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Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this shift towards modern nonlinear regularization methods, including their analysis, applications and issues for future research. In particular we will discuss variational methods and techniques derived from them, since they have attracted much recent interest and link to other fields, such as image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions and learning theory.

The continuous increase of nitrous oxide (N ₂O) abundance in the atmosphere is a global concern. Multiple pathways of N ₂O production occur in soil, but their significance and dependence on oxygen (O ₂) availability and nitrogen (N) fertilizer source are poorly understood. We examined N ₂O and nitric oxide (NO) production under 21%, 3%, 1%, 0.5%, and 0% (vol/vol) O ₂ concentrations following urea or ammonium sulfate [(NH ₄) ₂SO ₄] additions in loam, clay loam, and sandy loam soils that also contained ample nitrate. The contribution of the ammonia (NH ₃) oxidation pathways (nitrifier nitrification, nitrifier denitrification, and nitrification-coupled denitrification) and heterotrophic denitrification (HD) to N ₂O production was determined in 36-h incubations in microcosms by ¹⁵N- ¹⁸O isotope and NH ₃ oxidation inhibition (by 0.01% acetylene) methods. Nitrous oxide and NO production via NH ₃ oxidation pathways increased as O ₂ concentrations decreased from 21% to 0.5%. At low (0.5% and 3%) O ₂ concentrations, nitrifier denitrification contributed between 34% and 66%, and HD between 34% and 50% of total N ₂O production. Heterotrophic denitrification was responsible for all N ₂O production at 0% O ₂. Nitrifier denitrification was the main source of N ₂O production from ammonical fertilizer under low O ₂ concentrations with urea producing more N ₂O than (NH ₄) ₂SO ₄ additions. These findings challenge established thought attributing N ₂O emissions from soils with high water content to HD due to presumably low O ₂ availability. Our results imply that management practices that increase soil aeration, e.g., reducing compaction and enhancing soil structure, together with careful selection of fertilizer sources and/or nitrification inhibitors, could decrease N ₂O production in agricultural soils.

In order to discover the advantages and disadvantages of different fertilization regimes and identify the best management practice of fertilization in greenhouse fields, soil enzyme activities involved in carbon (C) transformations, soil chemical characteristics, and crop yields were monitored after long-term (20-year) fertilization regimes, including no fertilizer (CK), 300 kg N ha-1 and 600 kg N ha-1 as urea (N1 and N2), 75 Mg ha-1 horse manure compost (M), and M with either 300 or 600 kg N ha-1 urea (MN1 and MN2). Compared with CK, fertilization increased crop yields by 31% (N2) to 69% (MN1). However, compared with CK, inorganic fertilization (especially N2) also caused soil acidification and salinization. In the N2 treatment, soil total organic carbon (TOC) decreased from 14.1±0.27 g kg-1 at the beginning of the long-term experiment in 1988 to 12.6±0.11 g kg-1 (P<0.05). Compared to CK, N1 and N2 exhibited higher soil α-galactosidase and β-galactosidase activities, but lower soil α-glucosidase and β-glucosidase activities (P<0.05), indicating that inorganic fertilization had different impacts on these C transformation enzymes. Compared with CK, the M, MN1 and MN2 treatments exhibited higher enzyme activities, soil TOC, total nitrogen, dissolved organic C, and microbial biomass C and N. The fertilization regime of the MN1 treatment was identified as optimal because it produced the highest yields and increased soil quality, ensuring sustainability. The results suggest that inorganic fertilizer alone, especially in high amounts, in greenhouse fields is detrimental to soil quality.

Transcranial direct current stimulation (tDCS) is a noninvasive brain stimulation technique which modifies neural excitability by providing weak currents through scalp electrodes. The aim of this study is to introduce and analyze a novel optimization method for safe and well-targeted multiarray tDCS. For optimization, we consider an optimal control problem for a Laplace equation with Neumann boundary conditions with control and pointwise gradient state constraints. We prove well-posedness results for the proposed methods and provide computer simulation results in a highly realistic six-compartment geometry-adapted hexahedral head model. For discretization of the proposed minimization problem the finite element method is employed and the existence of at least one minimizer to the discretized optimization problem is shown. For numerical solution of the corresponding discretized problem we employ the alternating direction method of multipliers and comprehensively examine the cortical current flow field with regard to focality, target intensity, and orientation. The numerical results reveal that the optimized current flow fields show significantly higher focality and, in most cases, higher directional agreement to the target vector in comparison to standard bipolar electrode montages.

