In this paper we present a new reduced basis technique for parametrized nonlinear scalar conservation laws in presence of shocks. The essential ingredients are an efficient algorithm to approximate the shock curve, a procedure to detect the smooth components of the solution at the two sides of the shock, and a suitable interpolation strategy to reconstruct such smooth components during the online stage. The approach we propose is based on some theoretical properties of the solution to the problem. Some numerical examples prove the effectiveness of the proposed strategy.

We propose a regression model for data spatially distributed over general two-dimensional Riemannian manifolds. This is a generalized additive model with a roughness penalty term involving a differential operator computed over the non-planar domain. By virtue of a semiparametric framework, the model allows inclusion of space-varying covariate information. Estimation can be performed by conformally parameterizing the non-planar domain and then generalizing existing models for penalized spatial regression over planar domains. The conformal coordinates and the estimation problem are both computed with a finite element approach.

We present a theoretical analysis of the CORSING ( COmpRessed SolvING ) method for the numerical approximation of partial differential equations based on compressed sensing. In particular, we show that the best $s$ -term approximation of the weak solution of a PDE with respect to a system of $N$ trial functions, can be recovered via a Petrov-Galerkin approach using $m \ll N$ test functions. This recovery is guaranteed if the local $a$ -coherence associated with the bilinear form and the selected trial and test bases fulfills suitable decay properties. The fundamental tool of this analysis is the restricted inf-sup property, i.e., a combination of the classical inf-sup condition and the well-known restricted isometry property of compressed sensing.

Surrogate solutions and surrogate models for complex problems in many fields of science and engineering represent an important recent research line towards the construction of the best trade-off between modeling reliability and computational efficiency. Among surrogate models, hierarchical model (HiMod) reduction provides an effective approach for phenomena characterized by a dominant direction in their dynamics. HiMod approach obtains 1D models naturally enhanced by the inclusion of the effect of the transverse dynamics. HiMod reduction couples a finite element approximation along the mainstream with a locally tunable modal representation of the transverse dynamics. In particular, we focus on the pointwise HiMod reduction strategy, where the modal tuning is performed on each finite element node. We formalize the pointwise HiMod approach in an unsteady setting, by resorting to a model discontinuous in time, continuous and hierarchically reduced in space (c[M( $$\mathbf{M}$$ )G(s)]-dG(q) approximation). The selection of the modal distribution and of the space–time discretization is automatically performed via an adaptive procedure based on an a posteriori analysis of the global error. The final outcome of this procedure is a table, named HiMod lookup diagram, that sets the time partition and, for each time interval, the corresponding 1D finite element mesh together with the associated modal distribution. The results of the numerical verification confirm the robustness of the proposed adaptive procedure in terms of accuracy, sensitivity with respect to the goal quantity and the boundary conditions, and the computational saving. Finally, the validation results in the groundwater experimental setting are promising. The extension of the HiMod reduction to an unsteady framework represents a crucial step with a view to practical engineering applications. Moreover, the results of the validation phase confirm that HiMod approximation is a viable approach.