonstrate the robustness of the proposed error indicators

robust stochastic approximation approach to stochastic programming in this paper we consider optimization problems where the objective function is given in a form of the expectation a basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals expectations cannot be computed with high accuracy the aim of this paper is to compare two computational approaches based on monte carlo sampling techniques namely the stochastic approximation sa and the sample average approximation saa methods both approaches the sa and saa methods have a long history current opinion is that the saa method can efficiently use a specific say linear structure of the considered problem while the sa approach is a crude subgradient method which often performs poorly in practice we intend to demonstrate that a properly modified sa approach can be competitive and even significantly outperform the saa method for a certain class of convex stochastic problems we extend the analysis to the case of convex concave stochastic saddle point problems and present in our opinion highly encouraging results of numerical experiments

on the generalization ability of on line learning algorithms in this paper it is shown how to extract a hypothesis with small risk from the ensemble of hypotheses generated by an arbitrary on line learning algorithm run on an independent and identically distributed i i d sample of data using a simple large deviation argument we prove tight data dependent bounds for the risk of this hypothesis in terms of an easily computable statistic m sub n associated with the on line performance of the ensemble via sharp pointwise bounds on m sub n we then obtain risk tail bounds for kernel perceptron algorithms in terms of the spectrum of the empirical kernel matrix these bounds reveal that the linear hypotheses found via our approach achieve optimal tradeoffs between hinge loss and margin size over the class of all linear functions an issue that was left open by previous results a distinctive feature of our approach is that the key tools for our analysis come from the model of prediction of individual sequences i e a model making no probabilistic assumptions on the source generating the data in fact these tools turn out to be so powerful that we only need very elementary statistical facts to obtain our final risk bounds

rates of convergence for minimum contrast estimators we shall present here a general study of minimum contrast estimators in a nonparametric setting although our results are also valid in the classical parametric case for independent observations these estimators include many of the most popular estimators in various situations such as maximum likelihood estimators least squares and other estimators of the regression function estimators for mixture models or deconvolution the main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters optimal rates depending on entropy conditions are already known at least for some of the models involved and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large but under some circumstances large entropies or changes in the entropy structure due to local perturbations the resulting the rates are only suboptimal counterexamples are constructed which show that the phenomenon is real for non parametric maximum likelihood or regression this proves that under purely metric assumptions our theorem is optimal and that minimum contrast estimators happen to be suboptimal

adaptive system optimization using random directions stochastic approximation we present new algorithms for simulation optimization using random directions stochastic approximation rdsa these include first order gradient as well as second order newton schemes we incorporate both continuous valued as well as discrete valued perturbations into both types of algorithms the former are chosen to be independent and identically distributed i i d symmetric uniformly distributed random variables r v while the latter are i i d asymmetric bernoulli r v s our newton algorithm with a novel hessian estimation scheme requires n dimensional perturbations and three loss measurements per iteration whereas the simultaneous perturbation newton search algorithm of 1 requires 2n dimensional perturbations and four loss measurements per iteration we prove the asymptotic unbiasedness of both gradient and hessian estimates and asymptotic strong convergence for both first order and second order schemes we also provide asymptotic normality results which in particular establish that the asymmetric bernoulli variant of newton rdsa method is better than 2spsa of 1 numerical experiments are used to validate the theoretical results

on density estimation in the view of kolmogorov s ideas in approximation theory the paper deals with upper and lower bounds for the quality of probability density estimation connections are established between these problems and the theory of approximation of functions particularly it is demonstrated how some of kolmogorov s concepts work

rational approximations to fractional powers of self adjoint positive operators we investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function we provide accurate error bounds by exploiting classical results in approximation theory involving pade approximants the analysis improves some existing results and the numerical experiments proves its accuracy

high order time stepping fourier spectral method for multi dimensional space fractional reaction diffusion equations abstract this paper introduces a high order time stepping scheme that is based on using fourier spectral in space and a fourth order diagonal pade approximation to the matrix exponential function for solving multi dimensional space fractional reaction diffusion equations the resulting time stepping scheme is developed based on an exponential time differencing approach such that it alleviates solving a large non linear system at each time step while maintaining the stability of the scheme the non locality of the fractional operator in some other numerical schemes for these equations leads to full and dense matrices this scheme is able to overcome such computational inefficiency due to the full diagonal representation of the fractional operator it also attains spectral convergence for multiple spatial dimensions the stability of the scheme is discussed through the investigation of the amplification symbol and plotting its stability regions which provides an indication of the stability of the method the convergence analysis is performed empirically to show that the scheme is fourth order accurate in time as expected numerical experiments on reaction diffusion systems with application to pattern formation are discussed to show the effect of the fractional order in space fractional reaction diffusion equations and to validate the effectiveness of the scheme

