Abstract Established reduced‐order modeling (ROM) methods, for instance, Galerkin‐projection, approximate the solution by linearly projecting high‐dimensional spaces to a lower‐dimensional space spanned reduced basis. However, accuracy of these methods may be insufficient complex and multiscale simulations due restriction linear space. Alternatively, autoencoders (AEs) can used nonlinear dimens...

The main objective of the following document is to present an approximate numerical solution Korteweg de Vries (KdV) equation, since it has a wide field applications, such as acoustics, molecular biology, optics, and quantum mechanics. For this, basic concepts associated with method that will be used obtain presented initially, in our case, finite element particular Galerkin method. A further u...

In this paper a systematic modeling and control approach for flow problems is considered. A nonlinear Galerkin model is obtained from the partial differential equations (PDEs) describing the flow; and a Linear Parameter Varying (LPV) model is constructed to approximate the Galerkin model, where the parameter variation of the LPV model is controller by an adaptation mechanism. The LPV model is t...

Complementarity problems and variational inequalities arise in a wide variety of areas, including machine learning, planning, game theory, and physical simulation. In all of these areas, to handle large-scale problem instances, we need fast approximate solution methods. One promising idea is Galerkin approximation, in which we search for the best answer within the span of a given set of basis f...

A steady-state and transient finite element model has been developed to approximate, with simple triangular elements, the two-dimensional advection—diffusion equation for practical river surface flow simulations. Essentially, the space—time Crank—Nicolson—Galerkin formulation scheme was used to solve for a given conservative flow-field. Several kinds of point sources and boundary conditions, na...

For the Fredholm integral equation u=T u+f on the real line, fast solvers are designed on the basis of a discretized wavelet Galerkin method with the Sloan improvement of the Galerkin solution. The Galerkin system is solved by GMRES or by the Gauss elimination method. Our concept of the fast solver includes the requirements that the parameters of the approximate solution un can be determined in...

This work presents a unified analysis of Discontinuous Galerkin methods to approximate Friedrichs’ systems. A general set of boundary conditions is identified to guarantee existence and uniqueness of solutions to these systems. A formulation enforcing the boundary conditions weakly is proposed. This formulation is the starting point for the construction of Discontinuous Galerkin methods formula...

We present a novel microscopic tridomain model describing the electrical activity in cardiac tissue with dynamical gap junctions. The system consists of three PDEs modeling conduction intra-and extra-cellular domains, supplemented by nonlinear ODE for dynamics ion channels and establish global existence uniqueness weak solutions to our model. solution, which constitutes main result this paper, ...

We apply adaptive wavelet methods to boundary value problems with random coefficients, discretized by wavelets or frames in the spatial domain and tensorized polynomials in the parameter domain. Greedy algorithms control the approximate application of the fully discretized random operator, and the construction of sparse approximations to this operator. We suggest a power iteration for estimatin...

In this paper, we study the properties of approximate solutions to a doubly nonlinear and degenerate diffusion equation, known in the literature as the diffusive wave approximation of the shallow water equations (DSW), using a numerical approach based on the Galerkin finite element method. This equation arises in shallow water flow models when special assumptions are used to simplify the shallo...