Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains

In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. We consider commonly used model problems including the Laplace equation, the elasticity equation, and the Stokes system in perforated regions. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the sizes of the perforations, and configurations. Typical modeling approaches extract average properties in each coarse region, that encapsulate many perforations, and formulate a coarse-grid problem. In some applications, the coarse-grid problem can have a different form from the fine-scale problem, e.g. the coarse-grid system corresponding to a Stokes system in perforated domains leads to Darcy equations on a coarse grid. In this paper, we present a general offline/online procedure, which can adequately and adaptively represent the local degrees of freedom and derive appropriate coarse-grid equations. Our approaches start with the offline procedure, which constructs multiscale basis functions in each coarse region and formulates coarse-grid equations. We presented the offline simulations without the analysis and adaptive procedures, which are needed for accurate and efficient simulations. The main contributions of this paper are (1) the rigorous analysis of the offline approach, (2) the development of the online procedures and their analysis, and (3) the development of adaptive strategies. We present an online procedure, which allows adaptively incorporating global information and is important for a fast convergence when combined with the adaptivity. We present online adaptive enrichment algorithms for the three model problems mentioned above. Our methodology allows adding and guides constructing new online multiscale basis functions adaptively in appropriate regions. We present the convergence analysis of the online adaptive enrichment algorithm for the Stokes system. In particular, we show that the online procedure has a rapid convergence with a rate related to the number of offline basis functions, and one can obtain fast convergence by a sufficient number of offline basis functions, which are computed in the offline stage. The convergence theory can also be applied to the Laplace equation and the elasticity equation. To illustrate the performance of our method, we present numerical results with both small and large perforations. We see that only a few (1 or 2) online iterations can significantly improve the offline solution.     

A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients

We propose a multiscale multilevel Monte Carlo(MsMLMC) method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage,we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallestscale of the solution. We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients. Moreover, we provide convergence analysis of the proposed method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.     

Reduced basis methods with adaptive snapshot computations

We use asymptotically optimal adaptive numerical methods (here specifically a wavelet scheme) for snapshot computations within the offline phase of the Reduced Basis Method (RBM). The resulting discretizations for each snapshot (i.e., parameter-dependent) do not permit the standard RB ‘truth space’, but allow for error estimation of the RB approximation with respect to the exact solution of the considered parameterized partial differential equation. The residual-based a posteriori error estimators are computed by an adaptive dual wavelet expansion, which allows us to compute a surrogate of the dual norm of the residual. The resulting adaptive RBM is analyzed. We show the convergence of the resulting adaptive greedy method. Numerical experiments for stationary and instationary problems underline the potential of this approach.     

Stable enrichment and local preconditioning in the particle-partition of unity method

This paper is concerned with the stability and approximation properties of enriched meshfree and generalized finite element methods. In particular we focus on the particle-partition of unity method (PPUM) yet the presented results hold for any partition of unity based enrichment scheme. The goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h-version independent of the regularity of the solution. Hence, we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme. To this end we enrich our PPUM function space in an enrichment zone hierarchically near the singularities of the solution. This initial enrichment however can lead to a severe ill-conditioning and can compromise the stability of the discretization. To overcome the ill-conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable and well-conditioned basis independent of the employed initial enrichment. The construction of this preconditioner is of linear complexity with respect to the number of discretization points. We obtain optimal error bounds for an enriched PPUM discretization with local preconditioning that are independent of the regularity of the solution globally and within the employed enrichment zone we observe a kind of super-convergence. The results of our numerical experiments clearly show that our enriched PPUM with local preconditioning recovers the optimal convergence rate of O(h ^p ) of the uniform h-version globally. For the considered model problems from linear elastic fracture mechanics we obtain an improved convergence rate of O(h ^p+δ ) with d 3 \frac12{\delta\geq\frac{1}{2}} for p = 1. The convergence rate of our multilevel solver is essentially the same for a purely polynomial approximation and an enriched approximation.     

A Dirichlet–Neumann reduced basis method for homogeneous domain decomposition problems

Reduced basis methods allow efficient model reduction of parametrized partial differential equations. In the current paper, we consider a reduced basis method based on an iterative Dirichlet–Neumann coupling for homogeneous domain decomposition of elliptic PDEʼs. We gain very small basis sizes by an efficient treatment of problems with a-priori known geometry. Moreover iterative schemes may offer advantages over other approaches in the context of parallelization. We prove convergence of the iterative reduced scheme, derive rigorous a-posteriori error bounds and provide a full offline/online decomposition. Different methods for basis generation are investigated, in particular a variant of the POD-Greedy procedure. Experiments confirm the rigor of the error estimators and identify beneficial basis construction procedures.     

A new adaptive trust region algorithm for optimization problems

It is well known that trust region methods are very effective for optimization problems. In this article, a new adaptive trust region method is presented for solving unconstrained optimization problems. The proposed method combines a modified secant equation with the BFGS updated formula and an adaptive trust region radius, where the new trust region radius makes use of not only the function information but also the gradient information. Under suitable conditions, global convergence is proved, and we demonstrate the local superlinear convergence of the proposed method. The numerical results indicate that the proposed method is very efficient.     

