In this contribution, an adaptive method for Model Order Reduction (MOR) is presented simulating physically non-linear material behavior of micro-heterogeneous structures. The purely online and includes a Galerkin projection, which performed using projection operator initially gained by Proper Orthogonal Decomposition (POD). By the number degrees freedom to solve decreased drastically can be pr...

We apply the reduced basis method to solve Navier-Stokes equations in parametrized domains. Special attention is devoted to the treatment of the parametrized non-linear transport term in the reduced basis framework, including the case of non-affine parametric dependence that is treated by an empirical interpolation method. This method features (i) a rapid global convergence owing to the propert...

In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second order method for hyp...

Locally mass conservative methods for subsurface ow are presented and applied to the unstable miscible displacement problem arising in porous media. The pressure equation is solved either by the discontinuous Galerkin method or the mixed nite element method. The concentration equation is solved using a higher order Godunov method. Theoretical estimates are given for the approximation of the pre...

In this paper we introduce a projection method for the space of probability distributions based on the differential geometric approach to statistics. This method is based on a direct L2 metric as opposed to the usual Hellinger distance and the related Fisher Information metric. We explain how this apparatus can be used for the nonlinear filtering problem, in relationship also to earlier project...

A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization / L2 projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields are constrained using node-by-node transforma...

A conservative transport scheme based on the discontinuous Galerkin (DG) method has been developed for the cubed sphere. Two different central projection methods, equidistant and equiangular, are employed for mapping between the inscribed cube and the sphere. These mappings divide the spherical surface into six identical subdomains, and the resulting grid is free from singularities. Two standar...

In this paper we propose a state estimation approach for linear parabolic Partial Differential Equations (PDE) with uncertain parameters. It is based on an extension of the Galerkin projection method. The extended method models projection coefficients, representing the state of the PDE in some basis, by means of a Differential-Algebraic Equation (DAE). The original estimation problem for the PD...

Reduced-order modeling approaches for gas flow in dual-porosity dual-permeability porous media are studied based on the proper orthogonal decomposition (POD) method combined with Galerkin projection. The typical modeling approach for non-porous-medium liquid flow problems is not appropriate for this compressible gas flow in a dual-continuum porous media. The reason is that non-zero mass transfe...

We propose in this paper a novel two-stage Projection Correction Modeling (PCM) framework for image reconstruction from (non-uniform) Fourier measurements. PCM consists of a projection stage (P-stage) motivated by the multi-scale Galerkin method and a correction stage (C-stage) with an edge guided regularity fusing together the advantages of total variation (TV) and total fractional variation (...