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The quality of the Dniester River Basin has been seriously impacted by the chemicals released by agriculture, industry, and wastewater discharges. To assess its current chemical pollution status, a transboundary monitoring campaign was conducted in May 2019. Thirteen surface water, 13 sediment, and three biota samples were collected and analyzed using generic sample preparation methods for the determination of organic substances by liquid chromatography high-resolution mass spectrometry (LC-HRMS) and metals by inductively coupled plasma mass spectrometry (ICP-MS). Wide-scope target and suspect screening resulted in detection of Water Framework Directive (WFD) priority substances and emerging contaminants, whereas the raw data were stored in NORMAN Digital Sample Freezing Platform (DSFP) for future retrospective screening. Furthermore, risk assessment was performed to prioritize detected substances and propose a draft list of river basin–specific pollutants. All studied metals (As, Hg, Zn, Cu, Cr, Cd, Pb, Ni) were detected in the surface water and sediments. In total, 139 organic contaminants belonging to various chemical classes (pesticides, pharmaceuticals, drugs of abuse, stimulants, sweeteners, industrial chemicals, and their transformation products) were detected. The highest cumulative concentration of contaminants was observed in surface water from the Byk River, a tributary of the Dniester (Moldova). Concentrations of WFD priority substances diuron and mercury and EU Watch List neonicotinoid compounds imidacloprid and thiamethoxam exceeded their environmental quality standards (EQS), whereas concentrations of 23 emerging substances exceeded their predicted no-effect concentration (PNEC) at minimum one site. Emerging contaminants telmisartan, metolachlor, terbuthylazine, and 4-acetamidoantipyrine were prioritized as potential river basin–specific pollutants.

Extending upon Daubechies et al. (Constr. Approx. 20:399–463, 2004) and Runborg (Multiscale Methods in Science and Engineering, pp. 205–224, 2005), we provide the theoretical analysis of normal multi-scale transforms for curves with general linear predictor S, and a more flexible choice of normal directions. The main parameters influencing the asymptotic properties (convergence, decay estimates for detail coefficients, smoothness of normal re-parametrization) of this transform are the smoothness of the curve, the smoothness of S, and its order of exact polynomial reproduction. Our results give another indication why approximating S may not be the first choice in compression applications of normal multi-scale transforms.

Extending upon the work of Cohen, Dyn, and Matei (Appl. Comput. Harmon. Anal. 15:89–116, 2003) and of Amat and Liandrat (Appl. Comput. Harmon. Anal. 18:198–206, 2005), we present a new general sufficient condition for the Lipschitz stability of nonlinear subdivision schemes and multiscale transforms in the univariate case. It covers the special cases (weighted essentially nonoscillatory scheme, piecewise polynomial harmonic transform) considered so far but also implies the stability in some new cases (median interpolating transform, power-p schemes, etc.). Although the investigation concentrates on multiscale transforms $$\bigl\{v^0,d^1,\ldots,d^J\bigr\}\longmapsto v^J,\quad J\ge1,$$ in l ∞(ℤ) given by a stationary recursion of the form $$v^{j}=Sv^{j-1}+d^{j},\quad j\ge1,$$ involving a nonlinear subdivision operator S acting on l ∞(ℤ), the approach is extendable to other nonlinear multiscale transforms and norms, as well.

We prove the optimality of hierarchical and BPX-type preconditioners for finite element discretizations with nonconforming P1 finite elements. Such preconditioners were proposed about 15 years ago, and until now only suboptimal estimates of their preconditioning properties have been available. The main new tool is an improved Bernstein type inequality for an associated subdivision process generated by the prolongations which allows us to give an asymptotically optimal upper bound for the spectrum of the preconditioned systems.

Calibration problems connected with the determination of C20-C44 petroleum hydrocarbons by gas chromatography – flame ionization detection (GC-FID) and IR spectrometric methods were studied in this paper. It has been found that the mass responses and mass response factors of FID determined by capillary GC-FID as well as by IR spectrometry, at wave number 2959 and 2924 cm-1, of high boiling petroleum distillation fractions (in the range C20-C46) are not constant. Moreover, it was found that mass response factors for differentpetroleum oils that have the same boiling temperature interval might differ up to 1.5 fold both for FID and IRD methods, dependingon their hydrocarbon group composition. However, calibration of the IR spectrometric method using the empirical equations proposed in DIN 38 409, Part 18, norm (for the determination of total petroleumhydrocarbons in waters polluted with high boiled petroleum fractions)gave results which overlapped within ±10% of the expected values.