a unified view of some numerical methods for fractional diffusion abstract in recent years a number of numerical methods for the solution of fractional laplace and more generally fractional diffusion problems have been proposed the approaches are quite diverse and include among others the use of best uniform rational approximations quadrature for dunford taylor like integrals finite element approaches for a localized elliptic extension into a space of increased dimensions and time stepping methods for a parabolic reformulation of the fractional differential equation a systematic comparison both theoretical and experimental of these approaches has thus far been lacking a main contribution of the present work is the observation that all approaches mentioned above can in fact be interpreted as realizing different rational approximations of a univariate function over the spectrum of the original non fractional diffusion operator while this is obvious for some of the methods it is a new result in particular for extension based and time stepping approaches this observation allows us to cast all described methods into a unified theoretical and computational framework which has a number of benefits theoretically it enables us to develop new convergence proofs for several of the studied methods clarifies similarities and differences between the approaches suggests how to design new and improved methods and allows a direct comparison of the relative performance of the various methods practically it provides a single simple to implement efficient and fully parallel algorithm for the realization of all studied methods for instance this immediately yields a fast and memory efficient way of realizing all tensor product extension methods and lets us parallelize the otherwise inherently sequential time stepping approach finally we present a detailed numerical study comparing all investigated methods for various fractional exponents and draw conclusions from the results the comparison is made fair by the central insight that the computational effort of all these methods depends only on a single parameter the degree of the underlying rational approximation as a point of comparison we also test a simple rational approximation method based on a black box direct rational approximation algorithm which performs very well in practice

the black box multigrid numerical homogenization algorithm in mathematical models of flow through porous media the coefficients typically exhibit severe variations in two or more significantly different length scales consequently the numerical treatment of these problems relies on ahomogenizationorupscalingprocedure to define an approximate coarse scale problem that adequately captures the influence of the fine scale structure inherent in such a procedure is a compromise between its computational cost and the accuracy of the resulting coarse scale solution although techniques that balance the conflicting demands of accuracy and efficiency exist for a few specific classes of fine scale structure e g fine scale periodic this is not the case in general in this paper we propose a new efficient numerical approach for thehomogenizationof the permeability in models of single phase saturated flow our approach is motivated by the observation that multiple length scales are captured automatically by robust multilevel iterative solvers such as dendy sblack box multigrid in particular the operator induced variational coarsening in black box multigrid produces coarse grid operators that capture the essential coarse scale influence of the medium s fine scale structure we derive an explicit local cell based approximate expression for the symmetric 2 2 homogenized permeability tensor that is defined implicitly by the black box coarse grid operator the effectiveness of this black box multigrid numerical homogenization method is demonstrated through numerical examples

exponentiated gradient versus gradient descent for linear predictors we consider two algorithm for on line prediction based on a linear model the algorithms are the well known gradient descent gd algorithm and a new algorithm which we call eg they both maintain a weight vector using simple updates for the gd algorithm the update is based on subtracting the gradient of the squared error made on a prediction the eg algorithm uses the components of the gradient in the exponents of factors that are used in updating the weight vector multiplicatively we present worst case loss bounds for eg and compare them to previously known bounds for the gd algorithm the bounds suggest that the losses of the algorithms are in general incomparable but eg has a much smaller loss if only a few components of the input are relevant for the predictions we have performed experiments which show that our worst case upper bounds are quite tight already on simple artificial data

finite element approximation of diffusion equations with convolution terms approximation of solutions to diffusion equations with memory represented by convolution integral terms is considered such problems arise from modeling of flows in fissured media convergence of the method is proved and results of numerical experiments confirming the theoretical results are presented the advantages of implementation of the algorithm in a multiprocessing environment are discussed

a real distinct poles rational approximation of generalized mittag leffler functions and their inverses applications to fractional calculus abstract the generalized mittag leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives respectively however their computational complexities have made them difficult to deal with numerically we propose a real distinct pole rational approximation of the generalized mittag leffler function under some mild conditions this approximation is proven and empirically shown to be l acceptable due to the complete monotonicity property of the mittag leffler function we derive a rational approximation for the inverse generalized mittag leffler function these approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order several applications are presented such as complementary error function solution of fractional differential equations and the ultraslow diffusion model using the structural derivative

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

high order time stepping methods for two dimensional riesz fractional nonlinear reaction diffusion equations abstract anomalous diffusion and long range spatial interactions in anisotropic media could be captured by considering the riesz fractional derivatives rather than the classical laplacian while analytical solutions of the resulting fractional reaction diffusion models may not be available the numerical methods for approximating them are challenging in this paper fourth order l stable etdrk04 and a stable etdrk22 linearly implicit predictor corrector type time stepping methods are presented these methods are implemented on two dimensional riesz fractional nonlinear reaction diffusion equations with smooth and non smooth initial data three types of nonlinear reaction diffusion models are considered allen cahn equation with cubic nonlinearity fisher s equation with quadratic nonlinearity and enzyme kinetics equation with rational nonlinearity fourth order temporal convergence rate of the methods is proved analytically and computed numerically profiles of the numerical solutions corresponding to different orders and rates of diffusion are included computational efficiency of the etdrk04 and etdrk22 methods over the well known cox matthews etdrk4 is presented the superiority of the provided methods in terms of computational accuracy efficiency and reliability is demonstrated through the numerical experiments

