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In understanding and redesigning the function of proteins in modern biochemistry, protein engineers are increasingly focusing on exploring regions in proteins called loops. Analyzing various characteristics of these regions helps the experts design the transfer of the desired function from one protein to another. This process is denoted as loop grafting. We designed a set of interactive visualizations that provide experts with visual support through all the loop grafting pipeline steps. The workflow is divided into several phases, reflecting the steps of the pipeline. Each phase is supported by a specific set of abstracted 2D visual representations of proteins and their loops that are interactively linked with the 3D View of proteins. By sequentially passing through the individual phases, the user shapes the list of loops that are potential candidates for loop grafting. Finally, the actual in-silico insertion of the loop candidates from one protein to the other is performed, and the results are visually presented to the user. In this way, the fully computational rational design of proteins and their loops results in newly designed protein structures that can be further assembled and tested through in-vitro experiments. We showcase the contribution of our visual support design on a real case scenario changing the enantiomer selectivity of the engineered enzyme. Moreover, we provide the readers with the experts' feedback.

We consider the method of mappings for performing shape optimization for unsteady fluid-structure interaction (FSI) problems. In this work, we focus on the numerical implementation. We model the optimization problem such that it takes several theoretical results into account, such as regularity requirements on the transformations and a differential geometrical point of view on the manifold of shapes. Moreover, we discretize the problem such that we can compute exact discrete gradients. This allows for the use of general purpose optimization solvers. We focus on an FSI benchmark problem to validate our numerical implementation. The method is used to optimize parts of the outer boundary and the interface. The numerical simulations build on FEniCS, dolfin-adjoint and IPOPT. Moreover, as an additional theoretical result, we show that for a linear special case the adjoint attains the same structure as the forward problem but reverses the temporal flow of information.

Metal energy carriers recently gained growing interest in research as a promising storage and transport material for renewable electricity. Within the development of a metal-fueled circular energy economy, research involves a model hierarchy spanning from micro to macro scales, making the transfer of information among different levels of complexity a crucial task for the implementation of the new technology. Chemical reactor networks (CRNs) are models of reduced complexity and a promising approach to accomplish the scale-bridging task. This holds if valid information from CRNs can be obtained on a much denser set of operating conditions than available from experiments and elaborated simulation methods like Computational Fluid Dynamics (CFD). An approach for CRN calibration from recent literature, including model error quantification, is further developed to construct a CRN model of a laboratory reactor for flash ironmaking, using data from the literature. By introducing a meta model of a CRN parameter, a simple CRN model on an extended set of operating conditions has successfully been calibrated. This way, the employed coupled calibration and uncertainty quantification framework has proven promising for the task of scale-bridging in the model hierarchy under investigation., Comment: 35 pages, 14 Figures, updated preprint, submitted for peer-review. Corrected a bug and enhanced model. Overall structure of results and main outcomes remain the same

In this note, we revisit the problem of the pressure-driven transport of a meniscus through a narrow cylindrical capillary or pore. This generic process finds many applications in science and technology. As it is known that Direct Numerical Simulations of moving contact line problems are highly demanding in terms of computational costs, simplified models in the form of ordinary differential equations offer an interesting alternative to perform a mathematical optimization of the flow. Blake and De Coninck studied the pressure-driven transport of a meniscus and identified two major competing mechanisms. While a hydrophilic surface is favorable to enhance the spontaneous imbibition into the pore, the friction is known to be significantly reduced on a hydrophobic surface. Blake and De Coninck showed that, depending on the applied pressure difference, there exists an optimal wettability that minimizes the time required to move the meniscus over a certain distance. We revisit this problem and derive analytical solutions in the limiting cases of negligible inertia and negligible contact line friction., Comment: 12 pages, 4 figures

This work features the optimization of a Permanent Magnet Synchronous Motor using 2D nonlinear simulations in an Isogeometric Analysis framework. The rotor and stator designs are optimized for both geometric parameters and surface shapes via modifications of control points. The scaling laws for magnetism are employed to allow for axial and radial scaling, enabling a thorough optimization of all critical machine parameters for multiple operating points. The process is carried out in a gradient-based fashion with the objectives of lowering motor material cost, torque ripple and losses. It is shown that the optimization can be efficiently conducted for many optimization variables and all objective values can be reduced.

