Unique continuation problems and stabilised finite element methods

Numerical analysis for partial differential equations (PDEs) traditionally considers problems that are well-posed in the continuum, for example the boundary value problem for Poisson's equation. Computational methods such as the finite element method (FEM) then discretise the problem and provide numerical solutions. However, when a part of the boundary is inaccessible for measurements or no information is given on the boundary at all, the continuum problem might be ill-posed and solving it, in this case, requires regularisation. In this thesis we consider the unique continuation problem with (possibly noisy) data given in an interior subset of the domain. This is an ill-posed problem also known as data assimilation and is related to the elliptic Cauchy problem. It arises often in inverse problems and control theory. We will focus on two PDEs for which the stability of this problem depends on the physical parameters: the Helmholtz and the convection--diffusion equations. We first prove conditional stability estimates that are explicit in the wave number and in the Péclet number, respectively, by using Carleman inequalities. Under a geometric convexity assumption, we obtain that for the Helmholtz equation the stability constants grow at most linearly in the wave number. Then we present a discretise-then-regularise approach for the unique continuation problem. We cast the problem into PDE-constrained optimisation with discrete weakly consistent regularisation. The regularisation is driven by stabilised FEMs and we focus on the interior penalty stabilisation. For the Helmholtz and diffusion-dominated problems, we apply the continuum stability estimates to the approximation error and prove convergence rates by controlling the residual through stabilisation. For convection-dominated problems, we perform a different error analysis and obtain sharper weighted error estimates along the characteristics of the convective field through the data region, with quasi-optimal convergence rates. The results are illustrated by numerical examples

A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime

The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on a form suitable for use in numerical analysis and with explicit dependence on the P\'eclet number. A stabilized finite element method is then proposed and analysed. An upper bound on the condition number is first derived. Combining the stability estimates on the continuous problem with the numerical stability of the method, we then obtain error estimates in local $H^1$- or $L^2$-norms that are optimal with respect to the approximation order, the problem's stability and perturbations in data. The convergence order is the same for both norms, but the $H^1$-estimate requires an additional divergence assumption for the convective field. The theory is illustrated in some computational examples.Comment: 21 pages, 6 figures; in v2 we added two remarks and an appendix on psiDOs, and made some minor correction

Unique continuation for the Helmholtz equation using stabilized finite element methods

In this work we consider the computational approximation of a unique continuation problem for the Helmholtz equation using a stabilized finite element method. First conditional stability estimates are derived for which, under a convexity assumption on the geometry, the constants grow at most linearly in the wave number. Then these estimates are used to obtain error bounds for the finite element method that are explicit with respect to the wave number. Some numerical illustrations are given.Comment: corrected typos; included suggestions from reviewer

A stabilized finite element method for inverse problems subject to the convection-diffusion equation. II: convection-dominated regime

We consider the numerical approximation of the ill-posed data assimilation problem for stationary convection-diffusion equations and extend our previous analysis in [Numer. Math. 144, 451--477, 2020] to the convection-dominated regime. Slightly adjusting the stabilized finite element method proposed for dominant diffusion, we draw upon a local error analysis to obtain quasi-optimal convergence along the characteristics of the convective field through the data set. The weight function multiplying the discrete solution is taken to be Lipschitz and a corresponding super approximation result (discrete commutator property) is proven. The effect of data perturbations is included in the analysis and we conclude the paper with some numerical experiments.Comment: 25 pages, 16 figure

Recognition and Combinatorial Optimization Algorithms for Bipartite Chain Graphs

In this paper we give a recognition algorithm in O(n(n+m)) time for bipartite chain graphs, and directly calculate the density of such graphs. For their stability number and domination number, we give algorithms comparable to the existing ones. We point out some applications of bipartite chain graphs in chemistry and approach the Minimum Chain Completion problem

Optimal finite element approximation of unique continuation

We consider finite element approximations of ill-posed elliptic problems with conditional stability. The notion of {\emph{optimal error estimates}} is defined including both convergence with respect to mesh parameter and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the finite element approximation space. A proof is given that no finite element approximation can converge at a better rate than that given by the definition, justifying the concept. A recently introduced class of finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be quasi optimal in the sense of our definition

On Polar, Trivially Perfect Graphs

During the last decades, different types of decompositions have been processed in the field of graph theory. In various problems, for example in the construction of recognition algorithms, frequently appears the so-called weakly decomposition of graphs.Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. Recognizing a polar graph is known to be NP-complete. For this class of graphs, polynomial algorithms for the maximum stable set problem are unknown and algorithms for the dominating set problem are also NP-complete.In this paper we characterize the polar graphs using the weakly decomposition, give a polynomial time algorithm for recognizing graphs that are both trivially perfect and polar, and directly calculate the domination number. For the stability number and clique number, we give polynomial time algorithms.

Changes of carbon-isotope ratios in soil organic matter relative to parent vegetation and site specificity

Investigating the correlation between biodiversity and ecosystem function in natural environments using carbon-isotope composition (δ13C) allows distinguishing the nutrient cycling pattern and anthropogenic effects incorporation in plants and soil processes. The mechanisms behind the isotopic composition of soil organic matter (SOM) and parent vegetation in relation to the context of site-specificity was approached in this work. Formation of SOM can be affected by the presence of a high concentration of heavy metals in soils. Still, no systematic studies were performed in most of the industrial sites to support this hypothesis.  In order to explore this incomplete understood influence, investigation of carbon isotope signatures (d13C) variations in soil organic matter were performed in two industrial areas from Romania (Copșa Mică industrial platform and Baia Sprie mining zone). The current study, also, investigated the C:N ratio variation, as well as the influence of N speciation regarding d13C values of SOM. The decrease in C:N ratio indicated an increasing effect of the microbial products on SOM matter at increasing depth, for both regions, while an increase of the denitrification processes with depth was found for both areas. For the most appropriate depth (20-40 cm), the soil from Baia Sprie region was more enriched in 13C comparing with the soil from Copsa Mica region, and this higher isotope fractionation of SOM might be due to a higher carbon content, respectively a higher nitrogen content of Baia Sprie soil. It was concluded that the SOM of the surface soil in the two investigated regions has an 13C isotopic composition similar to the plant remains from which it was formed, offering an integrated value of plant material, time and the local origin and providing useful markers of tree isotopic composition

Development of the traceability system in the viticultural sector of Romania for improving food safety

Traceability in a food sector can be obtained by developing traceability systems. These systems refer to recording all important data referring to the evolution of a product along the production and supply chain. The benefits derived from the application of this model are great for those involved in the wine production chain (producer, processor and distributor), and for the consumers’ health. Thus, consumers are certain of the safety of the wines they buy, and their right for free choice is ensured by the transparency of the production and marketing chain. Wine producers and processors are more efficient from the viewpoint of products logistics. They might receive fewer complaints from consumers and apply the socalled due diligence defense – the proof they fulfilled their tasks, and they could finally use traceability in marketing to differentiate the products. The benefits for a better protection of public health and traceability help preventing frauds, when the authenticity of certain wines cannot be traced by analyses