Analysis of a class of nonlinear and non-separable multiscale representations

In this paper, we introduce a particular class of nonlinear and non-separable multiscale representations which embeds most of these representations. After motivating the introduction of such a class on one-dimensional examples, we investigate the multi-dimensional and non-separable case where the scaling factor is given by a non-diagonal dilation matrix M. We also propose new convergence and stability results in L ^p and Besov spaces for that class of nonlinear and non-separable multiscale representations. We end the paper with an application of the proposed study to the convergence and the stability of some nonlinear multiscale representations.     

Smoothness characterization and stability of nonlinear and non-separable multiscale representations

The aim of the paper is the construction and the analysis of nonlinear and non-separable multiscale representations for multivariate functions defined using a non-diagonal dilation matrix M. We show that a function in L^p or Besov spaces can be characterized by means of its multiscale representation. We also study the stability of these representations, a key issue to design adaptive algorithms.     

Denoising using nonlinear multiscale representations

The goal of this paper is to present some numerical results for the one-dimensional denoising problem by using the nonlinear multiscale representations. We introduce modified thresholding strategies in this new context which give significant significant improvements for one-dimensional denoising problems. To cite this article: B. Matei, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Stability of Manifold-Valued Subdivision Schemes and Multiscale Transformations

Linear subdivision schemes can be adapted in various ways so as to operate in nonlinear geometries such as Lie groups or Riemannian manifolds. It is well known that along with a linear subdivision scheme a multiscale transformation is defined. Such transformations can also be defined in a nonlinear setting. We show the stability of such nonlinear multiscale transforms. To do this we introduce a new kind of proximity condition which bounds the difference of the differential of a nonlinear subdivision scheme and a linear one. It turns out that—unlike the generic nonlinear case and modulo some minor technical assumptions—in the manifold-valued setting, convergence implies stability of the nonlinear subdivision scheme and associated nonlinear multiscale transformations.     

Multiscale methods for elliptic homogenization problems

In this article we study two families of multiscale methods for numerically solving elliptic homogenization problems. The recently developed multiscale finite element method [Hou and Wu, J Comp Phys 134 (1997), 169–189] captures the effect of microscales on macroscales through modification of finite element basis functions. Here we reformulate this method that captures the same effect through modification of bilinear forms in the finite element formulation. This new formulation is a general approach that can handle a large variety of differential problems and numerical methods. It can be easily extended to nonlinear problems and mixed finite element methods, for example. The latter extension is carried out in this article. The recently introduced heterogeneous multiscale method [Engquist and Engquist, Comm Math Sci 1 (2003), 87–132] is designed for efficient numerical solution of problems with multiscales and multiphysics. In the second part of this article, we study this method in mixed form (we call it the mixed heterogeneous multiscale method). We present a detailed analysis for stability and convergence of this new method. Estimates are obtained for the error between the homogenized and numerical multiscale solutions. Strategies for retrieving the microstructural information from the numerical solution are provided and analyzed. Relationship between the multiscale finite element and heterogeneous multiscale methods is discussed. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Representation of Fourier Integral Operators Using Shearlets

Traditional methods of time-frequency and multiscale analysis, such as wavelets and Gabor frames, have been successfully employed for representing most classes of pseudodifferential operators. However, these methods are not equally effective in dealing with Fourier Integral Operators in general. In this article, we show that the shearlets, recently introduced by the authors and their collaborators, provide very efficient representations for a large class of Fourier Integral Operators. The shearlets are an affine-like system of well-localized waveforms at various scales, locations and orientations, which are particularly efficient in representing anisotropic functions. Using this approach, we prove that the matrix representation of a Fourier Integral Operator with respect to a Parseval frame of shearlets is sparse and well-organized. This fact recovers a similar result recently obtained by Candès and Demanet using curvelets, which illustrates the benefits of directional multiscale representations (such as curvelets and shearlets) in the study of those functions and operators where traditional multiscale methods are unable to provide the appropriate geometric analysis in the phase space. The second author was supported in part by a National Science Foundation grant DMS 0604561.     

Approximation order from stability for nonlinear subdivision schemes

This paper proves approximation order properties of various nonlinear subdivision schemes. Building on some recent results on the stability of nonlinear multiscale transformations, we are able to give very short and concise proofs. In particular we point out an interesting connection between stability properties and approximation order for nonlinear subdivision schemes.     

Multiscale analysis and numerical simulation for stability of the incompressible flow of a Maxwell fluid

How to predict the stability of a small-scale flow subject to perturbations is a significant multiscale problem. It is difficult to directly study the stability by the theoretical analysis for the incompressible flow of a Maxwell fluid because of its analytical complexity. Here, we develop the multiscale analysis method based on the mathematical homogenization theory in the stress–stream function formulation. This method is used to derive the homogenized equation which governs the transport of the large-scale perturbations. The linear stabilities of the large-scale perturbations are analyzed theoretically based on the linearized homogenized equation, while the effect of the nonlinear terms on the linear stability results is discussed numerically based on the nonlinear homogenized equation. The agreements between the multiscale predictions and the direct numerical simulations demonstrate the multiscale analysis method is effective and credible to predict stabilities of flows.     

Generalized Cole–Hopf Transformation

On the basis of generalization of the Cole–Hopf transformation for parabolic equations with a source, we obtain some new representations of solutions and coefficients of nonlinear parabolic equations of mathematical physics which in fact are differential-algebraic identities. These representations can be used in studying the multidimensional direct and inverse problems.     

Negative derivatives and special functions

The formalism underlying the concept of negative derivatives is proved to be a powerful tool to study new series expansions of special functions defined by integral representations. Its importance is also discussed within the context of multiple integrations and of integral equations.     

