A Gauss-Jacobi Kernel Compression Scheme for Fractional Differential Equations

A scheme for approximating the kernel $w$ of the fractional $\alpha$-integral by a linear combination of exponentials is proposed and studied. The scheme is based on the application of a composite Gauss-Jacobi quadrature rule to an integral representation of $w$. This results in an approximation of $w$ in an interval $[\delta,T]$, with $0<\delta$, which converges rapidly in the number $J$ of quadrature nodes associated with each interval of the composite rule. Using error analysis for Gauss-Jacobi quadratures for analytic functions, an estimate of the relative pointwise error is obtained. The estimate shows that the number of terms required for the approximation to satisfy a prescribed error tolerance is bounded for all $\alpha\in(0,1)$, and that $J$ is bounded for $\alpha\in(0,1)$, $T>0$, and $\delta\in(0,T)$

Error Estimates for Adaptive Spectral Decompositions

Adaptive spectral (AS) decompositions associated with a piecewise constant function, $u$, yield small subspaces where the characteristic functions comprising $u$ are well approximated. When combined with Newton-like optimization methods, AS decompositions have proved remarkably efficient in providing at each nonlinear iteration a low-dimensional search space for the solution of inverse medium problems. Here, we derive $L^2$-error estimates for the AS decomposition of $u$, truncated after $K$ terms, when $u$ is piecewise constant and consists of $K$ characteristic functions over Lipschitz domains and a background. Numerical examples illustrate the accuracy of the AS decomposition for media that either do, or do not, satisfy the assumptions of the theory

Adaptive spectral decompositions for inverse medium problems

Inverse medium problems involve the reconstruction of a spatially varying unknown medium from available observations by exploring a restricted search space of possible solutions. Standard grid-based representations are very general but all too often computationally prohibitive due to the high dimension of the search space. Adaptive spectral (AS) decompositions instead expand the unknown medium in a basis of eigenfunctions of a judicious elliptic operator, which depends itself on the medium. Here the AS decomposition is combined with a standard inexact Newton-type method for the solution of time-harmonic scattering problems governed by the Helmholtz equation. By repeatedly adapting both the eigenfunction basis and its dimension, the resulting adaptive spectral inversion (ASI) method substantially reduces the dimension of the search space during the nonlinear optimization. Rigorous estimates of the AS decomposition are proved for a general piecewise constant medium. Numerical results illustrate the accuracy and efficiency of the ASI method for time-harmonic inverse scattering problems, including a salt dome model from geophysics

Estimation embarquée des efforts latéraux et de la dérive d'un véhicule : validation expérimentale

National audienceLes principales préoccupations de la sécurité de conduite sont la compréhension et la prévention des situations critiques. Un examen attentif du nombre d'accidents révèle que la perte du contrôle du véhicule est l'une des causes principales des accidents routiers. L'amélioration de la stabilisation du véhicule est possible lorsque ses paramètres dynamiques sont connus. Certains paramètres fondamentaux de la dynamique, tels que les efforts de contact pneumatiques/chaussée, l'angle de dérive et l'adhérence, ne sont pas disponibles sur les véhicules de série ; par conséquence, ces variables doivent être estimées. L'observateur proposé dans cette étude est de type filtre de Kalman, il est basé sur la réponse dynamique d'un véhicule équipé par des capteurs standards. Cet article décrit le procédé d'estimation et présente des évaluations expérimentales. Les résultats expérimentaux acquis avec le véhicule du laboratoire INRETS-MA prouvent l'exactitude et le potentiel de cette approche

High-Order Accurate Local Schemes for Fractional Differential Equations

High-order methods inspired by the multi-step Adams methods are proposed for systems of fractional differential equations. The schemes are based on an expansion in a weighted space. To obtain the schemes this expansion is terminated after terms. We study the local truncation error and its behavior with respect to the step-size h and P. Building on this analysis, we develop an error indicator based on the Milne device. Methods with fixed and variable step-size are tested numerically on a number of problems, including problems with known solutions, and a fractional version on the Van der Pol equation

On Wave Splitting, Source Separation and Echo Removal with Absorbing Boundary Conditions

Starting from classical absorbing boundary con- ditions (ABC), we propose a method for the sep- aration of time-dependent wave fields given mea- surements of the total wave field. The method is local in space and time, deterministic, and makes no prior assumptions on the frequency spectrum and the location of sources or physical bound- aries. By using increasingly higher order ABC, the method can be made arbitrarily accurate and is, in that sense, exact. Numerical examples il- lustrate the usefulness for source separation and echo removal

Increasing 3D Matrix Rigidity Strengthens Proliferation and Spheroid Development of Human Liver Cells in a Constant Growth Factor Environment

International audienceMechanical forces influence the growth and shape of virtually all tissues and organs. Recent studies show that increased cell contractibility, growth and differentiation might be normalized by modulating cell tensions. Particularly, the role of these tensions applied by the extracellular matrix during liver fibrosis could influence the hepatocarcinogenesis process. The objective of this study is to determine if 3D stiffness could influence growth and phenotype of normal and transformed hepatocytes and to integrate extracellular matrix (ECM) stiffness to tensional homeostasis. We have developed an appropriate 3D culture model: hepatic cells within three-dimensional collagen matrices with varying rigidity. Our results demonstrate that the rigidity influenced the cell phenotype and induced spheroid clusters development whereas in soft matrices, Huh7 transformed cells were less proliferative, well-spread and flattened. We confirmed that ERK1 played a predominant role over ERK2 in cisplatin-induced death, whereas ERK2 mainly controlled proliferation. As compared to 2D culture, 3D cultures are associated with epithelial markers expression. Interestingly, proliferation of normal hepatocytes was also induced in rigid gels. Furthermore, biotransformation activities are increased in 3D gels, where CYP1A2 enzyme can be highly induced/activated in primary culture of human hepatocytes embedded in the matrix. In conclusion, we demonstrated that increasing 3D rigidity could promote proliferation and spheroid developments of liver cells demonstrating that 3D collagen gels are an attractive tool for studying rigidity-dependent homeostasis of the liver cells embedded in the matrix and should be privileged for both chronic toxicological and pharmacological drug screening

High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

High-order adaptive methods for fractional differential equations are proposed. The methods rely on a kernel reduction method for the approximation and localization of the history term. To avoid complications typical to multistep methods, we focus our study on 1-step methods and approximate the local part of the fractional integral by integral deferred correction to enable high order accuracy. We present numerical results obtained with both implicit and the explicit methods applied to different problems