a conservative local multiscale model reduction technique for stokes flows in heterogeneous perforated domains abstract in this paper we present a new multiscale model reduction technique for the stokes flows in heterogeneous perforated domains the challenge in the numerical simulations of this problem lies in the fact that the solution contains many multiscale features and requires a very fine mesh to resolve all details in order to efficiently compute the solutions some model reductions are necessary to obtain a reduced model we apply the generalized multiscale finite element approach which is a framework allowing systematic construction of reduced models based on this general framework we will first construct a local snapshot space which contains many possible multiscale features of the solution using the snapshot space and a local spectral problem we identify dominant modes in the snapshot space and use them as the multiscale basis functions our basis functions are constructed locally with non overlapping supports which enhances the sparsity of the resulting linear system in order to enforce the mass conservation we propose a hybridized technique and uses a lagrange multiplier to achieve mass conservation we will mathematically analyze the stability and the convergence of the proposed method in addition we will present some numerical examples to show the performance of the scheme we show that with a few basis functions per coarse region one can obtain a solution with excellent accuracy

multiscale model reduction for transport and flow problems in perforated domains abstract convection dominated transport phenomenon is important for many applications in these applications the transport velocity is often a solution of heterogeneous flow problems which results to a coupled flow and transport phenomena in this paper we consider a coupled flow stokes problem and transport unsteady convection diffusion problem in perforated domains perforated domains see fig 1 represent void space outside hard inclusions as in porous media filters and so on we construct a coarse scale solver based on generalized multiscale finite element method gmsfem for a coupled flow and transport the main idea of the gmsfem is to develop a systematic approach for computing multiscale basis functions we use a mixed formulation and appropriate multiscale basis functions for both flow and transport to guarantee a mass conservation for the transport problem we use petrov galerkin mixed formulation which provides a stability as a first approach we use the multiscale flow solution in constructing the basis functions for the transport equation in the second approach we construct multiscale basis functions for coupled flow and transport without solving global flow problem the novelty of this approach is to construct a coupled multiscale basis function numerical results are presented for computations using offline basis we also present an algorithm for adaptively adding online multiscale basis functions which are computed using the residual information numerical examples using online gmsfem show the speed up of convergence

large scale averaging analysis of single phase flow in fractured reservoirs the method of large scale averaging is introduced to derive and analyze the most general form of the dual porosity model of single phase flow in naturally fractured reservoirs the dual porosity model contains the usual equations based on darcy s law and the coupling terms representing the fluid transfer between the matrix and the fractures a transient closure problem is developed to obtain and analyze the fracture and matrix permeability tensors and the fluid transfer terms the spatial averaging theorem is presented from a mathematical point of view and proved rigorously by the distribution theory the problem of wellposedness of the dual porosity model is also considered the techniques developed here are not restricted to either regular geometric fractures or spatially periodic reservoirs

a simple lemma on greedy approximation in hilbert space and convergence rates for projection pursuit regression and neural network training a general convergence criterion for certain iterative sequences in hilbert space is presented for an important subclass of these sequences estimates of the rate of convergence are given under very mild assumptions these results establish an 0 1 f4n nonsampling convergence rate for projection pursuit regression and neural network training where n represents the number of ridge functions neurons or coefficients in a greedy basis expansion

approximation dans les espaces metriques et theorie de l estimation we investigate the relations between the speed of estimation and the metric structure of the parameter space especially in the case when its metric dimension is infinite given some distance d on generally hellinger distance in the case of n i i d variables we consider the minimax risk for n observations r n q mathop inf sup limits t n theta in theta text ie theta d q theta t n t n being any estimate of we shall look for functions r such that for positive constants c 1 q and c 2 q c 1 r q n r n q c2 r q n r n is the speed of estimation and we shall show under fairly general conditions including i i d variables and regular cases of markov chains and stationnary gaussian processes that r n is determined up to multiplicative constants by the metric structure of we shall also give a construction for some sort of universal estimates the risk of which is bounded by c 2 r q n in all cases where the preceding theory applies

least squares mixed generalized multiscale finite element method abstract in this paper we present an approximation of elliptic problems with multiscale and high contrast diffusion coefficients a mixed formulation is considered such that both pressure and velocity are approximated simultaneously this formulation arises naturally in many applications such as flows in porous media due to the multiscale nature of the solutions using model reduction is required to efficiently obtain approximate solutions there are many multiscale approaches for elliptic problems in mixed formulation these approaches include numerical homogenization and mixed multiscale finite element method which aim to obtain a coarse accurate representation of the velocity without using an accurate representation for pressure it has been a challenging task to construct a method giving accurate representation for both pressure and velocity the goal in this paper is to construct multiscale basis functions for both pressure and velocity we will apply the framework of generalized multiscale finite element method gmsfem and design systematic strategies for the construction of basis the construction involves snapshot spaces and dimension reduction via local spectral problems the mixed formulation is minimized in the sense of least squares the compatibility condition for multiscale finite element spaces of the pressure and velocity is not required in the least squares mixed form this gives more flexibility for the construction of multiscale basis functions for velocity and pressure convergence analysis is carried out for the least squares mixed gmsfem several numerical examples for various permeability fields are presented to show the performance of the presented method the numerical results show that the least squares mixed gmsfem can give accurate approximation for both pressure and velocity using only a few basis functions per coarse element

