multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

on multiscale methods in petrov galerkin formulation in this work we investigate the advantages of multiscale methods in petrov galerkin pg formulation in a general framework the framework is based on a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space with good approximation properties and a high dimensional remainder space which only contains negligible fine scale information the multiscale space can then be used to obtain accurate galerkin approximations as a model problem we consider the poisson equation we prove that a petrov galerkin formulation does not suffer from a significant loss of accuracy and still preserve the convergence order of the original multiscale method we also prove inf sup stability of a pg continuous and a discontinuous galerkin finite element multiscale method furthermore we demonstrate that the petrov galerkin method can decrease the computational complexity significantly allowing for more efficient solution algorithms as another application of the framework we show how the petrov galerkin framework can be used to construct a locally mass conservative solver for two phase flow simulation that employs the buckley leverett equation to achieve this we couple a pg discontinuous galerkin finite element method with an upwind scheme for a hyperbolic conservation law

constraint energy minimizing generalized multiscale discontinuous galerkin method abstract numerical simulation of flow problems and wave propagation in heterogeneous media has important applications in many engineering areas however numerical solutions on the fine grid are often prohibitively expensive and multiscale model reduction techniques are introduced to efficiently solve for an accurate approximation on the coarse grid in this paper we propose an energy minimization based multiscale model reduction approach in the discontinuous galerkin discretization setting the main idea of the method is to extract the non decaying component in the high conductivity regions by identifying dominant modes with small eigenvalues of local spectral problems and define multiscale basis functions in coarse oversampled regions by constraint energy minimization problems the multiscale basis functions are in general discontinuous on the coarse grid and coupled by interior penalty discontinuous galerkin formulation the minimal degree of freedom in representing high contrast features is achieved through the design of local spectral problems which provides the most compressed local multiscale space we analyze the method for solving darcy flow problem and show that the convergence is linear in coarse mesh size and independent of the contrast provided that the oversampling size is appropriately chosen numerical results are presented to show the performance of the method for simulation on flow problem and wave propagation in high contrast heterogeneous media

bounds for the variation of the roots of a polynomial and the eigenvalues of a matrix we derive some new bounds for the distance between the roots of two polynomials in terms of their coefficients and for the distance between the eigenvalues of two matrices in terms of the norm of their difference

multiscale hybrid mixed method for the stokes and brinkman equations the method abstract the multiscale hybrid mixed mhm method is extended to the stokes and brinkman equations with highly heterogeneous coefficients the approach is constructive we first propose an equivalent dual hybrid formulation of the original problem using a coarse partition of the heterogeneous domain faces may be not aligned with jumps in the data then the exact velocity and the pressure are characterized as the solution of a global face problem and the solutions of local independent stokes or brinkman problems at the continuous level owing to this decomposition the one level mhm method stems from the standard galerkin approach for the lagrange multiplier space basis functions are responsible for upscaling the unresolved scales of the medium into the global formulation they are the exact solution of the local problems with prescribed neumann boundary conditions on faces driven by the lagrange multipliers we make the mhm method effective by adopting the unusual stabilized finite element method to solve the local problems approximately as such equal order interpolation turns out to be an option for the velocity the pressure and the lagrange multipliers the numerical solutions share the important properties of the continuum such as local equilibrium with respect to external forces and local mass conservation several academic and highly heterogeneous tests infer that the method achieves super convergence for the velocity as well optimal convergence for the pressure and also for the stress tensor in their natural norms

on total variation minimization and surface evolution using parametric maximum flows in a recent paper boykov et al lncs vol 3953 pp 409 422 2006 propose an approach for computing curve and surface evolution using a variational approach and the geo cuts method of boykov and kolmogorov international conference on computer vision pp 26 33 2003 we recall in this paper how this is related to well known approaches for mean curvature motion introduced by almgren et al siam journal on control and optimization 31 2 387 438 1993 and luckhaus and sturzenhecker calculus of variations and partial differential equations 3 2 253 271 1995 and show how the corresponding problems can be solved with sub pixel accuracy using parametric maximum flow techniques this provides interesting algorithms for computing crystalline curvature motion possibly with a forcing term

