On the regularizing power of multigrid-type algorithms

We consider the deblurring problem of noisy and blurred images in the case of known space invariant point spread functions with four choices of boundary conditions. We combine an algebraic multigrid previously defined ad hoc for structured matrices related to space invariant operators (Toeplitz, circulants, trigonometric matrix algebras, etc.) and the classical geometric multigrid studied in the partial differential equations context. The resulting technique is parameterized in order to have more degrees of freedom: a simple choice of the parameters allows us to devise a quite powerful regularizing method. It defines an iterative regularizing method where the smoother itself has to be an iterative regularizing method (e.g., conjugate gradient, Landweber, conjugate gradient for normal equations, etc.). More precisely, with respect to the smoother, the regularization properties are improved and the total complexity is lower. Furthermore, in several cases, when it is directly applied to the system $A{\bf f}={\bf g}$, the quality of the restored image is comparable with that of all the best known techniques for the normal equations $A^TA{\bf f}=A^T{\bf g}$, but the related convergence is substantially faster. Finally, the associated curves of the relative errors versus the iteration numbers are ``flatter'' with respect to the smoother (the estimation of the stop iteration is less crucial). Therefore, we can choose multigrid procedures which are much more efficient than classical techniques without losing accuracy in the restored image (as often occurs when using preconditioning). Several numerical experiments show the effectiveness of our proposals

Analysis of Multigrid Preconditioning for Implicit PDE Solvers for Degenerate Parabolic Equations

Abstract. In this paper an implicit numerical method designed for nonlinear degenerate parabolic equations is proposed. A convergence analysis and the study of the related computa-tional cost are provided. In fact, due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required. The chosen scheme is the Newton method and its con-vergence is proven under mild assumptions. Every step of the Newton method implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate multigrid preconditioned Krylov methods. Numerical experiments for the validation of our analysis complement this contribution

Can eccentric exercise of the lower limb be made more efficiently, a pilot study

Abstract Background: Eccentric Exercise has been shown to be more effective in building muscle and healing damaged tissue than concentric or isometric exercise. It has also been shown to be effective in increasing motor control. But the duration of therapeutic exercise in physical therapy is limited by insurance to 30-60 minutes a day. Objectives: Four standard therapy eccentric exercises of the lower limbs were compared (toe raise, ball exercise, side lying eccentric exercise and incline board exercise) to a trainer called the BTE Eccentron to see if the efficiency of exercise could be increased using one exercise session to meet or beat the four individual exercises. Subjects and Methods: The study examined eight randomly selected participants with no known medical conditions (neurological or orthopedic) that would preclude their participation (age=24.1+/-2.1 years height=168.9+/-6.4 cm BMI=23.2+/-3.2). EMG was used to assess muscle recruitment in each exercise. The muscles studies were the gastrocnemius, hamstring, hip adductors, and quadriceps muscles. Results: Muscle use on the eccentron was almost double that of the other exercises. Thus, making therapy more efficient. One single exercise bout showed more muscle activation during eccentric exercise than the other four exercises, with an average muscle use almost 4 times higher on the eccentron. Conclusion: The Eccentron offers a considerable advantage for clinical treatment making exercise and neuromuscular training more efficient