an upscaling method using coefficient splitting and its applications to elliptic pdes in this paper we develop an upscaling method using coefficient splitting techniques green s function is constructed using the differential operator associated with the first part of the splitting an effective upscaling coefficient is recursively calculated by green s function the computation of the upscaling process involves some independent steps combining the proposed upscaling method with the stochastic collocation method we present a stochastic space reduction collocation method where the stochastic collocation method is performed on a lower dimension stochastic space than the full dimension stochastic space we thoroughly analyze the convergence of the proposed upscaling method for both deterministic and stochastic elliptic pdes computation complexity is also addressed for the stochastic upscaling method a number of numerical tests are presented to confirm the convergence analysis

estimation of integral functionals of a density let be a smooth function of k 2 variables we shall investigate in this paper the rates of convergence of estimators of t f f f x f x f k x x dx when f belongs to some class of densities of smoothness s we prove that when 2k 1 4 one can define an estimator t n of t f based on n i i d observations of density f on the real line which converges at the semiparametric rate 1 n on the other hand when s 2k 1 4 t f cannot be estimated at a rate faster than n with 4 s k 4s 1 we shall also provide some extensions to the multidimensional case those results extend previous works of levit of bickel and ritov and of donoho and nussbaum on estimation of quadratic functionals

numerical solution of fractional diffusion reaction problems based on bura abstract the paper is devoted to the numerical solution of algebraic systems of the type a q i u f 0 1 q 0 u f r n where a is a symmetric and positive definite matrix we assume that a is obtained by finite difference approximation of a second order diffusion problem in r d d 1 2 so that a q i approximates the related fractional diffusion reaction operator or could be a result of a time stepping procedure in solving time dependent sub diffusion problems we also assume that a method of optimal complexity for solving linear systems with matrices a c i c 0 is available we analyze and study numerically a class of solution methods based on the best uniform rational approximation bura of a certain scalar function in the unit interval the first such method originally proposed in harizanov et al 2018 for numerical solution of fractional in space diffusion problems was based on the bura r of 1 in 0 1 through scaling of the matrix a by its largest eigenvalue then the bura of t in 1 is given by t 1 r t and correspondingly a 1 r a is used as an approximation of a further this method was improved in harizanov et al 2019 using the same concept but by scaling the matrix a by its smallest eigenvalue in this paper we consider the bura r of 1 q for 0 1 then we define the approximation of a q i 1 as r a we also propose an alternative method that uses bura of to produce certain uniform rational approximation ura of 1 q comprehensive numerical experiments are used to demonstrate the computational efficiency and robustness of the new bura and ura methods

upscaling method for problems in perforated domains with non homogeneous boundary conditions on perforations using non local multi continuum method nlmc abstract in this paper we present an upscaling method for problems in perforated domains with non homogeneous boundary conditions on perforations our methodology is based on the recently developed non local multicontinuum method nlmc the main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non local effects we will construct multiscale basis functions for the coarse regions and additional multiscale basis functions for perforations with the aim of handling non homogeneous boundary conditions on perforations we start with describing our method for the laplace equation and then extending the framework for the elasticity problem and parabolic equations the resulting upscaled model has minimal size and the solution has physical meaning on the coarse grid we will present numerical results 1 for steady and unsteady problems 2 for laplace and elastic operators and 3 for neumann and robin non homogeneous boundary conditions on perforations numerical results show that the proposed method can provide good accuracy and provide significant reduction on the degrees of freedoms

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

iterative low rank approximation solvers for the extension method for fractional diffusion abstract we consider the numerical method for fractional diffusion problems which is based on an extension to a mixed boundary value problem for a local operator in a higher dimensional space we observe that when this problem is discretized using tensor product spaces as is commonly done the solution can be very well approximated by low rank tensors we provide some analysis to support this observation this fact motivates us to apply iterative low rank approximation algorithms in order to efficiently solve this extended problem in particular we employ a recently proposed greedy tucker approximation method as well as a more classical greedy rank one update method throughout all objects of interest are kept in suitable low rank approximations which dramatically reduces the required amount of memory compared to the full formulation of the extended problem our approach can be used for general non structured space discretizations if the space discretization itself has tensor product structure we can further decompose the problem in order to deal with even lower dimensional objects we also note that the approach can be directly applied to higher order discretizations both in space and the extended variable a further contribution of our work is a rank one version of a diagonal preconditioner which can mitigate the severe ill conditioning of the extended problem arising due to mesh grading and singularity of the coefficient this preconditioner can be realized using a simple pre and postprocessing step and does not affect the computational effort of our algorithm in several numerical examples we demonstrate the convergence behavior of the proposed methods in particular the tucker approximation approach requires only a few iterations in order to reach the discretization error in all tested settings

rational krylov methods for functions of matrices with applications to fractional partial differential equations abstract in this paper we propose a new choice of poles to define reliable rational krylov methods these methods are used for approximating function of positive definite matrices in particular the fractional power and the fractional resolvent are considered because of their importance in the numerical solution of fractional partial differential equations the numerical experiments on some fractional partial differential equation models confirm that the proposed approach is promising

