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Data for complex plasma–wall interactions require long-running and expensive computer simulations. Furthermore, the number of input parameters is large, which results in low coverage of the (physical) parameter space. Unpredictable occasions of outliers create a need to conduct the exploration of this multi-dimensional space using robust analysis tools. We restate the Gaussian process (GP) method as a Bayesian adaptive exploration method for establishing surrogate surfaces in the variables of interest. On this basis, we expand the analysis by the Student-t process (TP) method in order to improve the robustness of the result with respect to outliers. The most obvious difference between both methods shows up in the marginal likelihood for the hyperparameters of the covariance function, where the TP method features a broader marginal probability distribution in the presence of outliers. Eventually, we provide first investigations, with a mixture likelihood of two Gaussians within a Gaussian process ansatz for describing either outlier or non-outlier behavior. The parameters of the two Gaussians are set such that the mixture likelihood resembles the shape of a Student-t likelihood.

We present a new framework for the reconstruction of climate indices based on proxy data such as tree rings. The framework is based on the supervised learning method Gaussian Process Regression (GPR) and designed to preserve the amplitude of past climate variability and to adequately handle noise-contaminated proxies and variable proxy availability in time. To this end, the GPR is performed in a modified input space, termed embedding space. We test the new framework for the reconstruction of the Atlantic Multi-decadal Variability (AMV) in a controlled environment with pseudoproxies derived from coupled climate-model simulations. In this test environment, the GPR outperforms benchmark reconstructions based on multi-linear Principle Component Regression. On AMV-relevant timescales, i.e., multi-decadal timescales, the GPR is able to reconstruct the true magnitude of variability even if the proxies contain a non-climatic noise signal and become sparser back in time. Thus, we conclude that the embedded GPR framework is a highly promising tool for climate-index reconstructions.

Journal of plasma physics 88(5), 895880502 (2022). doi:10.1017/S0022377822000769, Published by Cambridge Univ. Press, London

We present a case study for Bayesian analysis and proper representation of distributions and dependence among parameters when calibrating process-oriented environmental models. A simple water quality model for the Elbe River (Germany) is referred to as an example, but the approach is applicable to a wide range of environmental models with time-series output. Model parameters are estimated by Bayesian inference via Markov Chain Monte Carlo (MCMC) sampling. While the best-fit solution matches usual least-squares model calibration (with a penalty term for excessive parameter values), the Bayesian approach has the advantage of yielding a joint probability distribution for parameters. This posterior distribution encompasses all possible parameter combinations that produce a simulation output that fits observed data within measurement and modeling uncertainty. Bayesian inference further permits the introduction of prior knowledge, e.g., positivity of certain parameters. The estimated distribution shows to which extent model parameters are controlled by observations through the process of inference, highlighting issues that cannot be settled unless more information becomes available. An interactive interface enables tracking for how ranges of parameter values that are consistent with observations change during the process of a step-by-step assignment of fixed parameter values. Based on an initial analysis of the posterior via an undirected Gaussian graphical model, a directed Bayesian network (BN) is constructed. The BN transparently conveys information on the interdependence of parameters after calibration. Finally, a strategy to reduce the number of expensive model runs in MCMC sampling for the presented purpose is introduced based on a newly developed variant of delayed acceptance sampling with a Gaussian process surrogate and linear dimensionality reduction to support function-valued outputs.

Dynamics of many classical physics systems are described in terms of Hamilton’s equations. Commonly, initial conditions are only imperfectly known. The associated volume in phase space is preserved over time due to the symplecticity of the Hamiltonian flow. Here we study the propagation of uncertain initial conditions through dynamical systems using symplectic surrogate models of Hamiltonian flow maps. This allows fast sensitivity analysis with respect to the distribution of initial conditions and an estimation of local Lyapunov exponents (LLE) that give insight into local predictability of a dynamical system. In Hamiltonian systems, LLEs permit a distinction between regular and chaotic orbits. Combined with Bayesian methods we provide a statistical analysis of local stability and sensitivity in phase space for Hamiltonian systems. The intended application is the early classification of regular and chaotic orbits of fusion alpha particles in stellarator reactors. The degree of stochastization during a given time period is used as an estimate for the probability that orbits of a specific region in phase space are lost at the plasma boundary. Thus, the approach offers a promising way to accelerate the computation of fusion alpha particle losses.

