Model-based Reconstruction of Distributed Phenomena using Discretized Representations of Partial Differential Equations

Parameter identification and reconstruction for distributed phenomena based on hybrid density filter

“…It is important to note that the individual shape functions Ψ i (z) are defined in the entire solution domain Ω and the form strongly depends upon the used numerical method, e.g., orthogonal polynomials for finite-spectral method [4]. Due to the fact that the approximated solutionp(z, t) cannot satisfy the partial differential equation (4) everywhere in the region of interest, a residual R Ω remains.…”

“…For brevity, the input vector u * (t) and the boundary condition vector b * (t) can be combined to a so-called augmented input vector u(t). The interested reader should refer to [4] and [9] for more information on how to derive the system of ordinary differential equations (6).…”

“…To circumvent the restriction on the time step Δt, it is reasonable to integrate the set of ordinary differential equations by means of implicit methods, such as the Crank-Nicolson discretization as it was shown in [4]. These methods lead to a system model which is unconditionally stable for any time step Δt.…”

“…In addition, the challenge for the observation of distributed phenomena is that measurements are available only at discrete time steps and discrete locations, meaning that no information between the individual sensor nodes are available. In order to get meaningful and accurate information not only at the sensor nodes itself but also between these nodes, the model-based reconstruction of the distributed phenomenon is of major significance [4]. By exploiting additional background information of the phenomenon in form of a physical model, the accuracy of the reconstruction can be improved significantly, even at non-measurement points.…”

“…That means, the physical model characterizing the distributed phenomena has to be converted from a distributed-parameter form to a lumped-parameter form. This conversion can be achieved by methods for solving partial differential equations, such as finite-difference method, finite-element method [7], modal analysis [8] and finite-spectral method [4], [9].…”