Multiscale Basis Research Articles

Fast homogenization algorithm based on asymptotic theory and multiscale schemes

A time-frequency interpretation of the classical asymptotic theory of homogenization for elliptic PDE with periodic coefficients is presented and the relations with known multilevel/multiscale numerical schemes are investigated. We formulate a new fast iterative algorithm for the approximation of homogenized solutions based on the combination of these two apparently different approaches. The asymptotic homogenization process is interpreted as a migration to infinity of the frequencies related to microscale contributions and the discovering of those related to the homogenized solution. At different scale/frequency of the periodic coefficients of the operator, band-pass filters select only the contributions of the homogenized solution which is then composed as the limit of an iterative procedure. This novel method can be interpreted in case of finite difference discretizations as a generalized nonstationary subdivision scheme and its convergence and stability are discussed. In particular, stable compositions of the homogenized solution are investigated in relation with the contracting behavior of specific operators generated by reduction processes and Schur's complements of suitable matrices produced by discretizations via wavelets and multiscale bases.

Multiscale modelling and computation of fluid flow

AbstractMany problems of fundamental and practical importance have multiscale solutions. Direct numerical simulation of these multiscale problems is difficult due to the range of length scales in the underlying physical problems. Here, we describe two multiscale methods for computing nonlinear partial differential equations with multiscale solutions. The first method relies on constructing local multiscale bases for diffusion‐dominated problems. We demonstrate that such an approach can be used to upscale two‐phase flow in heterogeneous porous media. The second method is to construct semi‐analytic multiscale solutions local in space and time. We use these solutions to approximate the large‐scale solution for convection‐dominated transport. This approach overcomes the common difficulty due to the memory effect in deriving the averaged equations for convection‐dominated transport. Our multiscale analysis provides a useful guideline for designing effective numerical methods for incompressible flow. Copyright © 2005 John Wiley & Sons, Ltd.

Sparse grids

We present a survey of the fundamentals and the applications of sparse grids, with a focus on the solution of partial differential equations (PDEs). The sparse grid approach, introduced in Zenger (1991), is based on a higher-dimensional multiscale basis, which is derived from a one-dimensional multi-scale basis by a tensor product construction. Discretizations on sparse grids involve . That is why sparse grids are especially well-suited for problems of very high dimensionality.The sparse grid approach can be extended to nonsmooth solutions by adaptive refinement methods. Furthermore, it can be generalized from piecewise linear to higher-order polynomials. Also, more sophisticated basis functions like interpolets, prewavelets, or wavelets can be used in a straightforward way.We describe the basic features of sparse grids and report the results of various numerical experiments for the solution of elliptic PDEs as well as for other selected problems such as numerical quadrature and data mining.

An adaptive multiscale basis method for modelling discrete-time non-linear dynamical systems

Based on locally applicable autoregressive moving average with exogenous (ARMAX) models this paper proposes two new adaptive modelling schemes for discrete-time non-linear dynamic systems based on the concept of a multiscale which originates in approximation theory. A general result on the convergence of modelling based on the multiscale basis functions with a least square estimated is derived. With the advantage of the multiresolution nature of the multiscale basis, the new modelling scheme can be used to capture both the global and local characteristics of a non-linear system over distinct scales. The algorithms also provide a trade-off between the model structure/model size and the modelling error, one of which emphasizes the modelling error with variable model structure whilst the other stresses a desired model size with better modelling error. Data-based modelling examples are used to demonstrate the effectiveness of the multiscale modelling schemes with different multiscale basis.

Electronic Structure Calculations Using Self-Adaptive Multiscale Voronoi Basis Functions

The multiscale self-adaptivity of Voronoi basis functions is currently proving to be useful for the simulation of complex fluid systems involving structures on a number of distinct lengthscales. In this paper, we explore the possibility of extending the use of such multiscale basis functions to the framework of density functional theoretic electronic structure computations.

Multilevel Stabilization of Convection-Diffusion Problems by Variable-Order Inner Products

We are concerned with the task of stabilizing discrete approximations to convection–diffusion problems. We propose to consistently modify the exact variational formulation of the problem by adding a fractional order inner product, involving the residual of the equation. The inner product is expressed through a multilevel decomposition of its arguments, in terms of components along a multiscale basis. The order of the inner product locally varies from −1/2 to −1, depending on the value of a suitably-defined multiscale Peclet number. Numerical approximations obtained via the Galerkin method applied to the modified formulation are analyzed.