We present an approach to enhance the performance and flexibility of the Bayesian inference of model parameters based on observations of the measured data. Going beyond the usual surrogate-enhanced Monte-Carlo or optimization methods that focus on a scalar loss, we place emphasis on a function-valued output of a formally infinite dimension. For this purpose, the surrogate models are built on a combination of linear dimensionality reduction in an adaptive basis of principal components and Gaussian process regression for the map between reduced feature spaces. Since the decoded surrogate provides the full model output rather than only the loss, it is re-usable for multiple calibration measurements as well as different loss metrics and, consequently, allows for flexible marginalization over such quantities and applications to Bayesian hierarchical models. We evaluate the method’s performance based on a case study of a toy model and a simple riverine diatom model for the Elbe river. As input data, this model uses six tunable scalar parameters as well as silica concentrations in the upper reach of the river together with the continuous time-series of temperature, radiation, and river discharge over a specific year. The output consists of continuous time-series data that are calibrated against corresponding measurements from the Geesthacht Weir station at the Elbe river. For this study, only two scalar inputs were considered together with a function-valued output and compared to an existing model calibration using direct simulation runs without a surrogate.

The analysis of defects and defect dynamics in crystalline materials is important for fundamental science and for a wide range of applied engineering. With increasing system size the analysis of molecular-dynamics simulation data becomes non-trivial. Here, we present a workflow for semi-automatic identification and classification of defects in crystalline structures, combining a new approach for defect description with several already existing open-source software packages. Our approach addresses the key challenges posed by the often relatively tiny volume fraction of the modified parts of the sample, thermal motion and the presence of potentially unforeseen atomic configurations (defect types) after irradiation. The local environment of any atom is converted into a rotation-invariant descriptive vector (‘fingerprint’), which can be compared to known defect types and also yields a distance metric suited for classification. Vectors which cannot be associated to known structures indicate new types of defects. As proof-of-concept we apply our method on an iron sample to analyze the defects caused by a collision cascade induced by a 10 keV primary-knock-on-atom. The obtained results are in good agreement with reported literature values. Program summary Program Title: Fingerprinting and Visualization Analyzer of Defects (FaVAD). CPC Library link to program files: https://doi.org/10.17632/bmv9kxkzg3.1 Developer’s repository link: http://gitlab.mpcdf.mpg.de/NMPP/favad.git Licensing provisions: GPLv3. Programming language: Python 3, Fortran 90, and C ++ . Nature of problem: The analysis of damage and damage evolution in crystalline materials is important for fundamental science and for a wide range of applied engineering. Defects in materials on an atomic level are commonly analyzed by Wigner–Seitz or topology Voronoi tessellation based methods. However, these approaches exhibit specific shortcoming, especially at elevated sample temperatures. In order to improve upon that, a more robust and quantifiable identification and classification approach of known as well as of unpredicted defect structures is desirable. Solution method: A fingerprint-like method is proposed to analyze in detail the damage in a material augmented with a probabilistic interpretation. It is based on the calculation of a descriptor vector for each atom in the sample. These vectors represent in a compact form the individual environments of the atoms. For standard types of defects (i.e. interstitial atoms) the corresponding descriptor vectors can be precomputed and used for rapid classification. Unexpected or less common defect types can be identified by applying a principal component analysis to the descriptor vectors. Vacancies in the material are identified by computing the radii of the largest empty spheres which can be embedded into the sample, followed by a thresholding process. This new method is easy to use and requires only modest computational resources. Finally, the classified defects are visualized using the open source software VisIt. Additional comments including restrictions and unusual features: The descriptor vectors are computed using the command line interface of QUIP with the Gaussian Approximation potential (GAP) package. The analysis of the sample is done using Python scripts which make extensive use of the NumPy package. A modified KDTree2 code is employed to calculate the location of single vacancies and voids. The program VisIt is used for the visualization of the classified point defects in the sample. We provide a Dockerfile for automatically creating a portable Docker container which installs all the programs together with a Python script to analyze as an example a damaged iron molecular dynamics sample. A shell script to install the programs locally in a Linux-based server or desktop environment is included also.

A Gaussian-process surrogate model based on already acquired data is employed to approximate an unknown target surface. In order to optimally locate the next function evaluations in parameter space a whole variety of utility functions are at one’s disposal. However, good choice of a specific utility or a certain combination of them prepares the fastest way to determine a best surrogate surface or its extremum for lowest amount of additional data possible. In this paper, we propose to consider the global (integrated) variance as an utility function, i.e., to integrate the variance of the surrogate over a finite volume in parameter space. It turns out that this utility not only complements the tool set for fine tuning investigations in a region of interest but expedites the optimization procedure in toto.