a new certification framework for the port reduced static condensation reduced basis element method abstract in this paper we introduce a new certification framework for the port reduced static condensation reduced basis element pr scrbe method which has been developed for the simulation of large component based applications such as bridges or acoustic waveguides in an offline computational stage we construct a library of interoperable parametrized reference components in the subsequent online stage we instantiate and connect the components at the interfaces ports to form a system of components to compute a truth finite element approximation of the say coercive elliptic partial differential equation on the component based system we use a domain decomposition approach for an efficient simulation we employ two different types of model reduction a reduced basis rb approximation within the interior of the component huynh et al 2013 and empirical port reduction eftang and patera 2013 on the ports where the components connect we demonstrate the well posedness of the pr scrbe approximation and introduce a new certification framework to assess the quality of the port reduction we use conservative fluxes we adapt the standard estimators from rb methods to the scrbe setting to derive an a posteriori error estimator for the rb error contribution in order to combine the a posteriori estimators for both error contributions and derive a rigorous a posteriori error estimator for pr scrbe we adapt techniques from multi scale methods and component mode synthesis finally we prove that the effectivity of the derived estimator can be bounded we provide numerical experiments for heat conduction and linear elasticity to show that the derived a posteriori error estimator provides an effective estimator moreover we demonstrate the applicability of the introduced certification framework by analyzing the computational online costs

bounds for vertical points of solutions of prescribed mean curvature type equations i on obtient des conditions sous lesquelles des solutions radiales d equations de type capillaire generalisees presentent un premier point vertical avec des bornes et des estimations asymptotiques pour ces points

on the multi level splitting of finite element spaces in this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases we show that the condition number of such a stiffness matrix behaves like o log 2 where is the condition number of the stiffness matrix with respect to a nodal basis in the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like o left left log frac 1 h right 2 right instead of o left left frac 1 h right 2 right for a nodal basis the proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization especially we do not need the quasiuniformity of the employed triangulations as the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis our results mean that the method of conjugate gradients needs onlyo log n steps ando n log n computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures heren denotes the dimension of the finite element space and of the discrete linear problem to be solved

efficient implementation of the localized orthogonal decomposition method abstract in this paper we present algorithms for an efficient implementation of the localized orthogonal decomposition method lod the lod is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales we show how the method can be implemented in a fairly standard finite element framework and discuss its realization for different types of problems such as linear elliptic problems with rough coefficients and linear eigenvalue problems

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

the total variation flow in rn the purpose of this chapter is to prove existence and uniqueness of the minimizing total variation flow in n frac partial u partial t div left frac du left du right right in left 0 infty right times mathbb r n 3 1 coupled with the initial condition u 0 x u 0 x text x in mathbb r n 3 2 when u0 l loc 1 l n

a reduced basis localized orthogonal decomposition in this work we combine the framework of the reduced basis method rb with the framework of the localized orthogonal decomposition lod in order to solve parametrized elliptic multiscale problems the idea of the lod is to split a high dimensional finite element space into a low dimensional space with comparably good approximation properties and a remainder space with negligible information the low dimensional space is spanned by locally supported basis functions associated with the node of a coarse mesh obtained by solving decoupled local problems however for parameter dependent multiscale problems the local basis has to be computed repeatedly for each choice of the parameter to overcome this issue we propose an rb approach to compute in an offline stage lod for suitable representative parameters the online solution of the multiscale problems can then be obtained in a coarse space thanks to the lod decomposition and for an arbitrary value of the parameters thanks to a suitable interpolation of the selected rb the online rb lod has a basis with local support and leads to sparse systems applications of the strategy to both linear and nonlinear problems are given

hierarchical multiscale finite element method for multi continuum media abstract simulation in media with multiple continua where each continuum interacts with every other is often challenging due to multiple scales and high contrast one needs some type of model reduction one of the approaches is a multi continuum technique where every process in each continuum is modeled separately and an interaction term is added direct numerical simulation in multiscale media is usually not practicable for this reason one constructs the corresponding homogenized equations which approximate the solutions to the multiscale equations when the microscopic scales tend to 0 computing the effective coefficients of the homogenized equations can be expensive because one needs to solve local cell problems for a large number of macroscopic points the paper considers a two scale two continuum system where the interaction terms between the continua are scaled as o 1 2 where is the microscopic scale we prove that in the homogenization limit we obtain the same limit for both continua we derive the homogenized equation for the limit function and prove the homogenization convergence rigorously the homogenized coefficients are established from solutions of the cell problems which are systems of equations of a similar form as the two continuum system we develop a hierarchical approach for solving these cell problems at a dense network of macroscopic points with an essentially optimal computation cost the method employs the fact that neighboring representative volume elements rves share similar features and effective properties of the neighboring rves are close to each other the hierarchical approach reduces computation cost by using different levels of resolution for cell problems at different macroscopic points solutions of the cell problems which are solved with a higher level of resolution are employed to correct the solutions at neighboring macroscopic points that are computed by approximation spaces with a lower level of resolution the method requires a hierarchy of macroscopic grid points and corresponding nested approximation spaces with different levels of resolution each level of macroscopic points is assigned to an approximation finite element fe space which is used to solve the cell problems at the macroscopic points from that level we prove rigorously that this hierarchical method achieves the same level of accuracy as that of the full solve where cell problems at every macroscopic point are solved using fe spaces with the highest level of resolution but at the essentially optimal computation cost numerical implementation that computes effective permeabilities of a two scale multicontinuum system via the numerical solutions of the cell problems supports the analytical results