generalized multiscale finite element method for multicontinua unsaturated flow problems in fractured porous media abstract in this paper we present a multiscale method for simulations of the multicontinua unsaturated flow problems in heterogeneous fractured porous media the mathematical model is described by the system of richards equations for each continuum that is coupled by the specific transfer term to illustrate the idea of our approach we consider a dual continua background model with discrete fractures networks that is generalized as a multicontinua model for unsaturated fluid flow in the complex heterogeneous porous media we present fine grid approximation based on the finite element method and discrete fracture model dfm approach in this model we construct an unstructured fine grid that takes into account complex fracture geometries for two and three dimensional formulations due to construction of the unstructured grid the fine grid approximation leads to the very large system of equations for reduction of the discrete system size we develop a multiscale method for coarse grid approximation of the coupled problem using generalized multiscale finite element method gmsfem in this method we construct coupled multiscale basis functions that are used to construct highly accurate coarse grid approximation the multiscale method allowed us to capture detailed interactions between multiple continua the adaptive approach is investigated where we consider two approaches for multiscale basis functions construction 1 based on the spectral characteristics of the local problems and 2 using simplified multiscale basis functions we investigate accuracy of the proposed method for the several test problems in two and three dimensional formulations we present a comparison of the relative error for different number of basis functions and for adaptive approach numerical results illustrate that the presented method provides accurate solution of the unsaturated multicontinua problem on the coarse grid with huge reduction of the discrete system size

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

linearly implicit predictor corrector methods for space fractional reaction diffusion equations with non smooth initial data abstract in this paper linearly implicit predictor corrector methods are proposed for solving space fractional reaction diffusion equations with non smooth initial data the methods are based on matrix transfer technique for spatial discretization and are shown to be unconditionally stable it is observed that the linearly implicit predictor corrector method derived by using 1 1 pade approximation to matrix exponential function incurs oscillatory behavior for some time steps these oscillations are due to high frequency components present in the solution and are diminished as the order of the space fractional derivative decreases slow diffusion we present a priori reliability constraint to avoid these unwanted oscillations and generalize the constraints for all m m pade approximants m z to the matrix exponential functions these time stepping constraints are seen to be dependent on the order of the space fractional derivative the linearly implicit predictor corrector method based on the 0 2 pade approximations to the matrix exponential function is shown to be oscillation free for any time step error estimates are obtained for the methods and are theoretically shown to be second order convergent computational complexity of the algorithms is discussed for solving multidimensional space fractional reaction diffusion systems several numerical experiments are performed to support our theoretical observations and to show the effectiveness reliability and efficiency of the methods

probability inequalities for likelihood ratios and convergence rates of sieve mles let y 1 y n be independent identically distributed with density p 0 and let f be a space of densities we show that the supremum of the likelihood ratios i 1 n p y i p 0 y i where the supremum is over p f with p l 2 p 0 1 2 2 e is exponentially small with probability exponentially close to 1 the exponent is proportional to ne 2 the only condition required for this to hold is that e exceeds a value determined by the bracketing hellinger entropy of f a similar inequality also holds if we replace f by f n and p 0 by q n where q n is an approximation to p 0 in a suitable sense these results are applied to establish rates of convergence of sieve mles furthermore weak conditions are given under which the optimal rate e n defined by h e n f ne n 2 where h f is the hellinger entropy of f is nearly achievable by sieve estimators

universal donsker classes and metric entropy when mathfrak f is a universal donsker class then for independent indetically distributed i i d observation mathbf x 1 ldots mathbf x n with an unknown law p for any mathfrak f i in mathfrak f i 1 ldots m quad n 1 2 left mathfrak f 1 left mathbf x 1 right ldots mathfrak f i left mathbf x n right right 1 leq i leq m is asymptotically normal with mean vector n 1 2 left int mathfrak f i left mathbf x n right d mathbf p left x right right 1 leq i leq m and covariance matrix int mathfrak f i mathfrak f j d mathbf p int mathfrak f id mathbf p int mathfrak f jd mathbf p uniformly for mathfrak f i in mathfrak f then for certain statistics formed frome the mathfrak f i left mathbf x k right even where mathfrak f i may be chosen depending on the mathbf x k there will be asymptotic distribution as n rightarrow infty for example for mathbf x 2 statistics where f i are indicators of disjoint intervals depending suitably on mathbf x 1 ldots mathbf x n whose union is the real line mathbf x 2 quadratic forms have limiting distributions roy 1956 and watson 1958 which may however not be mathbf x 2 distributions and may depend on p chernoff and lehmann 1954 universal donsker classes of sets are up to mild measurability conditions just classes satisfying the vapnik cervonenkis comdinatorial conditions defined later in this section donsker the vapnik cervonenkis combinatorial conditions defined later in this section durst and dudley 1981 and dudley 1984 chapter 11 the use of such classes allows a variety of extensions of the roy watson results to general multidimensional sample spaces pollard 1979 and moore and subblebine 1981 vapnik and cervonenkis 1974 indicated application of their families of sets to classification pattern recognition problems more recently the classes have been applied to tree structured classifiacation breiman friedman olshen and stone 1984 chapter 12