We present simple models of trajectories in space, both in 2D and in 3D. The first examples, which model bicircular moves in the same direction, are classical curves (epicycloids, etc.). Then, we explore bicircular moves in reverse direction and tricircular moves in 2D and 3D, to explore complex visualisations of extraplanetary movements. These moves are studied in a plane setting. Then, adding increasing complexity, we explore them in a non planar setting (which is a closer model of the real situation). The exploration is followed by using these approaches for creating mathematical art in 2D and 3D printed objects, providing new ways of mathematical representations. Students' activities are organized around this exploration., Comment: In Proceedings ADG 2023, arXiv:2401.10725

In structural optimization, both parameters and shape are relevant for the model performance. Yet, conventional optimization techniques usually consider either parameters or the shape separately. This work addresses this problem by proposing a simple yet powerful approach to combine parameter and shape optimization in a framework using Isogeometric Analysis (IGA). The optimization employs sensitivity analysis by determining the gradients of an objective function with respect to parameters and control points that represent the geometry. The gradients with respect to the control points are calculated in an analytical way using the adjoint method, which enables straightforward shape optimization by altering of these control points. Given that a change in a single geometry parameter corresponds to modifications in multiple control points, the chain rule is employed to obtain the gradient with respect to the parameters in an efficient semi-analytical way. The presented method is exemplarily applied to nonlinear 2D magnetostatic simulations featuring a permanent magnet synchronous motor and compared to designs, which were optimized using parameter and shape optimization separately. It is numerically shown that the permanent magnet mass can be reduced and the torque ripple can be eliminated almost completely by simultaneously adjusting rotor parameters and shape. The approach allows for novel designs to be created with the potential to reduce the optimization time substantially.

Recently, the stochastic Polyak step size (SPS) has emerged as a competitive adaptive step size scheme for stochastic gradient descent. Here we develop ProxSPS, a proximal variant of SPS that can handle regularization terms. Developing a proximal variant of SPS is particularly important, since SPS requires a lower bound of the objective function to work well. When the objective function is the sum of a loss and a regularizer, available estimates of a lower bound of the sum can be loose. In contrast, ProxSPS only requires a lower bound for the loss which is often readily available. As a consequence, we show that ProxSPS is easier to tune and more stable in the presence of regularization. Furthermore for image classification tasks, ProxSPS performs as well as AdamW with little to no tuning, and results in a network with smaller weight parameters. We also provide an extensive convergence analysis for ProxSPS that includes the non-smooth, smooth, weakly convex and strongly convex setting.

Forecast quality should be assessed in the context of what is possible in theory and what is reasonable to expect in practice. Often, one can identify an approximate upper bound to a probabilistic forecast's sharpness, which sets a lower, not necessarily achievable, limit to error metrics. In retail forecasting, a simple, but often unconquerable sharpness limit is given by the Poisson distribution. When evaluating forecasts using traditional metrics such as Mean Absolute Error, it is hard to judge whether a certain achieved value reflects unavoidable Poisson noise or truly indicates an overdispersed prediction model. Moreover, every evaluation metric suffers from precision scaling: Perhaps surprisingly, the metric's value is mostly defined by the selling rate and by the resulting rate-dependent Poisson noise, and only secondarily by the forecast quality. For any metric, comparing two groups of forecasted products often yields "the slow movers are performing worse than the fast movers" or vice versa, the na\"ive scaling trap. To distill the intrinsic quality of a forecast, we stratify predictions into buckets of approximately equal predicted value and evaluate metrics separately per bucket. By comparing the achieved value per bucket to benchmarks, we obtain an intuitive visualization of forecast quality, which can be summarized into a single rating that makes forecast quality comparable among different products or even industries. The thereby developed scaling-aware forecast rating is applied to forecasting models used on the M5 competition dataset as well as to real-life forecasts provided by Blue Yonder's Demand Edge for Retail solution for grocery products in Sainsbury's supermarkets in the United Kingdom. The results permit a clear interpretation and high-level understanding of model quality by non-experts., Comment: 39 pages, 11 figures; improved introduction and outlook

