SUMMARYWe present a Parametrized-Background Data-Weak (PBDW) formulation of the variational data assimilation (state estimation) problem for systems modeled by partial differential equations. The main contributions are a constrained optimization weak framework informed by the notion of experimentally observable spaces; a priori and a posteriori error estimates for the field and associated linear-functional outputs; Weak Greedy construction of prior (background) spaces associated with an underlying potentially high-dimensional parametric manifold; stability-informed choice of observation functionals and related sensor locations; and finally, output prediction from the optimality saddle in O(M 3 ) operations, where M is the number of experimental observations. We present results for a synthetic Helmholtz acoustics model problem to illustrate the elements of the methodology and confirm the numerical properties suggested by the theory. To conclude, we consider a physical raised-box acoustic resonator chamber: we integrate the PBDW methodology and a Robotic Observation Platform to achieve real-time in situ state estimation of the time-harmonic pressure field; we demonstrate the considerable improvement in prediction provided by the integration of a best-knowledge model and experimental observations; we extract even from these results with real data the numerical trends indicated by the theoretical convergence and stability analyses.

We present a space-time certified reduced basis method for long-time integration of parametrized parabolic equations with quadratic nonlinearity which admit an affine decomposition in parameter but with no restriction on coercivity of the linearized operator. We first consider a finite element discretization based on discontinuous Galerkin time integration and introduce associated Petrov-Galerkin space-time trial-and test-space norms that yield optimal and asymptotically mesh independent stability constants. We then employ an hp Petrov-Galerkin (or minimum residual) space-time reduced basis approximation. We provide the Brezzi-Rappaz-Raviart a posteriori error bounds which admit efficient offline-online computational procedures for the three key ingredients: the dual norm of the residual, an inf-sup lower bound, and the Sobolev embedding constant. The latter are based, respectively, on a more round-off resistant residual norm evaluation procedure, a variant of the successive constraint method, and a time-marching implementation of a fixed-point iteration of the embedding constant for the discontinuous Galerkin norm. Finally, we apply the method to a natural convection problem governed by the Boussinesq equations. The result indicates that the space-time formulation enables rapid and certified characterization of moderate-Grashof-number flows exhibiting steady periodic responses. However, the space-time reduced basis convergence is slow, and the Brezzi-Rappaz-Raviart threshold condition is rather restrictive, such that offline effort will be acceptable only for very few parameters.

Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx)We present a space-time certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval (0, T ] parametrized with respect to the Peclet number. We first introduce a Petrov-Galerkin space-time finite element discretization, which enjoys a favorable inf-sup constant that decreases slowly with Peclet number and final time T . We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi-Rappaz-Raviart a posteriori error bounds. We detail computational procedures that permit offline-online decomposition for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf-sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L 2 -error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T , unlike the exponentially growing estimate of the classical formulation for high Peclet number cases.

The Generalized Empirical Interpolation Method (GEIM) is an extension first presented in [1] of the classical empirical interpolation method (see [2], [3], [4]) where the evaluation at interpolating points is replaced by the evaluation at interpolating continuous linear functionals on a class of Banach spaces. As outlined in [1], this allows to relax the continuity constraint in the target functions and expand the application domain. A special effort has been made in this paper to understand the concept of stability condition of the generalized interpolant (the Lebesgue constant) by relating it in the first part of the paper to an inf-sup problem in the case of Hilbert spaces. In the second part, it will be explained how GEIM can be employed to monitor in real time physical experiments by combining the acquisition of measurements from the processes with their mathematical models (parameter-dependent PDE's). This idea will be illustrated through a parameter dependent Stokes problem in which it will be shown that the pressure and velocity fields can efficiently be reconstructed with a relatively low dimension of the interpolating spaces.

We examined the effects of habitual exercise plus nutritional intervention through consumption of milk fat globule membrane (MFGM), a milk component, on aging-related deficits in muscle mass and function in senescence-accelerated P1 mice. Combining wheel-running and MFGM (MFGMEx) intake significantly attenuated age-related declines in quadriceps muscle mass (control: 318 ± 6 mg; MFGMEx: 356 ± 9 mg; P < 0.05) and in contractile force (1.4-fold and 1.5-fold higher in the soleus and extensor digitorum longus muscles, respectively). Microarray analysis of genes in the quadriceps muscle revealed that MFGMEx stimulated neuromuscular development; this was supported by significantly increased docking protein-7 (Dok-7) and myogenin mRNA expression. Treatment of differentiating myoblasts with MFGM-derived phospholipid or sphingolipid fractions plus mechanical stretching also significantly increased Dok-7 mRNA expression. These findings suggest that habitual exercise plus dietary MFGM improves muscle function deficits through neuromuscular development, and that phospholipid and sphingolipid in MFGM contribute to its physiological actions.Electronic supplementary materialThe online version of this article (doi:10.1186/2193-1801-3-339) contains supplementary material, which is available to authorized users.