A parabolic inverse convection–diffusion–reaction problem solved using space–time localization and adaptivity

A linear algorithm for the minimal realization problem in physical coordinates with a non-invertible output matrix

“…and the inputs u(k) and the outputs y(k) remain unchanged, as is well known from system's theory (and can be easily verified). This transformation has many other invariants, see Appendix A, and especially the eigenvalues of the matrix A, as can be directly seen from (7). Now, to be more precise, let us call {A s , B s , C s } the matrices of the minimal realization found by subspace methods, and T f the (unknown) basis-change matrix that transforms this data-driven minimal realization to the minimal realization in the physical base (4).…”

“…We assume to know the true (physical) C f matrix, because it can be easily built from the definition of the state-vector in the physical-mathematical model of the real system and the definition of the system outputs y. However, unfortunately, the matrix T f ∈ R n x ×n x cannot be computed by simply using (7), since the linear system with multiple right-hand-side…”

“…There is a huge literature and interest in physical (grey-box) modelling, in applied sciences, in engineering applications, often involving complex mathematical problems like e.g. computational inverse problems [7] [8]. A well known approach for computing physical parameter estimates is to adopt nonlinear estimation procedures [9]; these, anyway, suffer from convergence problems, depending much from the initialization of the estimates [10] and from the illconditioning of the problem to be numerically solved.…”

Reconstructing unknown nonlinear boundary conditions in a time‐fractional inverse reaction–diffusion–convection problem

“…In the integer‐order case, Equation is used to model various problems. For instance, this equation is used to predict water quality in rivers, by measuring the quantity of organic matter contained . In the context of the heat conduction process, the nonlinear boundary conditions and show the case where the heat fluxes across the boundaries are functions of the temperature.…”