This paper is concerned with the solution of linear operator equations with a compact operator. These operators do not have a bounded inverse and therefore these kinds of equations have to be regularized before solution. The Arnoldi process provides a convenient way to reduce a compact operator to a nearby operator of finite rank and regularization is achieved with Tikhonov's method. This paper investigates properties of this simple solution approach. Computed examples illustrate the theory presented.

Achieving both high quality and cost-efficiency are two critical yet often conflicting objectives in manufacturing and maintenance processes. Quality standards vary depending on the specific application, while cost-effectiveness remains a constant priority. These competing objectives lead to multi-objective optimization problems, where algorithms are employed to identify Pareto-optimal solutions-compromise points which provide decision-makers with feasible parameter settings. The successful application of such optimization algorithms relies on the ability to model the underlying physical system, which is typically complex, through either physical or data-driven approaches, and to represent it mathematically. This paper applies three multi-objective optimization algorithms to determine optimal process parameters for high-velocity oxygen fuel (HVOF) thermal spraying. Their ability to enhance coating performance while maintaining process efficiency is systematically evaluated, considering practical constraints and industrial feasibility. Practical validation trials are conducted to verify the approximate theoretical solutions generated by the algorithms, ensuring their applicability and reliability in real-world scenarios. By exploring the performance of these diverse algorithms in an industrial setting, this study offers insights into their practical applicability, guiding both researchers and practitioners in enhancing process efficiency and product quality in the coating industry.

The development of thermal spray coatings for specific industrial and scientific applications is critical, particularly in the context of sustainable and economical production. This study employs a hybrid experimental-statistical approach to identify the influence of key process parameters and their interactions, based on a systematic design of experiments. Five factors were investigated: total gas flow (TGF), fuel-to-oxygen ratio (F2O), powder feed rate (PFR), standoff distance (SOD), and coating velocity (CV). The effects of these factors on in-flight particle properties, process performance, and coating characteristics were analyzed. Nondestructive evaluations, including deposition efficiency, surface roughness, and surface hardness, were directly compared with microstructural measurements. In contrast to previous studies, TGF emerged as the most influential parameter, exerting a stronger effect on particle properties, process performance, and coating characteristics than F2O. Significant interactions were identified, including the combined effects of TGF and F2O on roughness and hardness, TGF and PFR on deposition efficiency, and TGF and SOD on surface roughness. This investigation advances beyond validating known correlations by uncovering nuanced multidimensional interactions, offering a robust framework for optimizing WC-CoCr coatings. The findings contribute to the broader objective of enhancing the performance and sustainability of modern coating technologies through nondestructive methodologies.

We consider the efficient numerical minimization of Tikhonov functionals with nonlinear operators and non-smooth and non-convex penalty terms, which appear for example in variational regularization. For this, we consider a new class of subspace-containing derivative (SCD) semismooth Newton methods, which are based on a novel concept of graphical derivatives, and exhibit locally superlinear convergence. We present a detailed description of these methods, and provide explicit algorithms in the case of sparsity and total-variation penalty terms. The numerical performance of these methods is then illustrated on a number of tomographic imaging problems.