Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations

Abstract

<p style='text-indent:20px;'>We are interested in the numerical solution of coupled semilinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard space discretizations of the differential operators and illustrate how the resulting system of ordinary differential equations (ODEs) can be treated directly in matrix or tensor form. Moreover, in the framework of the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) we derive a two- and three-sided model order reduction strategy that is applied directly to the ODE system in matrix and tensor form respectively. We discuss how to integrate the reduced order model and, in particular, how to solve the tensor-valued linear system arising at each timestep of a semi-implicit time discretization scheme. We illustrate the efficiency of the proposed method through a comparison to existing techniques on classical benchmark problems such as the two- and three-dimensional Burgers equation.</p>

Adaptive POD-DEIM correction for Turing pattern approximation in reaction-diffusion PDE systems

“…To build the corrected form PODc (24) and POD-DEIMc (25) we choose R = 80 ≈ ρ sol . In Figure 8(c) we show the relative error E(u, r) for the unknown u in (19) for increasing values of the dimension of the reduced space r. Similar behaviour has been obtained for the variable v (not reported).…”

“…• the errors E(u, r) defined in (19), obtained for all the surrogate models proposed and for r ≤ r max where r max ≤ ρ sol ;…”

“…In the next simulations, the reduced system (12) will be solved always by the IMEX Euler method, but in vector form like in (6), because the matrix form and the Sylvester formulation in (8) are not still possible. We mention that recently a two-sided POD approach based on the matrix form has been proposed in [19,27].…”

“…For increasing values of r until r max = R = 45, we calculate the relative errors (19) for POD and POD-DEIM in the classical and corrected versions here introduced. The results are shown in Figure 4(c) for the variable u.…”

Discrete empirical interpolation for hyper‐reduction of hydro‐mechanical problems in groundwater flow through soil

“…The cost of assembling the operators scales with the dimension of the full‐order problem ($$N\rm _h$$), and by consequence, the efficiency gains that can be achieved with ROMs are limited. This drawback is often referred to as the lifting bottleneck, reflecting the fact that the evaluation of the nonlinear terms implied a lift back to the original dimension of the full‐order problem and then a projection to the reduced order 6 …”