two level schemes of cauchy problem method for solving fractional powers of elliptic operators abstract in this paper we develop and investigate numerical algorithms for solving equations of the first order for the fractional powers of discrete elliptic operators a y 0 1 for v h with v h a finite element or discrete approximation space the main goal is to consider two different approaches to construct efficient high order time stepping schemes for the implementation of the cauchy problem method vabishchevich 2015 the first approach is based on a solution of the related time dependent pde and the second approach is based on diagonal pade approximations to 1 x when the relation of the solution on two times levels is used the second and fourth order approximations are constructed by both approaches in order to increase the accuracy of approximations the graded time grid is constructed which compensates the singular behavior of the solution for t close to 0 results of numerical experiments are presented they agree well with the theoretical results

high order numerical schemes for solving fractional powers of elliptic operators abstract in many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes a general class of such models deals with nonlocal fractional power elliptic operators in order to solve these problems numerically it is proposed vabishchevich 2015 to consider equivalent local nonstationary initial value pseudo parabolic problems previously such problems were solved by using the standard implicit backward and symmetrical euler methods in this paper we use the one parameter family of three level finite difference schemes for solving the initial value problem for the first order nonstationary pseudo parabolic problem the fourth order approximation scheme is developed by selecting the optimal value of the weight parameter the results of the theoretical analysis are supplemented by results of extensive computational experiments

a simplified method for upscaling composite materials with high contrast of the conductivity a large class of industrial composite materials such as metal foams fibrous glass materials mineral wools and the like are widely used in insulation and advanced heat exchangers these materials are characterized by a substantial difference between the thermal properties of the highly conductive materials glass or metal and the insulator air as well as low volume fractions and complex network like structures of the highly conductive components in this paper we address the important issue for the engineering practice of developing fast reliable and accurate methods for computing the macroscopic upscaled thermal conductivities of such materials we assume that the materials have constant macroscopic thermal conductivity tensors which can be obtained by upscaling techniques based on the postprocessing of a number of linearly independent solutions of the steady state heat equation on representative elementary volumes revs we propose theoretically justify and computationally study a numerical method for computing the effective conductivities of materials for which the ratio delta of low and high conductivities satisfies delta ll1 we show that in this case one needs to solve the heat equation in the region occupied by the highly conductive media only further we prove that under certain conditions on the microscale geometry the proposed method gives an approximation that is mathcal o delta close to the upscaled conductivity finally we illustrate the accuracy and the limitations of the method on a number of numerical examples

rational approximation to the fractional laplacian operator in reaction diffusion problems this paper provides a new numerical strategy for solving fractional in space reaction diffusion equations on bounded domains under homogeneous dirichlet boundary conditions using the matrix transfer technique the fractional laplacian operator is replaced by a matrix which in general is dense the approach here presented is based on the approximation of this matrix by the product of two suitable banded matrices this leads to a semilinear initial value problem in which the matrices involved are sparse numerical results are presented to verify the effectiveness of the proposed solution strategy

approximation of density functions by sequences of exponential families 1 introduction consider the estimation of a probability density func tion p x defined on a bounded interval we approximate the logarithm of the density by a basis function expansion consisting of polynomials splines or trigonometric series the expansion yields a regular exponential family within which we estimate the density by the method of maximum likelihood this method of density estimation arises by application of the principle of maxi mum entropy or minimum relative entropy subject to empirical constraints we show that if the logarithm of the density has r square integrable deriva tives f idr log p12 ox then the sequence of density estimators p converges to p in the sense of relative entropy kullback leibler fp log p pn at rate opr 1 m2r m n as m x c and m2 n 0 in the spline and trigonometric cases and m3 n o 0 in the polynomial case where m is the dimension of the family and n is the sample size boundary conditions are assumed for the density in the trigonometric case this convergence rate specializes to op r n 2r 2r 1 by setting m nl 2r 1 when the log density is known to have degree of smoothness at least r analogous convergence results for the relative entropy are shown to hold in general for any class of log density functions and sequence of finite dimensional linear spaces having l2 and l approximation properties the approximation of log densities using polynomials has previously been considered by neyman 1937 to define alternatives for goodness of fit tests by good 1963 as an application of the method of maximum entropy or minimum relative entropy by crain 1974 1976a b 1977 who demonstrates existence and consistency of the maximum likelihood estimator and by mead and

a sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

exponential integrator methods for systems of non linear space fractional models with super diffusion processes in pattern formation abstract nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in science and engineering however solving these type of models is computationally demanding in this paper we propose an exponential integrator method for space fractional models as an attractive and easy to code alternative for other existing second order exponential integrator methods this scheme is based on using a real distinct poles discretization for the underlying matrix exponentials one of the major benefits of the proposed scheme is that the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency the scheme is established to be second order convergent and proven to be robust for nonlinear space fractional reaction diffusion problems involving non smooth initial data our approach is exhibited by solving a system of two dimensional problems which exhibits pattern formation and has applications in cell division empirically super diffusion processes are displayed by investigating the effect of the fractional power of the underlying laplacian operator on the pattern formation found in these models furthermore the superiority of our method over competing second order etd schemes bdf2 scheme and imex schemes is demonstrated

metric based upscaling we consider divergence form elliptic operators in dimension n ge 2 with l coefficients although solutions of these operators are only holder continuous we show that they are differentiable c1 with respect to harmonic coordinates it follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales this new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged homogenized quantities from subgrid scales into computational scales error bounds can be given and this method can also be used as a compression tool for differential operators 2006 wiley periodicals inc

