Correction to: The continuous stochastic gradient method: part II–application and numerics (Computational Optimization and Applications, (2023), 10.1007/s10589-023-00540-w) - Grieshammer, Max; Pflug, Lukas; Stingl, Michael; Uihlein, Andrian (Computational Optimization and Applications, 2023)
The original version of this article unfortunately contained an error in Reference section. The reference [2] did not have appropriate details of the paper to which it is referred, and therefore the article is published with incomplete reference details. The complete details of reference [2] is: Grieshammer, M., Pflug, L., Stingl, M., Uihlein, A.: The continuous stochastic gradient method: part I–convergence theory. Comput. Optim. Appl. (2023). The original article has been corrected.
CRIS-Team Scopus Importer:2023-12-22

Optimizing Color of Particulate Products - Uihlein, Andrian; Pflug, Lukas; Stingl, Michael (Proceedings in Applied Mathematics and Mechanics, 2023)
Optical properties of nanoparticles largely depend on their shape and material distribution. Given any such property, e.g., a desired color, the aim is to design an optimal nanoparticle, whose properties best match these targets. The corresponding nonlinear optimization problem is challenging due to numerous difficulties: The objective function is given by a multi-dimensional integral over wavelengths, directions of incoming light rays and nanoparticle design distributions, where the integrand depends on the extinction cross section of the nanoparticle design. Thus, classical full gradient schemes based on numerical integration prove ineffective due to their tremendous computational cost. On the other hand, known stochastic gradient descent methods cannot deal with the nonlinear fashion in which the design variables enter the objective function. Therefore, we introduce the continuous stochastic gradient (CSG) method, which does not compute a full gradient, but instead uses information from previous iterations in an optimal way to learn the exact gradient during the optimization process. Hence, CSG thematically lies in between a stochastic optimization scheme and a full gradient method, combining both the low computational cost of SG due to very few partial gradient evaluations (small “batch size”) as well as step size rules and line search techniques known from full gradient descent.<br

Predictive design of plasmonic color - Bleyer, Gudrun; Uihlein, Andrian; Sultan, Umair; Ickler, Maret; Yamashita, Ruri; Pflug, Lukas; Watanabe, Satoshi; Vogel, Nicolas (Journal of Colloid and Interface Science, 2025)
Predictive design promises direct access to nanomaterials with optimal properties, identified by a rigorous mathematical optimization. This strategy thus provides structural blueprints for synthesis and eliminates extensive parameter screening. Here, we establish predictive design of nanoparticle systems using the example of plasmonic coloration. The perceived color of nanoparticle dispersions is influenced by the interplay of the entire visible particle spectrum with the wavelength-dependent sensitivities of the human eye, and not a mere function of plasmon resonances. We use Mie theory to compute the far-field spectra of spherical gold and silver core–shell nanoparticles and convert them into color perceived by the human eye. Color maps that describe the minimum color difference relative to a predefined target are established as a function of concentration, core diameter and shell thickness, thus providing predictions of the optimal particle structure for a targeted color. We demonstrate the feasibility of this predictive design by successfully targeting multiple colors and discuss limitations and discrepancies between experiment and simulations. Our results showcase that predictive design holds promise for resource-efficient syntheses of functional nanoparticles as the optimal structural blueprints for a desired target property can be accurately predicted in silic
Keywords: Plasmonic color; Predictive design; Nanoparticles; Structure optimization; CIElab

Stabilized SQP Methods in Hilbert Spaces - Uihlein, Andrian; Wollner, Winnifried (Numerical Functional Analysis and Optimization, 2024)
Based on techniques by (S.J. Wright 1998) for finite-dimensional optimization, we investigate a stabilized sequential quadratic programming method for nonlinear optimization problems in infinite-dimensional Hilbert spaces. The method is shown to achieve fast local convergence even in the absence of a constraint qualification, generalizing the results obtained by (S.J. Wright 1998 and W.W. Hager 1999) in finite dimensions to this broader setting.<br
Keywords: Infinite dimensional spaces; stabilized SQP method