a combined finite element method for elliptic problems posted in domains with rough boundaries abstract a combined finite element method is presented in this paper to solve the elliptic problems posted in domains with rough boundaries solving these problems numerically is difficult because resolving the boundaries usually requires very fine meshes while good quality meshes often over refine unnecessarily the interior of the domain the basic idea of the proposed method is to use a fine mesh with size h in the vicinity of oscillating boundaries and a coarse mesh with size h h for other portions of the domain to reduce some unnecessary computational effort the transmission conditions across the fine coarse mesh interface are treated by the penalty technique the key point of the method lies in the new scheme employing a weighted average in the definition of the bilinear form which avoids the affection of the ratio h h in the error estimate we prove a quasi optimal convergence in terms of elements since there is no whole h 2 regularity in the domain with rough boundaries numerical results are provided for elliptic equations in domains with non oscillating or oscillating boundaries to illustrate the theoretical results

a characterization of convex calibrable sets in the main purpose of this paper is to characterize the calibrability of bounded convex sets in open image in new window by the mean curvature of its boundary extending the known analogous result in dimension 2 as a by product of our analysis we prove that any bounded convex set c of class c1 1 has a convex calibrable set k in its interior and and for any volume v k c the solution of the perimeter minimizing problem with fixed volume v in the class of sets contained in c is a convex set as a consequence we describe the evolution of convex sets in open image in new window by the minimizing total variation flow

localized orthogonal decomposition techniques for boundary value problems in this paper we propose a local orthogonal decomposition method lod for elliptic partial differential equations with inhomogeneous dirichlet and neumann boundary conditions for this purpose we present new boundary correctors which preserve the common convergence rates of the lod even if the boundary condition has a rapidly oscillating fine scale structure we prove a corresponding a priori error estimate and present numerical experiments we also demonstrate numerically that the method is reliable with respect to thin conductivity channels in the diffusion matrix accurate results are obtained without resolving these channels by the coarse grid and without using patches that contain the channels

the dirichlet problem for the total variation flow suppose that is an open bounded domain with a lipschitz boundary the purpose of this chapter is to study the dirichlet problem left begin gathered frac partial u partial t div left frac du left du right right in q left 0 infty right times omega hfill u left t x right phi left x right on s left 0 infty right times partial omega hfill u left 0 x right u 0 left x right in x in omega hfill end gathered right 5 1 where u0 l1 and l1 this evolution equation is related to the gradient descent method used to solve the problem begin gathered minimize int omega left du right int omega f udx int partial omega left u phi right d mathcal h n 1 hfill u in bv left omega right hfill end gathered 2 where f l1 l existence for this variational problem was proved in 118 theorem 1 4

variational multiscale error estimators for solid mechanics adaptive simulations an orthogonal subgrid scale approach abstract in this work we present a general error estimator for the finite element solution of solid mechanics problems based on the variational multiscale method the main idea is to consider a rich model for the subgrid scales as an error estimator the subscales are considered to belong to a space orthogonal to the finite element space orthogonal subgrid scales and we take into account their contribution both in the element interiors and on the element boundaries subscales on the element boundaries a simple analysis shows that the upper bound for the obtained error estimator is sharper than in other error estimators based on the variational multiscale method numerical examples show that the proposed error estimator is an accurate approximation for the energy norm error and can be used both in simple linear constitutive models and in more complex non linear cases

can a finite element method perform arbitrarily badly in this paper we construct elliptic boundary value problems whose standard finite element approximations converge arbitrarily slowly in the energy norm and show that adaptive procedures cannot improve this slow convergence we also show that the l 2 norm and the nodal point errors converge arbitrarily slowly with the l 2 norm two cases need to be distinguished and the usual duality principle does not characterize the error completely the constructed elliptic problems are one dimensional