approximation of least squares regression on nested subspaces on presente une methode generale pour approximer le biais et la variance dans une echelle de normes hilbertiennes

analysis of a two scale locally conservative subgrid upscaling for elliptic problems we present a two scale theoretical framework for approximating the solution of a second order elliptic problem the elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid but limits on computational power require that computations be performed on a coarse grid we consider the elliptic problem in mixed variational form over w times bf v subset l 2 times h rm div we base our scale expansion on local mass conservation over the coarse grid it is used to define a direct sum decomposition of w times bf v into coarse and subgrid subspaces w c times bf v c and delta w times delta bf v such that 1 nabla cdot bf v c w c and nabla cdot delta bf v delta w and 2 the space delta bf v is locally supported over the coarse mesh we then explicitly decompose the variational problem into coarse and subgrid scale problems the subgrid problem gives a well defined operator taking w c times bf v c to delta w times delta bf v which is localized in space and it is used to upscale that is to remove the subgrid from the coarse scale problem using standard mixed finite element spaces two scale mixed spaces are defined a mixed approximation is defined which can be viewed as a type of variational multiscale method or a residual free bubble technique a numerical green s function approach is used to make the approximation to the subgrid operator efficient to compute a mixed method pi operator is defined for the two scale approximation spaces and used to show optimal order error estimates

optimal rates for zero order convex optimization the power of two function evaluations we consider derivative free algorithms for stochastic and nonstochastic convex optimization problems that use only function values rather than gradients focusing on nonasymptotic bounds on convergence rates we show that if pairs of function values are available algorithms for d dimensional optimization that use gradient estimates based on random perturbations suffer a factor of at most sqrt d in convergence rate over traditional stochastic gradient methods we establish such results for both smooth and nonsmooth cases sharpening previous analyses that suggested a worse dimension dependence and extend our results to the case of multiple m ge 2 evaluations we complement our algorithmic development with information theoretic lower bounds on the minimax convergence rate of such problems establishing the sharpness of our achievable results up to constant sometimes logarithmic factors

stochastic computation based on orthogonal expansion of random fields abstract in solving stochastic differential equations recently a random variable based polynomial chaos rv pc method has been developed as a major numerical solver for many realistic random media problems the rv pc method however confronts a critical challenge of curse of dimensionality since a random field is represented by random variables the use of various optimal sampling techniques and conventional high dimensional methods still faces the curse of dimensionality distinguished from all the random variable based methods in this study a novel random field based orthogonal expansion rf oe method is proposed in aim to circumvent the curse of dimensionality for many physical systems whereas the input information is represented as random fields or stochastic processes e g seismic ocean wave wind load shock wave and geophysical media multiscale modeling of random media problems is selected as the benchmark problem to test the rf oe method especially the rf oe method provides a perfect matching with the higher order mehler s formula by replacing high dimensional random variable representations with a series of orthogonal expansion terms about an underlying random field process the rf oe method reduces the number of dimensions of a stochastic differential equation exponentially in the first example the rf oe method is verified with monte carlo simulation on a lognormal random media flow transport problem in the second example the rf oe method is applied to a time domain problem involving orthogonal expansion of random excitations in the conclusion the items for further development of the rf oe method are identified

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

mutual information metric entropy and cumulative relative entropy risk assume p theta theta epsilon theta is a set of probability distributions with a common dominating measure on a complete separable metric space y a state theta epsilon theta is chosen by nature a statistician obtains n independent observations y 1 dots y n from y distributed according to p theta for each time t between 1 and n based on the observations y 1 dots y t 1 the statistician produces an estimated distribution hat p t for p theta and suffers a loss l p theta hat p t the cumulative risk for the statistician is the average total loss up to time n of special interest in information theory data compression mathematical finance computational learning theory and statistical mechanics is the special case when the loss l p theta hat p t is the relative entropy between the true distribution p theta and the estimated distribution hat p t here the cumulative bayes risk from time 1 to n is the mutual information between the random parameter theta and the observations y 1 dots y n new bounds on this mutual information are given in terms of the laplace transform of the hellinger distance between pairs of distributions indexed by parameters in theta from these bounds on the cumulative minimax risk are given in terms of the metric entropy of theta with respect to the hellinger distance the assumptions required for these bounds are very general and do not depend on the choice of the dominating measure they apply to both finite and infinite dimensional theta they apply in some cases where y is infinite dimensional in some cases where y is not compact in some cases where the distributions are not smooth and in some parametric cases where asymptotic normality of the posterior distribution fails

on a numerical subgrid upscaling algorithm for stokes brinkman equations this paper discusses a numerical subgrid resolution approach for solving the stokes brinkman system of equations which is describing coupled flow in plain and in highly porous media various scientific and industrial problems are described by this system and often the geometry and or the permeability vary on several scales a particular target is the process of oil filtration in many complicated filters the filter medium or the filter element geometry are too fine to be resolved by a feasible computational grid the subgrid approach presented in this paper is aimed at describing how these fine details are accounted for by solving auxiliary problems in appropriately chosen grid cells on a relatively coarse computational grid this is done via a systematic and careful procedure of modifying and updating the coefficients of the stokes brinkman system in chosen cells this numerical subgrid approach is motivated from one side from homogenization theory from which we borrow the formulations for the so called cell problem and from the other side from the numerical upscaling approaches such as multiscale finite volume multiscale finite element etc results on the algorithm s efficiency both in terms of computational time and memory usage are presented comparison of the full fine grid solution when possible of the stokes brinkman system with the subgrid solution of the upscaled stokes brinkman system including effective permeabilities for the quasi porous cells are presented in order to evaluate the accuracy and the efficiency advantages and limitations of the considered subgrid approach are discussed