The exposure of metal surfaces to energetic particles originating for example from a plasma can lead to the formation of blisters. These features are caused by hydrogen-filled gas bubbles near the surface and can influence the diffusion and retention of hydrogen atoms in the bulk material during plasma exposure. No comprehensive theory of the underlying processes governing these effects has been developed so far. A major constraint to the investigation of blisters has been the effort that is required to identify and characterize a statistically relevant ensemble of blisters. Therefore, the analysis of a limited number of blisters in isolated observations before and after plasma exposure has been state of the art so far. In this article, a framework is presented that allows to automatically analyze blisters in image data with the aid of convolutional neural networks (CNNs) which are trained with artificial training data. It thereby resolves the limitations of existing analysis methods that prevented the large-scale and in-situ investigation of blistering up to now. Automated routines are used to identify and localize blisters, estimate important blister parameters such as size, and track individual blisters over time. Apart from discussing the principles underlying this approach and describing the implementation of the framework in detail, this article also presents first results acquired with a molybdenum sample that was exposed to a cold deuterium plasma. The results indicate a blister identification accuracy of the framework of more than 80% within the considered parameter ranges and overall demonstrate the feasibility of the approach in general.

Beryllium film deposition was studied with cavity samples in remote areas of the inner and outer JET divertor and below divertor tile 5 during the 2011-2012 campaign with the ITER-like wall. Predominantly beryllium films were formed inside the cavities with some additional carbon, the ratio Be/C was > 2. These deposited layers had high D/(Be+C) ratios of about 0.3. The formation of these films is mainly due to sticking of beryllium-containing particles with low sticking coefficients < 0.5. The observed surface loss probabilities depend on the position in the divertor. The particles responsible for film deposition originated from the location of in the divertor strike points. (C) 2016 Elsevier Ltd. For complete list of authors see http://dx.doi.org/10.1016/j.nme.2016.12.005

An analysis tool using Adaptive Kernel to solve an ill-posed inverse problem for a 2D model space is introduced. It is applicable for linear and non-linear forward models, for example in tomography and image reconstruction. While an optimisation based on a Gaussian Approximation is possible, it becomes intractable for more than some hundred kernel functions. This is because the determinant of the Hessian of the system has be evaluated. The SVD typically used for 1D problems fails with increasing problem size. Alternatively Stochastic Trace Estimation can be used, giving a reasonable approximation. An alternative to searching for the MAP solution is to integrate using Marcov Chain Monte Carlo without the need to determine the determinant of the Hessian. This also allows to treat problems where a linear approximation is not justified.

As key building blocks for modern data processing and analysis methods—ranging from AI, ML and UQ to model comparison, density estimation and parameter estimation—Bayesian inference and entropic concepts are in the center of this rapidly growing research area. [...]

The key input quantity to climate modelling and weather forecasts is the solar beam irradiance, i.e., the primary amount of energy provided by the sun. Despite its importance the absolute accuracy of the measurements are limited—which not only affects the modelling but also ground truth tests of satellite observations. Here we focus on the problem of improving instrument calibration based on dedicated measurements. A Bayesian approach reveals that the standard approach results in inferior results. An alternative approach method based on monomial based selection of regression functions, combined with model selection is shown to yield superior estimations for a wide range of conditions. The approach is illustrated on selected data and possible further enhancements are outlined.

We present an approach to construct structure-preserving emulators for Hamiltonian flow maps and Poincaré maps based directly on orbit data. Intended applications are in moderate-dimensional systems, in particular, long-term tracing of fast charged particles in accelerators and magnetic plasma confinement configurations. The method is based on multi-output Gaussian process (GP) regression on scattered training data. To obtain long-term stability, the symplectic property is enforced via the choice of the matrix-valued covariance function. Based on earlier work on spline interpolation, we observe derivatives of the generating function of a canonical transformation. A product kernel produces an accurate implicit method, whereas a sum kernel results in a fast explicit method from this approach. Both are related to symplectic Euler methods in terms of numerical integration but fulfill a complementary purpose. The developed methods are first tested on the pendulum and the Hénon–Heiles system and results compared to spectral regression of the flow map with orthogonal polynomials. Chaotic behavior is studied on the standard map. Finally, the application to magnetic field line tracing in a perturbed tokamak configuration is demonstrated. As an additional feature, in the limit of small mapping times, the Hamiltonian function can be identified with a part of the generating function and thereby learned from observed time-series data of the system’s evolution. For implicit GP methods, we demonstrate regression performance comparable to spectral bases and artificial neural networks for symplectic flow maps, applicability to Poincaré maps, and correct representation of chaotic diffusion as well as a substantial increase in performance for learning the Hamiltonian function compared to existing approaches.