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

a generalized multiscale finite element method for poroelasticity problems i in this paper we consider the numerical solution of poroelasticity problems that are of biot type and develop a general algorithm for solving coupled systems we discuss the challenges associated with mechanics and flow problems in heterogeneous media the two primary issues being the multiscale nature of the media and the solutions of the fluid and mechanics variables traditionally developed with separate grids and methods for the numerical solution we develop and implement a generalized multiscale finite element method gmsfem that solves problem on a coarse grid by constructing local multiscale basis functions the procedure begins with construction of multiscale bases for both displacement and pressure in each coarse block using a snapshot space and local spectral problems we construct a basis of reduced dimension finally after multiplying by a multiscale partitions of unity the multiscale basis is constructed in the offline phase and the coarse grid problem then can be solved for arbitrary forcing and boundary conditions we implement this algorithm on two heterogeneous media and compute error between the multiscale solution with the fine scale solutions randomized oversampling and forcing strategies are also tested

edge multiscale methods for elliptic problems with heterogeneous coefficients abstract in this paper we proposed two new types of edge multiscale methods motivated by 14 to solve partial differential equations pdes with high contrast heterogeneous coefficients edge spectral multiscale finte element method esmsfem and wavelet based edge multiscale finite element method wemsfem their convergence rates for elliptic problems with high contrast heterogeneous coefficients are demonstrated in terms of the coarse mesh size h the number of spectral basis functions and the level of the wavelet space l which are verified by extensive numerical tests

uniqueness of the cheeger set of a convex body abstract we prove that if c r n is an open bounded convex set then there is only one cheeger set inside c and it is convex a cheeger set of c is a set which minimizes the ratio perimeter over volume among all subsets of c

edge residuals dominate a posteriori error estimates for low order finite element methods we prove that up to higher order perturbation terms edge residuals yield global upper and local lower bounds on the error of linear finite element methods both in h1 and l2 norms we present two proofs one uses the standard l2 projection and the other relies on a new weighted clement type interpolation operator

a hierarchical finite element monte carlo method for stochastic two scale elliptic equations we consider two scale elliptic equations whose coefficients are random in particular we study two cases in the first case the coefficients are obtained from an ergodic dynamical system acting on a probability space and in the second the case the coefficients are periodic in the microscale but are random we suppose that the coefficients also depend on the macroscopic slow variables while the effective coefficient of the ergodic homogenization problem is deterministic to approximate it it is necessary to solve cell equations in a large but finite size truncated cube and compute an approximated effective coefficient from the solution of this equation this approximated effective coefficient is however realization dependent and the deterministic effective coefficient of the homogenization problem can be approximated by taking its expectation in the periodic random setting the effective coefficient for each realization is obtained from the solutions of cell equations which are posed in the unit cube but to compute its average by the monte carlo method we need to consider many uncorrelated realizations to accurately approximate the average straightforward employment of finite element approximation and the monte carlo method to compute this expectation with the same level of finite element resolution and the same number of monte carlo samples at every macroscopic point is prohibitively expensive we develop a hierarchical finite element monte carlo algorithm to approximate the effective coefficients at a dense hierarchical network of macroscopic points the method requires an optimal level of complexity that is essentially equal to that for computing the effective coefficient at one macroscopic point and achieves essentially the same accuracy the levels of accuracy for solving cell problems and for the monte carlo sampling are chosen according to the level in the hierarchy that the macroscopic points belong to solutions and the effective coefficients at the points where the cell problems are solved with higher accuracy and the effective coefficients are approximated with a larger number of monte carlo samples are employed as correctors for the effective coefficient at those points at which the cell problems are solved with lower accuracy and fewer monte carlo samples the method combines the hierarchical finite element method for solving cell problems at a dense network of macroscopic points with the optimal complexity developed in brown etal 2013 with a hierarchical monte carlo sampling algorithm that uses different number of samples at different macroscopic points depending on the level in the hierarchy that the macroscopic points belong to proof of concept numerical examples confirm the theoretical results

improving spectral variation bounds with chebyshev polynomials a new spectral variation bound is established for linear operators a and b on finite dimensional vector spaces this bound is v a b c a b 1n a b n 1 n with c 8 this result affirmatively answers a conjecture of friedland

darcy equation for random porous media many attempts have been made recently to deduce darcy equations for flows through porous media much has been done for models having a periodic microstructure this paper presents a mathematical analysis based on the more general and more realistic assumption that the microstructure of porous media is random and stochastically homogeneous for this type of random domain we consider homogenization of the poisson and stokes equations supplemented by homogeneous dirichlet boundary conditions on the random boundary with this approach we get the darcy equation in general as well as present details for the particular case of a checkerboard model of porous media 1996 john wiley sons inc