A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems

Inspired by the reduced basis approach and modern numerical multiscale methods, we present a new framework for an efficient treatment of heterogeneous multiscale problems. The new approach is based on the idea of considering heterogeneous multiscale problems as parametrized partial differential equations where the parameters are smooth functions. We then construct, in an offline phase, a suitable localized reduced basis that is used in an online phase to efficiently compute approximations of the multiscale problem by means of a discontinuous Galerkin method on a coarse grid. We present our approach for elliptic multiscale problems and discuss an a posteriori error estimate that can be used in the construction process of the localized reduced basis. Numerical experiments are given to demonstrate the efficiency of the new approach.     

Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems

We employ the reduced basis method as a surrogate model for the solution of optimal control problems governed by parametrized partial differential equations (PDEs) and develop rigorous a posteriori error bounds for the error in the optimal control and the associated error in the cost functional. The proposed bounds can be efficiently evaluated in an offline–online computational procedure. We present numerical results that confirm the validity of our approach.     

Optimally Adjusted Mixture Sampling and Locally Weighted Histogram Analysis

Consider the two problems of simulating observations and estimating expectations and normalizing constants for multiple distributions. First, we present a self-adjusted mixture sampling method, which accommodates both adaptive serial tempering and a generalized Wang–Landau algorithm. The set of distributions are combined into a labeled mixture, with the mixture weights depending on the initial estimates of log normalizing constants (or free energies). Then, observations are generated by Markov transitions, and free energy estimates are adjusted online by stochastic approximation. We propose two stochastic approximation schemes by Rao–Blackwellization of the scheme commonly used, and derive the optimal choice of a gain matrix, resulting in the minimum asymptotic variance for free energy estimation, in a simple and feasible form. Second, we develop an offline method, locally weighted histogram analysis, for estimating free energies and expectations, using all the simulated data from multiple distributions by either self-adjusted mixture sampling or other sampling algorithms. This method can be computationally much faster, with little sacrifice of statistical efficiency, than a global method currently used, especially when a large number of distributions are involved. We provide both theoretical results and numerical studies to demonstrate the advantages of the proposed methods.     

Multiscale finite element for problems with highly oscillatory coefficients

Summary. In this paper, we study a multiscale finite element method for solving a class of elliptic problems with finite number of well separated scales. The method is designed to efficiently capture the large scale behavior of the solution without resolving all small scale features. This is accomplished by constructing the multiscale finite element base functions that are adaptive to the local property of the differential operator. The construction of the base functions is fully decoupled from element to element; thus the method is perfectly parallel and is naturally adapted to massively parallel computers. We present the convergence analysis of the method along with the results of our numerical experiments. Some generalizations of the multiscale finite element method are also discussed. Received April 17, 1998 / Revised version received March 25, 2000 / Published online June 7, 2001     

Adaptive mesh point selection for the efficient solution of scalar IVPs

We discuss adaptive mesh point selection for the solution of scalar initial value problems. We consider a method that is optimal in the sense of the speed of convergence, and we aim at minimizing the local errors. Although the speed of convergence cannot be improved by using the adaptive mesh points compared to the equidistant points, we show that the factor in the error expression can be significantly reduced. We obtain formulas specifying the gain achieved in terms of the number of discretization subintervals, as well as in terms of the prescribed level of the local error. Both nonconstructive and constructive versions of the adaptive mesh selection are shown, and a numerical example is given.     

Multistep approximation algorithms: Improved convergence rates through postconditioning with smoothing kernels

We show how certain widely used multistep approximation algorithms can be interpreted as instances of an approximate Newton method. It was shown in an earlier paper by the second author that the convergence rates of approximate Newton methods (in the context of the numerical solution of PDEs) suffer from a “loss of derivatives”, and that the subsequent linear rate of convergence can be improved to be superlinear using an adaptation of Nash–Moser iteration for numerical analysis purposes; the essence of the adaptation being a splitting of the inversion and the smoothing into two separate steps. We show how these ideas apply to scattered data approximation as well as the numerical solution of partial differential equations. We investigate the use of several radial kernels for the smoothing operation. In our numerical examples we use radial basis functions also in the inversion step. This revised version was published online in June 2006 with corrections to the Cover Date.     

Approximation of weak adjoints by reverse automatic differentiation of BDF methods

We shed light on the relation between the discrete adjoints of multistep backward differentiation formula (BDF) methods and the solution of the adjoint differential equation. To this end, we develop a functional-analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations. We devise a Petrov-Galerkin finite element (FE) interpretation of the BDF method and its discrete adjoint scheme obtained by reverse internal numerical differentiation. We show how the FE approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation. We also show convergence of the discrete adjoints to the classical adjoints on the inner time interval. Finally, we give numerical results for non-adaptive and fully adaptive BDF schemes. The presented framework opens the way to carry over techniques on global error estimation from FE methods to BDF methods.     