A new stable collocation method for solving a class of nonlinear fractional delay differential equations

Numerical Algorithms - In this paper, a stable collocation method for solving the nonlinear fractional delay differential equations is proposed by constructing a new set of multiscale orthonormal...     

Operator estimates in nonlinear problems of reiterated homogenization

For a nonlinear elliptic equation of monotone type with multiscale coefficients, we obtain operator-type estimates for the difference between the solution and special approximations.     

Consistency,stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems

Summary. In an abstract framework we present a formalism which specifies the notions of consistency and stability of Petrov-Galerkin methods used to approximate nonlinear problems which are, in many practical situations, strongly nonlinear elliptic problems. This formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems. This theory is illustrated with the example: in a two dimensional domain with Dirichlet boundary conditions. Received June 10, 1992 / Revised version received February 28, 1994     

On a multiscale analysis of an inverse problem of nonlinear transfer law identification in periodic microstructure

This paper is devoted to the multiscale analysis of a homogenization inverse problem of the heat exchange law identification, which is governed by parabolic equations with nonlinear transmission conditions in a periodic heterogeneous medium. The aim of this work is to transform this inverse problem with nonlinear transmission conditions into a new one governed by a less complex nonlinear parabolic equation, while preserving the same form and physical properties of the heat exchange law that it will be identified, based on periodic homogenization theory. For this, we reformulate first the encountered homogenization inverse problem to an optimal control one. Then, we study the well-posedness of the state problem using the Leray–Schauder topological degrees and we also check the existence of the solution for the obtained optimal control problem. Finally, using the periodic homogenization theory and priori estimates, with justified choise of test functions, we reduce our inverse problem to a less complex one in a homogeneous medium.     

Multiscale recurrence analysis of long-term nonlinear and nonstationary time series

Recurrence analysis is an effective tool to characterize and quantify the dynamics of complex systems, e.g., laminar, divergent or nonlinear transition behaviors. However, recurrence computation is highly expensive as the size of time series increases. Few, if any, previous approaches have been capable of quantifying the recurrence properties from a long-term time series, while which is often collected in the real-time monitoring of complex systems. This paper presents a novel multiscale framework to explore recurrence dynamics in complex systems and resolve computational issues for a large-scale dataset. As opposed to the traditional single-scale recurrence analysis, we characterize and quantify recurrence dynamics in multiple wavelet scales, which captures not only nonlinear but also nonstationary behaviors in a long-term time series. The proposed multiscale recurrence approach was utilized to identify heart failure subjects from the 24-h time series of heart rate variability (HRV). It was shown to identify the conditions of congestive heart failure with an average sensitivity of 92.1% and specificity of 94.7%. The proposed multiscale recurrence framework can be potentially extended to other nonlinear dynamic methods that are computationally expensive for large-scale datasets.     

Stability of Nonlinear Subdivision and Multiscale Transforms

Extending upon the work of Cohen, Dyn, and Matei (Appl. Comput. Harmon. Anal. 15:89–116, 2003) and of Amat and Liandrat (Appl. Comput. Harmon. Anal. 18:198–206, 2005), we present a new general sufficient condition for the Lipschitz stability of nonlinear subdivision schemes and multiscale transforms in the univariate case. It covers the special cases (weighted essentially nonoscillatory scheme, piecewise polynomial harmonic transform) considered so far but also implies the stability in some new cases (median interpolating transform, power-p schemes, etc.). Although the investigation concentrates on multiscale transforms $\bigl\{v^0,d^1,\ldots,d^J\bigr\}\longmapsto v^J,\quad J\ge1,$ in ? _∞(?) given by a stationary recursion of the form $v^{j}=Sv^{j-1}+d^{j},\quad j\ge1,$ involving a nonlinear subdivision operator S acting on ? _∞(?), the approach is extendable to other nonlinear multiscale transforms and norms, as well.     

Reduction of computational errors using nonstandard finite difference models

A previously reported bifurcation technique is applied to the construction of nonstandard finite difference representations of systems of nonlinear differential equations. This technique provides a measure of the deviation between bifurcation parameters obtained from fixed step representations of the nonlinear system and the values of the parameters determined from computational experiments. Since this deviation or ‘error’ is characteristic of a particular scheme, we have used this measure to construct low-error nonstandard representations. We present results from several nonlinear test models which show that such nonstandard schemes yield orbits that followed closely the expected dynamics and also provide a large reduction in the computational error in comparison to standard numerical integration schemes. Finally, we outline a criteria for controlling possible numerical overflow in fixed step-size schemes.     

Multiscale analysis for ergodic schrödinger operators and positivity of Lyapunov exponents

A variant of multiscale analysis for ergodic Schrödinger operators is developed. This enables us to prove positivity of Lyapunov exponents, given initial scale estimates and an initial Wegner estimate. This postivivity is then applied to high-dimensional skew-shifts at small coupling, where initial conditions are checked using the Pastur-Figotin formalism.     

Development and application of multiscale models of acute viral infections in intervention research

The objective of this study is to advance two frontiers in multiscale modelling of acute viral infections, which are (a) the mathematical technology or technical frontier, where we present a new method for development of multiscale models of acute viral infections using influenza A virus (IAV) as a paradigm in which a new set of metrics to measure both individual level and community level infectiousness are introduced, and (b) the scientific applications frontier, where we demonstrate the implementation of multiscale modelling in evaluating the comparative effectiveness of IAV health interventions from efficacy data. The multiscale model is developed by integrating the within-host scale and the between-host scale. Using the example of IAV as a paradigm, we demonstrate the utility and process by which multiscale modelling can be used to evaluate the comparative effectiveness of health interventions that operate at different scale domains. The multiscale modelling is general enough to be applicable to other acute viral infections.     

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.