Anisotropic colloidal particles exhibit complex dynamics which play a crucial role in their functionality, transport and phase behaviour. In this work, we investigate the two-dimensional diffusion of smoothly curved colloidal rods -- also known as colloidal bananas -- as a function of their opening angle, {\alpha}. We measure the translational and rotational diffusion coefficients of the particles with opening angles ranging from 0{\deg} (straight rods) to nearly 360{\deg}(closed rings). In particular, we find that the anisotropic diffusion of the particles varies non-monotonically with their opening angle and that the axis of fastest diffusion switches from the long to the short axis of the particles when {\alpha}>180{\deg}. We also find that the rotational diffusion coefficient of nearly closed rings is approximately an order of magnitude higher than that of straight rods of the same length. Finally, we show that the experimental results are consistent with Slender Body Theory, indicating that the dynamical behavior of the particles arises primarily from their local drag anisotropy. These results highlight the impact of curvature on the Brownian Motion of elongated colloidal particles, which must be taken into account when seeking to understand the behaviour of curved colloidal particles., Comment: 5 pages (including references) and 4 figures

We present sMolBoxes, a dataflow representation for the exploration and analysis of long molecular dynamics (MD) simulations. When MD simulations reach millions of snapshots, a frame-by-frame observation is not feasible anymore. Thus, biochemists rely to a large extent only on quantitative analysis of geometric and physico-chemical properties. However, the usage of abstract methods to study inherently spatial data hinders the exploration and poses a considerable workload. sMolBoxes link quantitative analysis of a user-defined set of properties with interactive 3D visualizations. They enable visual explanations of molecular behaviors, which lead to an efficient discovery of biochemically significant parts of the MD simulation. sMolBoxes follow a node-based model for flexible definition, combination, and immediate evaluation of properties to be investigated. Progressive analytics enable fluid switching between multiple properties, which facilitates hypothesis generation. Each sMolBox provides quick insight to an observed property or function, available in more detail in the bigBox View. The case study illustrates that even with relatively few sMolBoxes, it is possible to express complex analyses tasks, and their use in exploratory analysis is perceived as more efficient than traditional scripting-based methods., Comment: 10 pages, 9 figures, IEEE VIS, TVCG

The sample average approximation (SAA) approach is applied to risk-neutral optimization problems governed by semilinear elliptic partial differential equations with random inputs. After constructing a compact set that contains the SAA critical points, we derive nonasymptotic sample size estimates for SAA critical points using the covering number approach. Thereby, we derive upper bounds on the number of samples needed to obtain accurate critical points of the risk-neutral PDE-constrained optimization problem through SAA critical points. We quantify accuracy using expectation and exponential tail bounds. Numerical illustrations are presented., Comment: 26 pages, 10 figures

We present a biologically inspired approach for path planning with dynamic obstacle avoidance. Path planning is performed in a condensed configuration space of a robot generated by self-organizing neural networks (SONN). The robot itself and static as well as dynamic obstacles are mapped from the Cartesian task space into the configuration space by precomputed kinematics. The condensed space represents a cognitive map of the environment, which is inspired by place cells and the concept of cognitive maps in mammalian brains. The generation of training data as well as the evaluation were performed on a real industrial robot accompanied by simulations. To evaluate the reactive collision-free online planning within a changing environment, a demonstrator was realized. Then, a comparative study regarding sample-based planners was carried out. So we could show that the robot is able to operate in dynamically changing environments and re-plan its motion trajectories within impressing 0.02 seconds, which proofs the real-time capability of our concept., Comment: Accepted to IROS2022. Supplementary video: https://youtu.be/CEkVDDg9ORw

We develop an implementable stochastic proximal point (SPP) method for a class of weakly convex, composite optimization problems. The proposed stochastic proximal point algorithm incorporates a variance reduction mechanism and the resulting SPP updates are solved using an inexact semismooth Newton framework. We establish detailed convergence results that take the inexactness of the SPP steps into account and that are in accordance with existing convergence guarantees of (proximal) stochastic variance-reduced gradient methods. Numerical experiments show that the proposed algorithm competes favorably with other state-of-the-art methods and achieves higher robustness with respect to the step size selection.