We give a refined analysis of the Holder regularity for the limit functions arising from a nonlinear pyramid algorithm for robust removal of non-Gaussian noise proposed by Donoho and Yu [6,7,17]. The synthesis part of this algorithm can be interpreted as a nonlinear triadic subdivision scheme where new points are inserted based on local quadratic polynomial median interpolation and imputation. We introduce the analogon of the Donoho–Yu scheme for dyadic refinement, and show that its limit functions are in C α for α>log 4(128/31)=1.0229... . In the triadic case, we improve the lower bound of α>log 2(135/121)=0.0997... previously obtained in [6] to α>log 3(135/53)=0.8510... . These lower bounds are relatively close to the anticipated upper bounds of log 2(16/7)=1.1982... in the dyadic, respectivly 1 in the triadic cases, and have been obtained by deriving recursive inequalities for the l∞ norm of second rather than first order differences of the sequences arising in the subdivision process.

The article extends upon previous work by Temlyakov, Konyagin, and Wojtaszczyk on comparing the error of certain greedy algorithms with that of best m-term approximation with respect to a general biorthogonal system in a Banach space X. We consider both necessary and sufficient conditions which cover most of the special cases previously considered. Some new results concerning the Haar system in L1, L∞, and BMO are also included.

The potential of sparse grid discretizations for solving boundary integral equations is studied for the screen problem on a square in $\mathbb{R}^3$ . Theoretical and numerical results on approximation rates, preconditioning, adaptivity and compression for piecewise constant and linear sparse grid spaces are obtained.

We give some estimates related to finite element multilevel splittings and Sobolev norms of negative order. Basically, results for the positive order case are carried over by duality. In particular, semi-orthogonal splittings based on piecewise constants are studied for Sobolev spaces of order −1/2. Numerical experiments are provided for the screen problem.

We describe tensor product type techniques to derive robust solvers for anisotropic elliptic model problems on rectangular domains in ℝ d . Our analysis is based on the theory of additive subspace correction methods and applies to finite element and prewavelet schemes. We present multilevel- and prewavelet-based methods that are robust for anisotropic diffusion operators with additional Helmholtz term. Furthermore, the resulting convergence rates are independent of the discretization level. Beside their theoretical foundation, we also report on the results of various numerical experiments to compare the different methods.

In recent years, it has been shown that many modern iterative algorithms (multigrid schemes, multilevel preconditioners, domain decomposition methods etc.) for solving problems resulting from the discretization of PDEs can be interpreted as additive (Jacobi-like) or multiplicative (Gauss-Seidel-like) subspace correction methods. The key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space \(V\) into a sum of subspaces \(V_j\) and the splitting of the variational problem on \(V\) into auxiliary problems on these subspaces. In this paper, we propose a modification of the abstract convergence theory of the additive and multiplicative Schwarz methods, that makes the relation to traditional iteration methods more explicit. The analysis of the additive and multiplicative Schwarz iterations can be carried out in almost the same spirit as in the traditional block-matrix situation, making convergence proofs of multilevel and domain decomposition methods clearer, or, at least, more classical. In addition, we present a new bound for the convergence rate of the appropriately scaled multiplicative Schwarz method directly in terms of the condition number of the corresponding additive Schwarz operator. These results may be viewed as an appendix to the recent surveys [X], [Ys].

LetA be any real symmetric positive definiten×n matrix, and κ(A) its spectral condition number. It is shown that the optimal convergence rate $$\rho _{SOR}^* = \mathop {\min }\limits_{0< \omega< 2} \rho (M_{SOR,\omega } )$$ of the successive overrelaxation (SOR) method satisfies $$\rho _{SOR}^* \leqslant 1 - \frac{1}{{\alpha _n \kappa (A)}}, \alpha _n \approx \log n.$$ This worst case estimate is asymptotically sharp asn→∞. The corresponding examples are given by certain Toeplitz matrices.

Based on the framework of subspace splitting and the additive Schwarz scheme, we give bounds for the condition number of multilevel preconditioners for sparse grid discretizations of elliptic model problems. For a BXP-like preconditioner we derive an estimate of the optimal orderO(1) and for a HB-like variant we obtain an estimate of the orderO(k 2 ·2 k/2 ), wherek denotes the number of levels employed. Furthermore, we confirm these results by numerically computed condition numbers.

We derive an optimal multilevel preconditioner for nonconforming P1 element discretizations of second order elliptic boundary value problems.

We develop a hierarchical basis multilevel method for symmetric second order elliptic boundary value problems in two-dimensional polygonal domains based on nonconforming P1 triangular elements. The main result is that the condition number of the hierarchical discretization is bounded byO(k) wherek is the number of refinement levels. For comparison, Yserentant's result yields the sharp boundO(k 2) for the corresponding hierarchical discretization with conforming linear elements.