a posteriori error control for the binary mumford shah model the binary mumford shah model is a widespread tool for image segmentation and can be considered as a basic model in shape optimization with a broad range of applications in computer vision ranging from basic segmentation and labeling to object reconstruction this paper presents robust a posteriori error estimates for a natural error quantity namely the area of the non properly segmented region to this end a suitable strictly convex and non constrained relaxation of the originally non convex functional is investigated and repin s functional approach for a posteriori error estimation is used to control the numerical error for the relaxed problem in the l norm in combination with a suitable cut out argument a fully practical estimate for the area mismatch is derived this estimate is incorporated in an adaptive meshing strategy two different adaptive primal dual finite element schemes and the most frequently used finite difference discretization are investigated and compared numerical experiments show qualitative and quantitative properties of the estimates and demonstrate their usefulness in practical applications

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

fast online generalized multiscale finite element method using constraint energy minimization abstract local multiscale methods often construct multiscale basis functions in the offline stage without taking into account input parameters such as source terms boundary conditions and so on these basis functions are then used in the online stage with a specific input parameter to solve the global problem at a reduced computational cost recently online approaches have been introduced where multiscale basis functions are adaptively constructed in some regions to reduce the error significantly in multiscale methods it is desired to have only 1 2 iterations to reduce the error to a desired threshold using generalized multiscale finite element framework 10 it was shown that by choosing sufficient number of offline basis functions the error reduction can be made independent of physical parameters such as scales and contrast in this paper our goal is to improve this using our recently proposed approach 4 and special online basis construction in oversampled regions we show that the error reduction can be made sufficiently large by appropriately selecting oversampling regions our numerical results show that one can achieve a three order of magnitude error reduction which is better than our previous methods we also develop an adaptive algorithm and enrich in selected regions with large residuals in our adaptive method we show that the convergence rate can be determined by a user defined parameter and we confirm this by numerical simulations the analysis of the method is presented

convergence of a discontinuous galerkin multiscale method we present a discontinuous galerkin multiscale method for second order elliptic problems and prove convergence we consider a heterogeneous and highly varying diffusion coefficient in l infty omega mathbb r d times d sym with uniform spectral bounds without any assumption on scale separation or periodicity the multiscale method uses a corrected basis that is computed on patches subdomains the error due to truncation of the corrected basis decreases exponentially with the size of the patches hence to achieve an algebraic convergence rate of the multiscale solution on a uniform mesh with mesh size h to a reference solution it is sufficient to choose the patch sizes mathcal o h log h we also discuss a way to further localize the corrected basis to elementwise support improved convergence rate can be achieved depending on the piecewise regularity of the forcing function linear convergence in energy norm and quadratic convergence in the l 2 norm is obtained independently of the

two level discretization techniques for ground state computations of bose einstein condensates this work presents a new methodology for computing ground states of bose einstein condensates based on finite element discretizations on two different scales of numerical resolution in a preprocessing step a low dimensional coarse generalized finite element space is constructed it is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties the nonlinear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low dimensional space without significant loss of accuracy when compared with the solution of the full fine scale problem the preprocessing step is independent of the types and numbers of bosons a postprocessing step further improves the accuracy of the method we present rigorous a priori error estimates that predict convergence rates h 3 for the ground state eigenfunction and h 4 for the corresponding eigenvalue without pre asymptotic effects h being the coarse scale discretization parameter numerical experiments indicate that these high rates may still be pessimistic

a two level overlapping schwarz method with energy minimizing multiscale coarse basis functions abstract a two level overlapping schwarz method is developed for second order elliptic problems with highly oscillatory and high contrast coefficients for which it is known that the standard coarse problem fails to give a robust preconditioner in this paper we develop energy minimizing multiscale finite element functions to form a more robust coarse problem first a local spectral problem is solved in each non overlapping coarse subdomain and dominant eigenfunctions are selected as auxiliary functions which are crucial for the high contrast case the required multiscale basis functions are then obtained by minimizing an energy subject to some orthogonality conditions with respect to the auxiliary functions due to an exponential decay property the minimization problem is solved locally on oversampling subdomains that are unions of a few coarse subdomains the coarse basis functions are therefore local and can be computed efficiently the resulting preconditioner is shown to be robust with respect to the contrast in the coefficients as well as the overlapping width in the subdomain partition numerical results are included to confirm the theory and show the performance

on a degenerate problem in the calculus of variations we establish the uniqueness of the solutions for a degenerate scalar problem in the multiple integrals calculus of variations the proof requires as a preliminary step the study of the regularity properties of the solutions and of their level sets we exploit the uniqueness and the regularity results to explore some of their qualitative properties in particular we emphasize the link between the supports of the solutions and the cheeger problem