Targeted color design of silver-gold alloy nanoparticles - Traore, Nabi; Spruck, Christina; Uihlein, Andrian; Pflug, Lukas; Peukert, Wolfgang (Nanoscale Advances, 2023)
This research article focuses on the targeted color design of silver-gold alloy nanoparticles (NPs), employing a multivariate optimization approach. NP synthesis involves interconnected process parameters, making independent variation challenging. Data-based property-process relationships are established to optimize optical properties effectively. We define a color target, employ a green chemical co-reduction method at room temperature and optimize process parameters accordingly. The CIEL*a*b* color space (Commission Internationale de l'Éclairage - International Commission on Illumination) and Euclidean distances facilitate accurate color matching to establish the property-process relationship. Concurrently, theoretical Mie calculations explore the structure-property relationship across particle sizes, concentrations, and molar gold contents. The theoretically optimal structure agrees very well with experimental particle structures at the property-process relationship's optimum. The data-driven property-process relationship provides valuable insights into the formation mechanism of a complex particle system, sheds light on the role of relevant process parameters and allows to evaluate the practically available property space. Model validation beyond the original grid demonstrates its robustness, yielding colors close to the target. Additionally, Design of Experiments (DoE) methods reduce experimental work by threefold with slight accuracy trade-offs. Our novel methodology for targeted color design demonstrates how data-based methods can be utilized alongside structure-property relationships to unravel property-process relationships in the design of complex nanoparticle systems and paves the way for future developments in targeted property design.
CRIS-Team Scopus Importer:2023-12-22

The Continuous Stochastic Gradient Method - Uihlein, Andrian (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2024)
We consider a class of optimization problems in which the objective function requires the computation of an expected value of a hard to evaluate property. Such structures arise in probabilistic settings, e.g., expected values in optimization under uncertainty, as well as fully deterministic contexts, e.g., optimizing a design for a (possibly infinite) set of scenarios. Therefore, both stochastic and deterministic optimization schemes have been considered to solve such problems. The latter typically rely on a fixed quadrature rule to approximate all appearing integrals numerically. As a result, the associated computational cost may render such approaches numerically infeasible in complex settings. Moreover, discretization can lead to artificial local solutions emerging in the problem, if the corresponding quadrature rule is not chosen carefully. Contrary, stochastic sample based methods do not suffer from this problem and provide a computationally cheap alternative. However, these schemes are typically limited to objective functions of very simple structure, i.e., minimizing an expected value. Settings in which the objective consists of several integrals, possibly coupled by nonlinear functions, are beyond the scope of such techniques. In this work, we consider continuous stochastic optimization approaches, specifically, a continuous stochastic gradient (CSG) scheme. To keep the numerical effort per iteration low, these methods also rely on random gradient samples instead of full integrals. However, unlike traditional stochastic sample based strategies, the gradient information is stored over the course of iterations. Thus, by calculating suitable design dependent integration weights, a convex combination of old gradient samples is used to build an approximation to the full gradient. We show that the approximation error almost surely converges to zero during the optimization process. As a result, we prove that CSG admits the same convergence results as a full gradient scheme, i.e., convergence for constant step sizes and line search techniques based on the continuous stochastic approximations. Moreover, it means that CSG is not restricted to the minimization of simple expected values and can instead deal with arbitrary combinations of integrals and Lipschitz continuous functions in the objective. In this sense, CSG can be considered as a hybrid of stochastic gradient methods and a classical gradient descent scheme. Several methods to obtain the aforementioned integration weights in practice are presented and discussed. Furthermore, we provide a full theoretical convergence analysis of CSG and perform numerical experiments, covering all important aspects of continuous stochastic optimization methods. For several large scale applications from the field of topology optimization, the proposed schemes are compared to different optimization techniques known from literature.<br

