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In this contribution, various sales forecast models for the German automobile market are developed and tested. Our most important criteria for the assessment of these models are the quality of the prediction as well as an easy explicability. Yearly, quarterly and monthly data for newly registered automobiles from 1992 to 2007 serve as the basis for the tests of these models. The time series model used consists of additive components: trend, seasonal, calendar and error component.

Molecular modeling is an important subdomain in the field of computational modeling, regarding both scientific and industrial applications. This is because computer simulations on a molecular level are a virtuous instrument to study the impact of microscopic on macroscopic phenomena. Accurate molecular models are indispensable for such simulations in order to predict physical target observables, like density, pressure, diffusion coefficients or energetic properties, quantitatively over a wide range of temperatures. Thereby, molecular interactions are described mathematically by force fields. The mathematical description includes parameters for both intramolecular and intermolecular interactions. While intramolecular force field parameters can be determined by quantum mechanics, the parameterization of the intermolecular part is often tedious. Recently, an empirical procedure, based on the minimization of a loss function between simulated and experimental physical properties, was published by the authors. Thereby, efficient gradient-based numerical optimization algorithms were used. However, empirical force field optimization is inhibited by the two following central issues appearing in molecular simulations: firstly, they are extremely time-consuming, even on modern and high-performance computer clusters, and secondly, simulation data is affected by statistical noise. The latter provokes the fact that an accurate computation of gradients or Hessians is nearly impossible close to a local or global minimum, mainly because the loss function is flat. Therefore, the question arises of whether to apply a derivative-free method approximating the loss function by an appropriate model function. In this paper, a new Sparse Grid-based Optimization Workflow (SpaGrOW) is presented, which accomplishes this task robustly and, at the same time, keeps the number of time-consuming simulations relatively small. This is achieved by an efficient sampling procedure for the approximation based on sparse grids, which is described in full detail: in order to counteract the fact that sparse grids are fully occupied on their boundaries, a mathematical transformation is applied to generate homogeneous Dirichlet boundary conditions. As the main drawback of sparse grids methods is the assumption that the function to be modeled exhibits certain smoothness properties, it has to be approximated by smooth functions first. Radial basis functions turned out to be very suitable to solve this task. The smoothing procedure and the subsequent interpolation on sparse grids are performed within sufficiently large compact trust regions of the parameter space. It is shown and explained how the combination of the three ingredients leads to a new efficient derivative-free algorithm, which has the additional advantage that it is capable of reducing the overall number of simulations by a factor of about two in comparison to gradient-based optimization methods. At the same time, the robustness with respect to statistical noise is maintained. This assertion is proven by both theoretical considerations and practical evaluations for molecular simulations on chemical example substances.

Automated force field optimisation of small molecules using a gradient-based workflow package
(2010)

In this study, the recently developed gradient-based optimisation workflow for the automated development of molecular models is for the first time applied to the parameterisation of force fields for molecular dynamics simulations. As a proof-of-concept, two small molecules (benzene and phosgene) are considered. In order to optimise the underlying intermolecular force field (described by the (12,6)-Lennard-Jones and the Coulomb potential), the energetic and diameter parameters ε and σ are fitted to experimental physical properties by gradient-based numerical optimisation techniques. Thereby, a quadratic loss function between experimental and simulated target properties is minimised with respect to the force field parameters. In this proof-of-concept, the considered physical target properties are chosen to be diverse: density, enthalpy of vapourisation and self-diffusion coefficient are optimised simultaneously at different temperatures. We found that in both cases, the optimisation could be successfully concluded by fulfillment of a pre-defined stopping criterion. Since a fairly small number of iterations were needed to do so, this study will serve as a good starting point for more complex systems and further improvements of the parametrisation task.

Computer simulations of chemical systems, especially systems of condensed matter, are highly important for both scientific and industrial applications. Thereby, molecular interactions are modeled on a microscopic level in order to study their impact on macroscopic phenomena. To be capable of predicting physical properties quantitatively, accurate molecular models are indispensable. Molecular interactions are described mathematically by force fields, which have to be parameterized. Recently, an automated gradient-based optimization procedure was published by the authors based on the minimization of a loss function between simulated and experimental physical properties. The applicability of gradient-based procedures is not trivial at all because of two reasons: firstly, simulation data are affected by statistical noise, and secondly, the molecular simulations required for the loss function evaluations are extremely time-consuming. Within the optimization process, gradients and Hessians were approximated by finite differences so that additional simulations for the respective modified parameter sets were required. Hence, a more efficient approach to computing gradients and Hessians is presented in this work. The method developed here is based on directional instead of partial derivatives. It is compared with the classical computations with respect to computation time. Firstly, molecular simulations are replaced by fit functions that define a functional dependence between specific physical observables and force field parameters. The goal of these simulated simulations is to assess the new methodology without much computational effort. Secondly, it is applied to real molecular simulations of the three chemical substances phosgene, methanol and ethylene oxide. It is shown that up to 75% of the simulations can be avoided using the new algorithm.