preconditioning markov chain monte carlo simulations using coarse scale models we study the preconditioning of markov chain monte carlo mcmc methods using coarse scale models with applications to subsurface characterization the purpose of preconditioning is to reduce the fine scale computational cost and increase the acceptance rate in the mcmc sampling this goal is achieved by generating markov chains based on two stage computations in the first stage a new proposal is first tested by the coarse scale model based on multiscale finite volume methods the full fine scale computation will be conducted only if the proposal passes the coarse scale screening for more efficient simulations an approximation of the full fine scale computation using precomputed multiscale basis functions can also be used comparing with the regular mcmc method the preconditioned mcmc method generates a modified markov chain by incorporating the coarse scale information of the problem the conditions under which the modified markov chain will converge to the correct posterior distribution are stated in the paper the validity of these assumptions for our application and the conditions which would guarantee a high acceptance rate are also discussed we would like to note that coarse scale models used in the simulations need to be inexpensive but not necessarily very accurate as our analysis and numerical simulations demonstrate we present numerical examples for sampling permeability fields using two point geostatistics the karhunen loeve expansion is used to represent the realizations of the permeability field conditioned to the dynamic data such as production data as well as some static data our numerical examples show that the acceptance rate can be increased by more than 10 times if mcmc simulations are preconditioned using coarse scale models

a combined finite element and oversampling multiscale petrov galerkin method for the multiscale elliptic problems with singularities in this paper we construct a combined finite element and oversampling multiscale petrov galerkin method fe omspgm to solve the multiscale problems which may have singularities in some special portions of the computational domain for example in the simulation of subsurface flow singularities lie in the porous media with channelized features or in near well regions since the solution behaves like the green function the basic idea of fe omspgm is to utilize the traditional finite element method fem directly on a fine mesh of the problematic part of the domain and using the petrov galerkin version of oversampling multiscale finite element method omspgm on a coarse mesh of the other part the transmission condition across the fe omspg interface is treated by the penalty technique the fe omspgm takes advantages of the fem and omspgm which uses much less dofs than the standard fem and may be more accurate than the omspgm for problems with singularities although the error analysis is carried out under the assumption that the oscillating coefficients are periodic our method is not restrict to the periodic case numerical examples with periodic and random highly oscillating coefficients as well as the multiscale problems on the l shaped domain and multiscale problems with high contrast channels or well singularities are presented to demonstrate the efficiency and accuracy of the proposed method

total variation and cheeger sets in gauss space abstract the aim of this paper is to study the isoperimetric problem with fixed volume inside convex sets and other related geometric variational problems in the gauss space in both the finite and infinite dimensional case we first study the finite dimensional case proving the existence of a maximal cheeger set which is convex inside any bounded convex set we also prove the uniqueness and convexity of solutions of the isoperimetric problem with fixed volume inside any convex set then we extend these results in the context of the abstract wiener space and for that we study the total variation denoising problem in this context

on boundary detachment phenomena for the total variation flow with dynamic boundary conditions abstract we combine the total variation flow suitable for crystal modeling and image analysis with the dynamic boundary conditions we analyze the behavior of facets at the parts of the boundary where these conditions are imposed we devote particular attention to the radially symmetric data we observe that the boundary layer detachment actually can happen at concave parts of the boundary

large solutions for nonlinear parabolic equations without absorption terms in this paper we give a suitable notion of entropy solution of parabolic p laplacian type equations with 1 p 2 which blows up at the boundary of the domain we prove existence and uniqueness of this type of solutions when the initial datum is locally integrable for 1 p 2 or integrable for p 1 i e the total variation flow case

a generalized multiscale finite element method for poroelasticity problems ii nonlinear coupling in this paper we consider the numerical solution of some nonlinear poroelasticity problems that are of biot type and develop a general algorithm for solving nonlinear coupled systems we discuss the difficulties associated with flow and mechanics in heterogenous media with nonlinear coupling the central issue being how to handle the nonlinearities and the multiscale scale nature of the media to compute an efficient numerical solution we develop and implement a generalized multiscale finite element method gmsfem that solves nonlinear problems on a coarse grid by constructing local multiscale basis functions and treating part of the nonlinearity locally as a parametric value after linearization with a picard iteration the procedure begins with construction of multiscale bases for both displacement and pressure in each coarse block by treating the staggered nonlinearity as a parametric value using a snapshot space and local spectral problems we construct an offline basis of reduced dimension from here an online parametric dependent space is constructed finally after multiplying by a multiscale partitions of unity the multiscale basis is constructed and the coarse grid problem then can be solved for arbitrary forcing and boundary conditions we implement this algorithm on a geometry with a linear and nonlinear pressure dependent permeability field and compute error between the multiscale solution with the fine scale solutions