Adaptive multiscale moment method (AMMM) for analysis of scattering from perfectly conducting plates

Adaptive multiscale moment method (AMMM) is presented for the analysis of scattering from a thin perfectly conducting plate. This algorithm employs the conventional moment method and a special matrix transformation, which is derived from the tensor products of the two one-dimensional (1-D) multiscale triangular basis functions that are used for expansion and testing functions in the conventional moment method. The special feature of these new basis functions introduced through this transformation is that they are orthogonal at the same scale except at the initial scale and not between scales. From one scale to another scale, the initial estimate for the solution can be predicted using this multiscale technique. Hence, the compression is applied directly to the solution and the size of the linear equations to be solved is reduced, thereby improving the efficiency of the conventional moment method. The basic difference between this methodology and the other techniques that have been presented so far is that we apply the compression not to the impedance matrix, but to the solution itself directly using an iterative solution methodology. The extrapolated results at the higher scale thus provide a good initial guess for the iterative method. Typically, when the number of unknowns exceeds a few thousand unknowns, the matrix solution time exceeds generally the matrix fill time. Hence, the goal of this method is directed in solving electrically larger problems, where the matrix solution time is of concern. Two numerical results are presented, which demonstrate that the AMMM is a useful method to analyze scattering from perfectly conducting plates.

Adaptive linear finite-element method for modelling nonlinear dynamic systems

This paper proposes a method of identifying nonlinear dynamic models with observation data (or a training data set) which exhibits a simple structure, adaptive input-space partition and fast convergence. The method employs the multiscale approximation concepts which have been introduced in numerical analysis motivated by wavelet analysis concepts. The partition or equivalently scaled basis functions are determined and selected adaptively in a sequenced and ordered manner. This method may be also considered as a single-layer neural network but with adaptive neural neurons. The number of multiscale basis functions required depends on the degree of nonlinearity of the system being modelled. The method is compared with the cerebellar model with interpolation (CEINT) and the cerebellar model articulation control (CMAC) methods and has been shown to achieve comparative modelling accuracies but with a reduced memory space and a concomitantly reduced training set.

Reproducing kernel hierarchical partition of unity, Part I—formulation and theory

This work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet functions are introduced. Based upon the built-in consistency conditions, the differential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with different combinations of the hierarchical kernels, a synchronized convergence effect may be observed. Being different from the conventional Legendre function based p-type hierarchical basis, the new hierarchical basis is an intrinsic pseudo-spectral basis, which can remain as a partition of unity in a local region, because the discrete wavelet kernels form a ‘partition of nullity’. These newly developed kernels can be used as the multi-scale basis to solve partial differential equations in numerical computation as a p-type refinement. Copyright © 1999 John Wiley & Sons, Ltd.

Reproducing kernel hierarchical partition of unity, Part I?formulation and theory

This work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet functions are introduced. Based upon the built-in consistency conditions, the differential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with different combinations of the hierarchical kernels, a synchronized convergence effect may be observed. Being different from the conventional Legendre function based p-type hierarchical basis, the new hierarchical basis is an intrinsic pseudo-spectral basis, which can remain as a partition of unity in a local region, because the discrete wavelet kernels form a ‘partition of nullity’. These newly developed kernels can be used as the multi-scale basis to solve partial differential equations in numerical computation as a p-type refinement. Copyright © 1999 John Wiley & Sons, Ltd.

Adaptive Multiscale Moment Method for Solving Two-Dimensional Fredholm Integral Equation of the First Kind - Abstract

The aim of this paper is to deal with the problem of numerical solution of two-dimensional Fredholm integral equation of the first kind. First, two-dimensional multiscale basis is introduced following the tensor product of the one-dimensional multiscale triangular basis. We also discuss the geometrical meaning of the coefficients of the multiscale basis and on the compression technique for representing the original two dimensional function of the multiscale basis. Second, by use of this kind of basis, the multiscale moment method for solving two-dimensional Fredholm integral equation of the first kind has been proposed. Furthermore, the adaptive algorithm of the multiscale moment method (AMMM) has been presented according to the characteristics of solution coefficients of the integral equation. From (V -1) -th scale to V-th scale, the approximate solution on the finer grid is predicted from the known solution on the coarse grid by use of a tensor B-spline interpolant product. Many of the numerical simulations are carried out to test the feasibility of the multiscale moment method and on how to implement the adaptive algorithm. It will be found that the adaptive algorithm can reduce the size of linear equations constructed from multiscale moment method for two-dimensional Fredholm inte

Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences

We present a multilevel approach for the solution of partial differential equations. It is based on a multiscale basis which is constructed from a one-dimensional multiscale basis by the tensor product approach. Together with the use of hash tables as data structure, this allows in a simple way for adaptive refinement and is, due to the tensor product approach, well suited for higher dimensional problems. Also, the adaptive treatment of partial differential equations, the discretization (involving finite differences) and the solution (here by preconditioned BiCG) can be programmed easily. We describe the basic features of the method, discuss the discretization, the solution and the refinement procedures and report on the results of different numerical experiments. — Author's Abstract

A-Posteriori Estimates and Adaptive Schemes for Transmission Problems

The results presented here are directed to Galerkin schemes with respect to stable multiscale bases discretizations for boundary integral equations which describe transmission problems. We derive a posteriori estimates which are reliable and efficient with respect to any desirable tolerance. Moreover, the convergence of an adaptive scheme is investigated. The underlying ideas are applicable to a wide class of elliptic problems, cf. [14]. Here further details concerning decay estimates and appropriate index-sets for a system of boundary integral equations are presented.

A Multiscale Moment Method for Solving Fredholm Integral Equation of the First Kind - Summary

The aim of this paper is to deal with the problem of numerical solution of Fredholm integral equation of the first kind. First, a new kind of basis functions has been proposed based on multiscale method and wavelet-like basis. Second, by using this basis, the multiscale moment method for solving Fredholm integral equation of the first kind in one dimension has been proposed. Furthermore, the adaptive algorithm of the multiscale moment method has been presented according to the characteristics of the integral equation. Many numerical simulations are carried out to test the feasibility of the multiscale method and the implemented adaptive algorithm. It is found that the two multiscaling algorithms can provide good results. The adaptive algorithm can reduce the orders of linear equations constructed by multiscale moment method for Fredholm integral equation. It is a very effective algorithm to solve integral equations.

Separability-based multiscale basis selection and feature extraction for signal and image classification

Algorithms for multiscale basis selection and feature extraction for pattern classification problems are presented. The basis selection algorithm is based on class separability measures rather than energy or entropy. At each level the "accumulated" tree-structured class separabilities obtained from the tree which includes a parent node and the one which includes its children are compared. The decomposition of the node (or subband) is performed (creating the children), if it provides larger combined separability. The suggested feature extraction algorithm focuses on dimensionality reduction of a multiscale feature space subject to maximum preservation of information useful for classification. At each level of decomposition, an optimal linear transform that preserves class separabilities and results in a reduced dimensional feature space is obtained. Classification and feature extraction is then performed at each scale and resulting "soft decisions" obtained for each area are integrated across scales. The suggested algorithms have been tested for classification and segmentation of one-dimensional (1-D) radar signals and two-dimensional (2-D) texture and document images. The same idea can be used for other tree structured local basis, e.g., local trigonometric basis functions, and even for nonorthogonal, redundant and composite basis dictionaries.

Tomographic reconstruction and estimation based on multiscale natural-pixel bases

We use a natural pixel-type representation of an object, originally developed for incomplete data tomography problems, to construct nearly orthonormal multiscale basis functions. The nearly orthonormal behavior of the multiscale basis functions results in a system matrix, relating the input (the object coefficients) and the output (the projection data), which is extremely sparse. In addition, the coarsest scale elements of this matrix capture any ill conditioning in the system matrix arising from the geometry of the imaging system. We exploit this feature to partition the system matrix by scales and obtain a reconstruction procedure that requires inversion of only a well-conditioned and sparse matrix. This enables us to formulate a tomographic reconstruction technique from incomplete data wherein the object is reconstructed at multiple scales or resolutions. In case of noisy projection data we extend our multiscale reconstruction technique to explicitly account for noise by calculating maximum a posteriori probability (MAP) multiscale reconstruction estimates based on a certain self-similar prior on the multiscale object coefficients. The framework for multiscale reconstruction presented can find application in regularization of imaging problems where the projection data are incomplete, irregular, and noisy, and in object feature recognition directly from projection data.