The continuous stochastic gradient method: part I–convergence theory - Grieshammer, Max; Pflug, Lukas; Stingl, Michael; Uihlein, Andrian (Computational Optimization and Applications, 2023)
In this contribution, we present a full overview of the continuous stochastic gradient (CSG) method, including convergence results, step size rules and algorithmic insights. We consider optimization problems in which the objective function requires some form of integration, e.g., expected values. Since approximating the integration by a fixed quadrature rule can introduce artificial local solutions into the problem while simultaneously raising the computational effort, stochastic optimization schemes have become increasingly popular in such contexts. However, known stochastic gradient type methods are typically limited to expected risk functions and inherently require many iterations. The latter is particularly problematic, if the evaluation of the cost function involves solving multiple state equations, given, e.g., in form of partial differential equations. To overcome these drawbacks, a recent article introduced the CSG method, which reuses old gradient sample information via the calculation of design dependent integration weights to obtain a better approximation to the full gradient. While in the original CSG paper convergence of a subsequence was established for a diminishing step size, here, we provide a complete convergence analysis of CSG for constant step sizes and an Armijo-type line search. Moreover, new methods to obtain the integration weights are presented, extending the application range of CSG to problems involving higher dimensional integrals and distributed data.
CRIS-Team Scopus Importer:2023-12-01
Keywords: Backtracking line search; Constant step size; Convergence analysis; Step size rule; Stochastic gradient scheme

The continuous stochastic gradient method: part II–application and numerics - Grieshammer, Max; Pflug, Lukas; Stingl, Michael; Uihlein, Andrian (Computational Optimization and Applications, 2024)
In this contribution, we present a numerical analysis of the continuous stochastic gradient (CSG) method, including applications from topology optimization and convergence rates. In contrast to standard stochastic gradient optimization schemes, CSG does not discard old gradient samples from previous iterations. Instead, design dependent integration weights are calculated to form a convex combination as an approximation to the true gradient at the current design. As the approximation error vanishes in the course of the iterations, CSG represents a hybrid approach, starting off like a purely stochastic method and behaving like a full gradient scheme in the limit. In this work, the efficiency of CSG is demonstrated for practically relevant applications from topology optimization. These settings are characterized by both, a large number of optimization variables and an objective function, whose evaluation requires the numerical computation of multiple integrals concatenated in a nonlinear fashion. Such problems could not be solved by any existing optimization method before. Lastly, with regards to convergence rates, first estimates are provided and confirmed with the help of numerical experiments.
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Keywords: 65K05; 90C06; 90C15; 90C30; Backtracking line search; Constant step size; Convergence analysis; Step size rule; Stochastic gradient scheme

Topology optimization of broadband acoustic transition section: a comparison between deterministic and stochastic approaches - Mousavi, Abbas; Uihlein, Andrian; Pflug, Lukas; Wadbro, Eddie (Structural and Multidisciplinary Optimization, 2024)
This paper focuses on the topology optimization of a broadband acoustic transition section that connects two cylindrical waveguides with different radii. The primary objective is to design a transition section that maximizes the transmission of a planar acoustic wave while ensuring that the transmitted wave exhibits a planar shape. Helmholtz equation is used to model linear wave propagation in the device. We utilize the finite element method to solve the state equation on a structured mesh of square elements. Subsequently, a material distribution topology optimization problem is formulated to optimize the distribution of sound-hard material in the transition section. We employ two different gradient-based approaches to solve the optimization problem: namely, a deterministic approach using the method of moving asymptotes (MMA), and a stochastic approach utilizing both stochastic gradient (SG) and continuous stochastic gradient (CSG) methods. A comparative analysis is provided among these methodologies concerning the design feasibility and the transmission performance of the optimized designs, and the computational efficiency. The outcomes highlight the effectiveness of stochastic techniques in achieving enhanced broadband acoustic performance with reduced computational demands and improved design practicality. The insights from this investigation demonstrate the potential of stochastic approaches in acoustic applications, especially when broadband acoustic performance is desired.
CRIS-Team Scopus Importer:2024-05-24
Keywords: Acoustic transition section; Material distribution approach; Stochastic methods; Topology optimization