An efficient adaptive analysis procedure for node-based smoothed point interpolation method (NS-PIM)

This paper presents an efficient adaptive analysis procedure being able to operate in the framework of the node-based smoothed point interpolation method (NS-PIM). The NS-PIM uses three-node triangular cells and is very easy to be implemented, which make it an ideal candidate for adaptive analysis. In the present adaptive procedure, a new error indicator is devised for NS-PIM settings; two ways are proposed to calculate the local critical value; a simple h-type local refinement scheme is adopted and Delaunay technology is used for regenerating optimal new mesh. A number of typical numerical examples involving stress concentration and solution singularities have been tested. The results demonstrate that the present procedure achieves much higher convergence rate results compared to the uniform refinement, and can obtain upper bound solution in strain energy.     

An outer approximate subdifferential method for piecewise affine optimization

Piecewise affine functions arise from Lagrangian duals of integer programming problems, and optimizing them provides good bounds for use in a branch and bound method. Methods such as the subgradient method and bundle methods assume only one subgradient is available at each point, but in many situations there is more information available. We present a new method for optimizing such functions, which is related to steepest descent, but uses an outer approximation to the subdifferential to avoid some of the numerical problems with the steepest descent approach. We provide convergence results for a class of outer approximations, and then develop a practical algorithm using such an approximation for the compact dual to the linear programming relaxation of the uncapacitated facility location problem. We make a numerical comparison of our outer approximation method with the projection method of Conn and Cornuéjols, and the bundle method of Schramm and Zowe. Received September 10, 1998 / Revised version received August 1999?Published online December 15, 1999     

DISCONTINUOUS GALERKIN METHODS AND THEIR ADAPTIVITY FOR THE TEMPERED FRACTIONAL (CONVECTION) DIFFUSION EQUATIONS

This paper focuses on the adaptive discontinuous Galerkin (DG) methods for the tempered fractional (convection) diffusion equations. The DG schemes with interior penalty for the diffusion term and numerical flux for the convection term are used to solve the equations, and the detailed stability and convergence analyses are provided. Based on the derived posteriori error estimates, the local error indicator is designed. The theoretical results and the effectiveness of the adaptive DG methods are, respectively, verified and displayed by the extensive numerical experiments. The strategy of designing adaptive schemes presented in this paper works for the general PDEs with fractional operators.     

An Adaptive Local (AL) Basis for Elliptic Problems with Complicated Discontinuous Coefficients

We develop a generalized finite element method for the discretization of elliptic partial differential equations in heterogeneous media. In [5] a semidiscrete method has been introduced to set up an adaptive local finite element basis (AL basis) on a coarse mesh with mesh size H which, typically, does not resolve the matrix of the media while the textbook finite element convergence rates are preserved. This method requires O(log(1/H)^d+1) basis functions per mesh point where d denotes the spatial dimension of the computational domain. We present a fully discrete version of this method, where the AL basis is constructed by solving finite-dimensional localized problems, and which preserves the optimal convergence rates. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

DISCRETE GALERKIN METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

In this article, we develop a fully Discrete Galerkin(DG) method for solving initial value fractional integro-differential equations(FIDEs). We consider Generalized Jacobi polynomials(GJPs) with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution. Numerical results are presented to demonstrate the effectiveness of the proposed method.     

Solving the Dym initial value problem in reproducing kernel space

We consider two numerical solution approaches for the Dym initial value problem using the reproducing kernel Hilbert space method. For each solution approach, the solution is represented in the form of a series contained in the reproducing kernel space, and a truncated approximate solution is obtained. This approximation converges to the exact solution of the Dym problem when a sufficient number of terms are included. In the first approach, we avoid to perform the Gram-Schmidt orthogonalization process on the basis functions, and this will decrease the computational time. Meanwhile, in the second approach, working with orthonormal basis elements gives some numerical advantages, despite the increased computational time. The latter approach also permits a more straightforward convergence analysis. Therefore, there are benefits to both approaches. After developing the reproducing kernel Hilbert space method for the numerical solution of the Dym equation, we present several numerical experiments in order to show that the method is efficient and can provide accurate approximations to the Dym initial value problem for sufficiently regular initial data after relatively few iterations. We present the absolute error of the results when exact solutions are known and residual errors for other cases. The results suggest that numerically solving the Dym initial value problem in reproducing kernel space is a useful approach for obtaining accurate solutions in an efficient manner.     

A Regularized Newton-Like Method for Nonlinear PDE

An adaptive regularization strategy for stabilizing Newton-like iterations on a coarse mesh is developed in the context of adaptive finite element methods for nonlinear PDE. Existence, uniqueness and approximation properties are known for finite element solutions of quasilinear problems assuming the initial mesh is fine enough. Here, an adaptive method is started on a coarse mesh where the finite element discretization and quadrature error produce a sequence of approximate problems with indefinite and ill-conditioned Jacobians. The methods of Tikhonov regularization and pseudo-transient continuation are related and used to define a regularized iteration using a positive semidefinite penalty term. The regularization matrix is adapted with the mesh refinements and its scaling is adapted with the iterations to find an approximate sequence of coarse-mesh solutions leading to an efficient approximation of the PDE solution. Local q-linear convergence is shown for the error and the residual in the asymptotic regime and numerical examples of a model problem illustrate distinct phases of the solution process and support the convergence theory.