Stable multiscale bases and local error estimation for elliptic problems

This paper is concerned with the analysis of adaptive multiscale techniques for the solution of a wide class of elliptic operator equations covering, in principle, singular integral as well as partial differential operators. The central objective is to derive reliable and efficient a-posteriori error estimators for Galerkin schemes which are based on stable multiscale bases. It is shown that the locality of corresponding multiresolution processes combined with certain norm equivalences involving weighted sequence norms of wavelet coefficients leads to adaptive space refinement strategies which are guaranteed to converge in a wide range of cases, again including operators of negative order.

Local Decomposition of Refinable Spaces and Wavelets

A convenient setting for studyingmultiscale techniquesin various areas of applications is usually based on a sequence of nested closed subspaces of some function space script F which is often referred to asmultiresolution analysis.The concept ofwaveletsis a prominent example where the practical use of such a multiresolution analysis relies on explicit representations of theorthogonaldifference between any two subsequent spaces. However, many applications prohibit the employment of a multiresolution analysis based on translation invariant spaces on all of open face Rs, say. It is then usually difficult to compute orthogonal complements explicitly. Moreover, certain applications suggest using other types of complements, in particular, those corresponding tobiorthogonal wavelets.The main objective of this paper is therefore to study possibly nonorthogonal but in a certain sensestableand evenlocaldecompositions of nested spaces and to develop tools which are not necessarily confined to the translation invariant setting. A convenient way ofparametrizingsuch decompositions is to reformulate them in terms of matrix relations. This allows one to characterizeallstable or local decompositions by identifying unique matrix transformations that carry one given decomposition into another one. It will be indicated how such a mechanism may help realizing several desirable features of multiscale decompositions and constructing stable multiscale bases with favorable properties. In particular, we apply these results to the identification of decompositions induced by local linear projectors. The importance of this particular application with regard to the construction of multiscale Riesz bases will be pointed out. Furthermore, we indicate possible specializations to orthogonal decompositions of spline spaces relative to nonuniform knot sequences, piecewise linear finite elements and principal shift invariant spaces. The common ground of all these examples as well as other situations of practical and theoretical interest is thatsome initialmultiscale basis is available. The techniques developed here can then be used to generate from such an initial decomposition other ones with desirable properties pertaining to moment conditions or stability properties.

A Multiresolution Method for Distributed Parameter Estimation

A multiresolution method for distributed parameter estimation (or inverse problems) is studied numerically. The identification of the coefficient of an elliptic equation in one dimension is considered as our model problem. First, multiscale bases are used to analyze the degree of ill-posedness of the inverse problem. Second, based on some numerical results, it is shown that the method of scale-by-scale multiresolution yields robust and fast convergence. Finally, it is shown how the method gives a natural regularization approach which is complementary to Tikhonov’s regularization.

Orthonormal wavelet representations for recognizing complex annotations

This paper describes a novel method of pattern recognition targeted for recognizing complex annotations found in paper documents. Our investigation is motivated by the high reliability required for accomplishing autonomous interpretation of maps and engineering drawings. The recognition problem is made difficult in part because characters and text may be expressed in arbitrary fonts and orientations. Our approach includes a novel incremental strategy based on the multiscale representation of wavelet decompositions. Our approach is motivated by biological mechanisms of the human visual system. Choosing wavelets that are simultaneously localized in both space and frequency, and decomposing a signal into a multiscale hierarchical basis with orientation selectivity, can provide a powerful methodology for pattern analysis. We evaluated several wavelets with different spatial-frequency characteristics and measured their performance in the context of character recognition. Wavelet bases are more attractive than traditional hierarchical bases because they are orthonormal, linear, continuous, and continuously invertible. The multiscale representation of wavelet transforms provides a mathematically coherent basis for multigrid techniques. In contrast to previous adhoc approaches, our method promises a practical solution embedded in a unified mathematical theory. A feasibility study is described in which more than 10000 patterns were recognized with an error rate of 2.6% by a neural network trained using multiscale representations from a class of 52 distinct alphanumeric patterns and graphical symbols. We observed a 10-fold reduction in the amount of information needed to represent each character for recognition. These results suggest that high reliability is possible at a reduced cost of representation.