Reductions in the protein and nitrogen content of plants grown under enhanced atmospheric CO 2 concentrations could adversely affect the quality of food grown in the future, but the mechanisms of change remain unclear. Now research investigating plant responses to enhanced levels of atmospheric CO 2 under field conditions finds that wheat nitrate assimilation was slower for elevated CO2 than for ambient CO 2 . Total protein and nitrogen concentrations in plants generally decline under elevated CO 2 atmospheres 1 , 2 . Explanations for this decline include that plants under elevated CO 2 grow larger, diluting the protein within their tissues 3 , 4 ; that carbohydrates accumulate within leaves, downregulating the amount of the most prevalent protein Rubisco 2 ; that carbon enrichment of the rhizosphere leads to progressively greater limitations of the nitrogen available to plants 4 ; and that elevated CO 2 directly inhibits plant nitrogen metabolism, especially the assimilation of nitrate into proteins in leaves of C 3 plants 5 . Recently, several meta-analyses have indicated that CO 2 inhibition of nitrate assimilation is the explanation most consistent with observations 6 , 7 , 8 . Here, we present the first direct field test of this explanation. We analysed wheat ( Triticum aestivum L.) grown under elevated and ambient CO 2 concentrations in the free-air CO 2 enrichment experiment at Maricopa, Arizona. In leaf tissue, the ratio of nitrate to total nitrogen concentration and the stable isotope ratios of organic nitrogen and free nitrate showed that nitrate assimilation was slower under elevated than ambient CO 2 . These findings imply that food quality will suffer under the CO 2 levels anticipated during this century unless more sophisticated approaches to nitrogen fertilization are employed.

Measurements in nanoscopic imaging suffer from blurring effects modeled with different point spread functions (PSF). Some apparatus even have PSFs that are locally dependent on phase shifts. Additionally, raw data are affected by Poisson noise resulting from laser sampling and “photon counts” in fluorescence microscopy. In these applications standard reconstruction methods (EM, filtered backprojection) deliver unsatisfactory and noisy results. Starting from a statistical modeling in terms of a MAP likelihood estimation we combine the iterative EM algorithm with total variation (TV) regularization techniques to make an efficient use of a-priori information. Typically, TV-based methods deliver reconstructed cartoon images suffering from contrast reduction. We propose extensions to EM-TV, based on Bregman iterations and primal and dual inverse scale space methods, in order to obtain improved imaging results by simultaneous contrast enhancement. Besides further generalizations of the primal and dual scale space methods in terms of general, convex variational regularization methods, we provide error estimates and convergence rates for exact and noisy data. We illustrate the performance of our techniques on synthetic and experimental biological data.

Mathematical models based on partial differential equations (PDEs) have become an integral part of quantitative analysis in most branches of science and engineering, recently expanding also towards biomedicine and socio-economic sciences. The application of PDEs in the latter is a promising field, but widely quite open and leading to a variety of novel mathematical challenges. In this introductory article of the Theme Issue, we will provide an overview of the field and its recent boosting topics. Moreover, we will put the contributions to the Theme Issue in an appropriate perspective.

PurposeAmendment of animal manures into eroded soils is an important approach to improving nutrient status and increasing the concentration of soil organic carbon (SOC). However, the contribution of the manure carbon to SOC and its variation along soil profile has not been quantified.Materials and methodsWe simulated soil erosion in a mollisol by removing the top soils of 0-, 5-, 10-, 20-, and 30-cm depth and compared SOC in soil profiles 10 years after either chemical fertilization alone or combined with cattle manure application.Results and discussionIncreasing erosion depth decreased SOC concentration and weakened soil aggregation. Compared to the chemical fertilization only, the addition of cattle manure significantly increased SOC accumulation and soil aggregation, which mainly occurred in 0–40-cm depths. The greatest effect of manure application was observed in the 10-cm erosion treatment. The application of cattle manure increased the 13C abundance in aggregates and bulk soil in the top 40 cm of soil profile. Using the natural 13C abundance method, we quantified the contribution of the cattle manure to SOC at 0–40-cm depths ranging from 1.1 to 8.4% across erosion treatments.ConclusionsThe greatest contribution of the manure-C to SOC occurred in surface layer with 10 cm of soil removal. The application of animal manures was recommended for restoring severely eroded soils.

We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regularization. We obtain rigorous convergence results and effective stopping criteria for the general procedure. The numerical results for denoising appear to give significant improvement over standard models, and preliminary results for deblurring/denoising are very encouraging.

Comprehensive assessment of the total greenhouse gas (GHG) budget of reduced tillage agricultural systems must consider emissions of nitrous oxide (N2O) and methane (CH4), each of which have higher global warming potentials than carbon dioxide (CO2). Tillage intensity may also impact nitric oxide (NO) emissions, which can have various environmental and agronomic impacts. In 2003 and 2004, we used chambers to measure N2O, CH4, and NO fluxes from plots that had been managed under differing tillage intensity since 1991. The effect of tillage on non‐CO2 GHG emissions varied, in both magnitude and direction, depending on fertilizer practices. Emissions of N2O following broadcast urea (BU) application were higher under no till (NT) and conservation tillage (CsT) compared to conventional tillage (CT). In contrast, following anhydrous ammonia (AA) injection, N2O emissions were higher under CT and CsT compared to NT. Emissions following surface urea ammonium nitrate (UAN) application did not vary with tillage. Total growing season non‐CO2 GHG emissions were equivalent to CO2 emissions of 0.15 to 1.9 Mg CO2 ha−1 yr−1 or 0.04 to 0.53 Mg soil‐C ha−1 yr−1 Emissions of N2O from AA‐amended plots were two to four times greater than UAN‐ and BU‐amended plots. Total NO + N2O losses in the UAN treatment were approximately 50% lower than AA and BU. This study demonstrates that N2O emissions can represent a substantial component of the total GHG budget of reduced tillage systems, and that interactions between fertilizer and tillage practices can be important in controlling non‐CO2 GHG emissions.