Liquid–liquid equilibria of dipropylene glycol dimethyl ether and water by molecular dynamics
(2011)

The Fraunhofer Institute for Algorithms and Scientific Computing (SCAI) has developed a software tool for the automated parameterization of force fields for molecular simulations using efficient gradient-based algorithms. This tool, combined with well-established simulation techniques, can quantitatively determine many physicochemical properties for given compounds.

GROW: A gradient-based optimization workflow for the automated development of molecular models
(2010)

Automated parameterization of intermolecular pair potentials using global optimization techniques
(2014)

In this work, different global optimization techniques are assessed for the automated development of molecular force fields, as used in molecular dynamics and Monte Carlo simulations. The quest of finding suitable force field parameters is treated as a mathematical minimization problem. Intricate problem characteristics such as extremely costly and even abortive simulations, noisy simulation results, and especially multiple local minima naturally lead to the use of sophisticated global optimization algorithms. Five diverse algorithms (pure random search, recursive random search, CMA-ES, differential evolution, and taboo search) are compared to our own tailor-made solution named CoSMoS. CoSMoS is an automated workflow. It models the parameters’ influence on the simulation observables to detect a globally optimal set of parameters. It is shown how and why this approach is superior to other algorithms. Applied to suitable test functions and simulations for phosgene, CoSMoS effectively reduces the number of required simulations and real time for the optimization task.

A central goal of molecular simulations is to predict physical or chemical properties such that costly and elaborate experiments can be minimized. The reliable generation of molecular models is a critical issue to do so. Hence, striving for semiautomated and fully automated parameterization of entire force fields for molecular simulations, the authors developed several modular program packages in recent years. The programs run with limited user interactions and can be executed in parallel on modern computer clusters. Various interlinked resolutions of molecular modeling are addressed: For intramolecular interactions, a force-field optimization package named Wolf2Pack has been developed that transfers knowledge gained from quantum mechanics to Newtonian-based molecular models. For intermolecular interactions, especially Lennard–Jones parameters, a modular optimization toolkit of programs and scripts has been created combining global and local optimization algorithms. Global optimization is performed by a tool named CoSMoS, while local optimization is done by the gradient-based optimization workflow named GROW or by a derivative-free method called SpaGrOW. The overall goal of all program packages is to realize an easy, efficient, and user-friendly development of reliable force-field parameters in a reasonable time. The various tools are needed and interlinked since different stages of the optimization process demand different courses of action. In this paper, the conception of all programs involved is presented and how they communicate with each other.

Ionic liquids are highly relevant for industrial applications as they stand out due to their special chemical and physical features, e.g. low vapor pressure, low melting point or extraordinary solution properties. The goal of this work is to study the capability of the three ionic liquids [C2MIM][NTf2], [C12MIM][NTf2] and [C2MIM][EtSO4] to diffuse through a POPC membrane (1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine). To achieve this, we used molecular simulation techniques, which on the one hand give insight into specific domains of the membrane and on the other hand compute partition coefficients and free energy profiles of solutes in lipid membranes, which cannot be measured by labor experiments. To be as accurate as possible we parameterized a new united atom force field for the ionic liquid of type 1-alkyl-3-methylimidazoliumethylsulfate [CnMIM][EtSO4] with n = 1,2,4,6,8. Like the other IL force field for [CnMIM][NTf2] (see Köddermann et al., ChemPhysChem 14, 3368–3374, 2013) used in this work, the new one was derived to reproduce experimental densities and self-diffusion coefficients. The new force field reproduces the experimental data extremely well. Using this force field, the influences of cation and anion exchanges as well as the variation of the chain length on the free energy could be analyzed. We performed umbrella-sampling to characterize the free energy profile of one ion pair, accompanied by a second one, in solution, at the membrane interface, and inside the membrane. In the outlook we present our intention to parameterize force fields in a systematic and user-friendly way. We will use the combination of two optimization toolkits, developed at SCAI: The global optimization toolkit CoSMoS and the local optimization techniques implemented in the software package GROW.

The paper describes methods for calculating chemical equilibria based on a constrained Gibbs free energy minimization. The methods allow the treatment of multicomponent systems with multiple phases, including gaseous phases, condensed phases, and stoichiometric phases. A special aspect is the detection and treatment of miscibility gaps. The underlying mathematical problem is described in detail together with the algorithmic approach for its solution. Results are presented for some test cases, including the computation of phase diagrams for ternary systems.