residual driven online generalized multiscale finite element methods the construction of local reduced order models via multiscale basis functions has been an area of active research in this paper we propose online multiscale basis functions which are constructed using the offline space and the current residual online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators we derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error this error decrease is independent of physical parameters such as the contrast and multiple scales in the problem the offline spaces are constructed using generalized multiscale finite element methods gmsfem we show that if one chooses a sufficient number of offline basis functions one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast we note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects using the residual information we can incorporate the distant information provided the offline approximation satisfies certain properties in the paper theoretical and numerical results are presented our numerical results show that if the offline space is sufficiently large in terms of the dimension such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues then the error reduction by adding online multiscale basis function is independent of the contrast we discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces

the ordered subsets mirror descent optimization method with applications to tomography we describe an optimization problem arising in reconstructing three dimensional medical images from positron emission tomography pet a mathematical model of the problem based on the maximum likelihood principle is posed as a problem of minimizing a convex function of several million variables over the standard simplex to solve a problem of these characteristics we develop and implement a new algorithm ordered subsets mirror descent and demonstrate theoretically and computationally that it is well suited for solving the pet reconstruction problem

stochastic convex optimization with bandit feedback this paper addresses the problem of minimizing a convex lipschitz function f over a convex compact set mathcal x under a stochastic bandit i e noisy zeroth order feedback model in this model the algorithm is allowed to observe noisy realizations of the function value f x at any query point x in mathcal x the quantity of interest is the regret of the algorithm which is the sum of the function values at algorithm s query points minus the optimal function value we demonstrate a generalization of the ellipsoid algorithm that incurs widetilde mathcal o rm poly d sqrt t regret since any algorithm has regret at least omega sqrt t on this problem our algorithm is optimal in terms of the scaling with t

analysis of the simulation of single phase flow through a naturally fractured reservoir a general form of the double porosity model for single phase flow through a naturally fractured reservoir is derived by explicitly considering fluid flow in individual matrix blocks the warren and root model is shown to be a crude approximation to this model the general model consists of a parabolic equation coupled to a series of parabolic equations it is shown that the coupling term can be viewed as a positive semidefinite perturbation of the time derivative and hence it is verified that the model is well posed a finite element method is presented to approximate the solution and optimal order l 2 error estimates are derived

randomized smoothing for stochastic optimization we analyze convergence rates of stochastic optimization algorithms for nonsmooth convex optimization problems by combining randomized smoothing techniques with accelerated gradient methods we obtain convergence rates of stochastic optimization procedures both in expectation and with high probability that have optimal dependence on the variance of the gradient estimates to the best of our knowledge these are the first variance based rates for nonsmooth optimization we give several applications of our results to statistical estimation problems and provide experimental results that demonstrate the effectiveness of the proposed algorithms we also describe how a combination of our algorithm with recent work on decentralized optimization yields a distributed stochastic optimization algorithm that is order optimal

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

two scale homogenization of non linear degenerate evolution equations using the notion of two scale convergence developed by allaire the homogenization of a degenerate non linear evolution equation with periodically oscillating coefficients is presented a two scale homogenized system is obtained as the limit of the periodic problem monotone operator methods and two scale convergence are employed to show that the solutions of the periodic problem converge to the unique solution of the homogenized system homogenized initial conditions are also obtained and the sense in which they hold for the homogenized initial value problem is made specific

micro structure models of diffusion in fissured media abstract two diffusion models are developed which recognize the local geometry of the individual cells or storage sites and the exchange of flux on the micro scale of these cells the cauchy problems for these model systems are shown to be resolved by holomorphic semigroups and various classical models are obtained as limits of these disributed micro structure models

information theory and superefficiency the asymptotic risk of efficient estimators with kullback leibler loss in smoothly parametrized statistical models is k 2n where k is the parameter dimension and n is the sample size under fairly general conditions we given a simple information theoretic proof that the set of parameter values where any arbitrary estimator is superefficient is negligible the proof is based on a result of rissanen that codes have asymptotic redundancy not smaller than k 2 log n except in a set of measure 0

multiscale model reduction for pore scale simulation of li ion batteries using gmsfem abstract in this work we consider electrochemical processes for the pore scale simulation of lithium ion batteries libs mathematical model consists of the coupled system of the equations for the concentration and electric potential we develop fine scale approximation using discontinuous galerkin approach where interface condition is imposed weakly we present novel multiscale model reduction technique based on the gmsfem where multiscale basis functions are constructed using information about variation of the medium at the micro level we present numerical results for two cases of the boundary conditions and compare errors for different coarse grids for testing the proposed computational multiscale method numerical results show that the multiscale basis functions can efficiently capture the information of the fine scale features of the medium with significant dimension reduction of the system and provide accurate solutions