Motion planning in the configuration space (C-space) induces benefits, such as smooth trajectories. It becomes more complex as the degrees of freedom (DOF) increase. This is due to the direct relation between the dimensionality of the search space and the DOF. Self-organizing neural networks (SONN) and their famous candidate, the Self-Organizing Map, have been proven to be useful tools for C-space reduction while preserving its underlying topology, as presented in [29]. In this work, we extend our previous study with additional models and adapt the approach from human motion data towards robots' kinematics. The evaluation includes the best performant models from [29] and three additional SONN architectures, representing the consequent continuation of this previous work. Generated Trajectories, planned with the different SONN models, were successfully tested in a robot simulation.

We analyze optimal control problems for two-phase Navier-Stokes equations with surface tension. Based on $L_p$-maximal regularity of the underlying linear problem and recent well-posedness results of the problem for sufficiently small data we show the differentiability of the solution with respect to initial and distributed controls for appropriate spaces resulting form the $L_p$-maximal regularity setting. We consider first a formulation where the interface is transformed to a hyperplane. Then we deduce differentiability results for the solution in the physical coordinates. Finally, we state an equivalent Volume-of-Fluid type formulation and use the obtained differentiability results to derive rigorosly the corresponding sensitivity equations of the Volume-of-Fluid type formulation. For objective functionals involving the velocity field or the discontinuous pressure or phase indciator field we derive differentiability results with respect to controls and state formulas for the derivative. The results of the paper form an analytical foundation for stating optimality conditions, justifying the application of derivative based optimization methods and for studying the convergence of discrete sensitivity schemes based on Volume-of-Fluid discretizations for optimal control of two-phase Navier-Stokes equations.

In PDE-constrained optimization, proper orthogonal decomposition (POD) provides a surrogate model of a (potentially expensive) PDE discretization, on which optimization iterations are executed. Because POD models usually provide good approximation quality only locally, they have to be updated during optimization. Updating the POD model is usually expensive, however, and therefore often impossible in a model-predictive control (MPC) context. Thus, reduced models of mediocre quality might be accepted. We take the view of a simplified Newton method for solving semilinear evolution equations to derive an algorithm that can serve as an offline phase to produce a POD model. Approaches that build the POD model with impulse response snapshots can be regarded as the first Newton step in this context. In particular, POD models that are based on impulse response snapshots are extended by adding a second simplified Newton step. This procedure improves the approximation quality of the POD model significantly by introducing a moderate amount of extra computational costs during optimization or the MPC loop. We illustrate our findings with an example satisfying our assumptions.

Deductive verification has been successful in verifying interesting properties of real-world programs. One notable gap is the limited support for floating-point reasoning. This is unfortunate, as floating-point arithmetic is particularly unintuitive to reason about due to rounding as well as the presence of the special values infinity and `Not a Number' (NaN). In this paper, we present the first floating-point support in a deductive verification tool for the Java programming language. Our support in the KeY verifier handles arithmetic via floating-point decision procedures inside SMT solvers and transcendental functions via axiomatization. We evaluate this integration on new benchmarks, and show that this approach is powerful enough to prove the absence of floating-point special values -- often a prerequisite for further reasoning about numerical computations -- as well as certain functional properties for realistic benchmarks.

Motivated by studies of the Greenberg-Hastings cellular automata (GHCA) as a caricature of excitable systems, in this paper we study kink-antikink dynamics in the perhaps simplest PDE model of excitable media given by the scalar reaction diffusion-type $\theta$-equations for excitable angular phase dynamics. On the one hand, we qualitatively study geometric kink positions using the comparison principle and the theory of terraces. This yields the minimal initial distance as a global lower bound, a well-defined sequence of collision data for kinks- and antikinks, and implies that periodic pure kink sequences are asymptotically equidistant. On the other hand, we study metastable dynamics of finitely many kinks using weak interaction theory for certain analytic kink positions, which admits a rigorous reduction to ODE. By blow-up type singular rescaling we show that distances become ordered in finite time, and eventually diverge. We conclude that diffusion implies a loss of information on kink distances so that the entropic complexity based on positions and collisions in the GHCA does not simply carry over to the PDE model., Comment: 40 pages, 9 figures