a multiscale method for linear elasticity reducing poisson locking we propose a generalized finite element method for linear elasticity equations with highly varying and oscillating coefficients the method is formulated in the framework of localized orthogonal decomposition techniques introduced by malqvist and peterseim 2014 assuming only l infinity coefficients we prove linear convergence in the h 1 norm also for materials with large lame parameter a the theoretical a priori error estimate is confirmed by numerical examples

coarse gradient langevin algorithms for dynamic data integration and uncertainty quantification the main goal of this paper is to design an efficient sampling technique for dynamic data integration using the langevin algorithms based on a coarse scale model of the problem we compute the proposals of the langevin algorithms using the coarse scale gradient of the target distribution to guarantee a correct and efficient sampling each proposal is first tested by a metropolis acceptance rejection step with a coarse scale distribution if the proposal is accepted in the first stage then a fine scale simulation is performed at the second stage to determine the acceptance probability comparing with the direct langevin algorithm the new method generates a modified markov chain by incorporating the coarse scale information of the problem under some mild technical conditions we prove that the modified markov chain converges to the correct posterior distribution we would like to note that the coarse scale models used in the simulations need to be inexpensive but not necessarily very accurate as our analysis and numerical simulations demonstrate we present numerical examples for sampling permeability fields using two point geostatistics karhunen loeve expansion is used to represent the realizations of the permeability field conditioned to the dynamic data such as the production data as well as the static data the numerical examples show that the coarse gradient langevin algorithms are much faster than the direct langevin algorithms but have similar acceptance rates

flux norm approach to finite dimensional homogenization approximations with non separated scales and high contrast we consider linear divergence form scalar elliptic equations and vectorial equations for elasticity with rough l r d coefficients a x that in particular model media with non separated scales and high contrast in material properties while the homogenization of pdes with periodic or ergodic coefficients and well separated scales is now well understood we consider here the most general case of arbitrary bounded coefficients for such problems we introduce explicit and optimal finite dimensional approximations of solutions that can be viewed as a theoretical galerkin method with controlled error estimates analogous to classical homogenization approximations in particular this approach allows one to analyze a given medium directly without introducing the mathematical concept of an e family of media as in classical homogenization we define the flux norm as the l 2 norm of the potential part of the fluxes of solutions which is equivalent to the usual h 1 norm we show that in the flux norm the error associated with approximating in a properly defined finite dimensional space the set of solutions of the aforementioned pdes with rough coefficients is equal to the error associated with approximating the set of solutions of the same type of pdes with smooth coefficients in a standard space for example piecewise polynomial we refer to this property as the transfer property a simple application of this property is the construction of finite dimensional approximation spaces with errors independent of the regularity and contrast of the coefficients and with optimal and explicit convergence rates this transfer property also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that can be extended to elasticity equations the proofs of these homogenization results are based on a new class of elliptic inequalities these inequalities play the same role in our approach as the div curl lemma in classical homogenization

localization of elliptic multiscale problems this paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients the basis functions are solutions of local problems on vertex patches the error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches hence on a uniform mesh of size h patches of diameter h log 1 h are sufficient to preserve a linear rate of convergence in h without pre asymptotic or resonance effects the analysis does not rely on regularity of the solution or scale separation in the coefficient this result motivates new and justifies old classes of variational multiscale methods see more at http www ams org journals mcom 2014 83 290 s0025 5718 2014 02868 8 sthash z2ccfxig dpuf

gamblets for opening the complexity bottleneck of implicit schemes for hyperbolic and parabolic odes pdes with rough coefficients implicit schemes are popular methods for the integration of time dependent pdes such as hyperbolic and parabolic pdes however the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to pdes with rough coefficients we present a generalization of gamblets introduced in 62 enabling the resolution of these implicit systems in near linear complexity and provide rigorous a priori error bounds on the resulting numerical approximations of hyperbolic and parabolic pdes these generalized gamblets induce a multiresolution decomposition of the solution space that is adapted to both the underlying hyperbolic and parabolic pde and the system of odes resulting from space discretization and to the time steps of the numerical scheme