Functionality of living cells is inherently linked to subunits with dimensions on the nanoscale. In case of osteoblasts the cell surface plays a particularly important role for adhesion and spreading which are crucial properties with regard to bone implants. Here we present a comprehensive characterization of the 3D nanomorphology of living as well as fixed osteoblastic cells using scanning ion conductance microscopy (SICM) which is a nanoprobing method largely avoiding forces. Dynamic ruffles are observed, manifesting themselves in characteristic membrane protrusions. They contribute to the overall surface corrugation which we systematically study by introducing the relative 3D excess area as a function of projected adhesion area. A clear anticorrelation is found upon analysis of ~40 different cells on glass as well as on amine covered surfaces. At the rim of lamellipodia characteristic edge heights between 100 nm and ~300 nm are observed. Power spectral densities of membrane fluctuations show frequency-dependent decay exponents in excess of -2 on living osteoblasts. We discuss the capability of apical membrane features and fluctuation dynamics in aiding assessment of adhesion and migration properties on a single-cell basis., Comment: 27 pages, 11 figures

In the modern drug discovery process, medicinal chemists deal with the complexity of analysis of large ensembles of candidate molecules. Computational tools, such as dimensionality reduction (DR) and classification, are commonly used to efficiently process the multidimensional space of features. These underlying calculations often hinder interpretability of results and prevent experts from assessing the impact of individual molecular features on the resulting representations. To provide a solution for scrutinizing such complex data, we introduce ChemVA, an interactive application for the visual exploration of large molecular ensembles and their features. Our tool consists of multiple coordinated views: Hexagonal view, Detail view, 3D view, Table view, and a newly proposed Difference view designed for the comparison of DR projections. These views display DR projections combined with biological activity, selected molecular features, and confidence scores for each of these projections. This conjunction of views allows the user to drill down through the dataset and to efficiently select candidate compounds. Our approach was evaluated on two case studies of finding structurally similar ligands with similar binding affinity to a target protein, as well as on an external qualitative evaluation. The results suggest that our system allows effective visual inspection and comparison of different high-dimensional molecular representations. Furthermore, ChemVA assists in the identification of candidate compounds while providing information on the certainty behind different molecular representations., Comment: Accepted for the IEEE VIS 2020 conference

We consider optimal control problems for a wide class of bilateral obstacle problems where the control appears in a possibly nonlinear source term. The non-differentiability of the solution operator poses the main challenge for the application of efficient optimization methods and the characterization of Bouligand generalized derivatives of the solution operator is essential for their theoretical foundation and numerical realization. In this paper, we derive specific elements of the Bouligand generalized differential if the control operator satisfies natural monotonicity properties. We construct monotone sequences of controls where the solution operator is G\^ateaux differentiable and characterize the corresponding limit element of the Bouligand generalized differential as being the solution operator of a Dirichlet problem on a quasi-open domain. In contrast to a similar recent result for the unilateral obstacle problem [RU19], we have to deal with an opposite monotonic behavior of the active and strictly active sets corresponding to the upper and lower obstacle. Moreover, the residual is no longer a nonnegative functional on H^{-1} and its representation as the difference of two nonnegative Radon measures requires special care. This necessitates new proof techniques that yield two elements of the Bouligand generalized differential. Also for the unilateral case we obtain an additional element to that derived in [RU19].

In this article we consider shape optimization problems as optimal control problems via the method of mappings. Instead of optimizing over a set of admissible shapes a reference domain is introduced and it is optimized over a set of admissible transformations. The focus is on the choice of the set of transformations, which we motivate from a function space perspective. In order to guarantee local injectivity of the admissible transformations we enrich the optimization problem by a nonlinear constraint. The approach requires no parameter tuning for the extension equation and can naturally be combined with geometric constraints on volume and barycenter of the shape. Numerical results for drag minimization of Stokes flow are presented.