adaptive variational multiscale methods based on a posteriori error estimation energy norm estimates for elliptic problems we develop a new adaptive multiscale finite element method using the variational multiscale framework together with a systematic technique for approximation of the fine scale part of the solution the fine scale is approximated by a sum of solutions to decoupled localized problems which are solved numerically on a fine grid partition of a patch of coarse grid elements the sizes of the patches of elements may be increased to control the error caused by localization we derive an a posteriori error estimate in the energy norm which captures the dependency of the crucial discretization parameters the coarse grid mesh size the fine grid mesh size and the sizes of the patches based on the a posteriori error estimate we present an adaptive algorithm that automatically tunes these parameters finally we show how the method works in practice by presenting various numerical examples

a posteriori error estimates for the virtual element method an posteriori error analysis for the virtual element method vem applied to general elliptic problems is presented the resulting error estimator is of residual type and applies on very general polygonal polyhedral meshes the estimator is fully computable as it relies only on quantities available from the vem solution namely its degrees of freedom and element wise polynomial projection upper and lower bounds of the error estimator with respect to the vem approximation error are proven the error estimator is used to drive adaptive mesh refinement in a number of test problems mesh adaptation is particularly simple to implement since elements with consecutive co planar edges faces are allowed and therefore locally adapted meshes do not require any local mesh post processing

an adaptive dynamically low dimensional approximation method for multiscale stochastic diffusion equations abstract in this paper we propose a dynamically low dimensional approximation method to solve a class of time dependent multiscale stochastic diffusion equations in cheng et al 2013 a dynamically bi orthogonal dybo method was developed to explore low dimensional structures of stochastic partial differential equations spdes and solve them efficiently however when the spdes have multiscale features in physical space the original dybo method becomes expensive to address this issue we construct multiscale basis functions within the framework of generalized multiscale finite element method gmsfem for dimension reduction in the physical space to further improve the accuracy we also perform online procedure to construct online adaptive basis functions in the stochastic space we use the generalized polynomial chaos gpc basis functions to represent the stochastic part of the solutions numerical results are presented to demonstrate the efficiency of the proposed method in solving time dependent pdes with multiscale and random features

an analysis of the nlmc upscaling method for high contrast problems abstract in this paper we propose simple multiscale basis functions with constraint energy minimization to solve elliptic problems with high contrast medium 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 in our method each coarse block is decomposed into various regions according to the contrast ratio and we require that the contrast ratio should be relatively small within each region the basis functions are constructed by solving a local problem defined on the oversampling domains and they have mean value one on the chosen region and zero mean otherwise numerical analysis shows that the resulting basis functions can be localizable and have a decay property the convergence of the multiscale solution is also proved finally some numerical experiments are carried out to illustrate the performances of the proposed method they show that the proposed method can solve problem with high contrast medium efficiently in particular if the oversampling size is large enough then we can achieve the desired error

on a crystalline variational problem part ii bv regularity and structure of minimizers on facets for a nonsmooth positively one homogeneous convex function n 0 it is possible to introduce the class n of smooth boundaries with respect to to define their mean curvature and to prove that for e n l e 9 based on these results we continue the analysis on the structure of e and on the regularity properties of we prove that a facet f of e is lipschitz up to negligible sets and that has bounded variation on f further properties of the jump set of are inspected in particular in three space dimensions we relate the sublevel sets of on f to the geometry of the wulff shape 1

overlapping schwarz methods with adaptive coarse spaces for multiscale problems in 3d we propose two variants of the overlapping additive schwarz method for the finite element discretization of scalar elliptic problems in 3d with highly heterogeneous coefficients the methods are efficient and simple to construct using the abstract framework of the additive schwarz method and an idea of adaptive coarse spaces in one variant the coarse space consists of finite element functions associated with the wire basket nodes and functions based on solving some generalized eigenvalue problems on the faces in the other variant it contains functions associated with the vertex nodes with functions based on solving some generalized eigenvalue problems on subdomain faces and subdomain edges the functions that constitute the coarse spaces are chosen adaptively and they correspond to the eigenvalues that are smaller than a given threshold the convergence rate of the preconditioned conjugate gradients method in both cases is shown to be independent of the variations in the coefficients for the sufficient number of eigenfunctions in the coarse space numerical results are given to support the theory

multiscale methods for problems with complex geometry we propose a multiscale method for elliptic problems on complex domains e g domains with cracks or complicated boundary for local singularities this paper also offers a discrete alternative to enrichment techniques such as xfem we construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account the corrections only need to be computed in regions surrounding fine scale geometric features we achieve linear convergence rate in the energy norm for the multiscale solution moreover the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh the analytical findings are verified in a series of numerical experiments