The simulation of complex physics models may lead to enormous computer running times. Since the simulations are expensive it is necessary to exploit the computational budget in the best possible manner. If for a few input parameter settings an output data set has been acquired, one could be interested in taking these data as a basis for finding an extremum and possibly an input parameter set for further computer simulations to determine it—a task which belongs to the realm of global optimization. Within the Bayesian framework we utilize Gaussian processes for the creation of a surrogate model function adjusted self-consistently via hyperparameters to represent the data. Although the probability distribution of the hyperparameters may be widely spread over phase space, we make the assumption that only the use of their expectation values is sufficient. While this shortcut facilitates a quickly accessible surrogate, it is somewhat justified by the fact that we are not interested in a full representation of the model by the surrogate but to reveal its maximum. To accomplish this the surrogate is fed to a utility function whose extremum determines the new parameter set for the next data point to obtain. Moreover, we propose to alternate between two utility functions—expected improvement and maximum variance—in order to avoid the drawbacks of each. Subsequent data points are drawn from the model function until the procedure either remains in the points found or the surrogate model does not change with the iteration. The procedure is applied to mock data in one and two dimensions in order to demonstrate proof of principle of the proposed approach.

Based on geometric invariance properties, we derive an explicit prior distribution for the parameters of multivariate linear regression problems in the absence of further prior information. The problem is formulated as a rotationally-invariant distribution of \(L\)-dimensional hyperplanes in \(N\) dimensions, and the associated system of partial differential equations is solved. The derived prior distribution generalizes the already known special cases, e.g., 2D plane in three dimensions.

A method to measure deuterium permeation through tungsten near room temperature under plasma loading is presented. The permeating deuterium is accumulated in a getter layer of zirconium, titanium or erbium, respectively, on the unexposed side of the sample. Subsequently, the amount of deuterium in the getter is measured ex-situ using nuclear reaction analysis. A cover layer system on the getter prevents direct loading of the getter with deuterium from the gas phase during plasma loading. In addition, it enables the distinction of deuterium in the getter and at the cover surface. The method appears promising to add additional permeation measurement capabilities to deuterium retention experiments, also in other plasma devices, without the need for a complex in-situ permeation measurement setup. Keywords: Deuterium, Plasma, Permeation, Tungsten, Getter, Ion-beam

Tungsten has established itself as the most suitable plasma-facing material for long-term operation in future magnetic-confinement fusion devices, but its properties make it a poor structural material and complicate the manufacturing of complex components. Recent advances in additive-manufacturing (AM) technology have begun to make the production of tungsten components with complex geometry more feasible. The design freedom afforded by AM could be leveraged to produce more resilient plasma-facing components (PFCs). A methodology to optimize the material distribution of composite PFCs was developed to reduce the maximum thermal stress caused by high heat fluxes. Its use was demonstrated for copper-infiltrated AM tungsten (Walt;subagt;AMalt;/subagt;/Cu) structures. Stress reductions of 50 - 85% are predicted under nominal load conditions. Optimized designs also reduce stress over a wide range of off-nominal conditions. The resulting optimized structures are composed of a spatially heterogeneous distribution of W and Cu comprising a broad range of composite mixtures. A sample manufacturable component was modelled based on optimization results.

Total reflection coefficients and break-up patterns for incident atomic carbon, C2, CH, CH2, CH3, CH4, C2H, C2H2, C2H3, C2H4, C2H5 and C2H6 from an amorphous hydrocarbon surface with a H:C ratio of 0.66 are calculated using the HCParcas molecular dynamics code with an empirical Brenner potential. Incident molecules are separately equilibrated at incident energies ranging from thermal up to the maximum energies at which the molecules can be structurally stabilized to mimic the conditions in an equilibrium plasma.

We present the results of erosion measurements on actively cooled tungsten samples at quasi-constant surface temperature conditions performed in the high heat flux facility GLADIS. The samples were exposed to a H beam at a central power density of 10 MW/m 2 up to a fluence of 10 26 m −2 . We observe a weak temperature dependence of the erosion yield. The data are compared with similar data obtained from loading with a H beam with He admixture. Both datasets are analysed in a probabilistic approach. We obtain activation energies of 0.04 eV and 0.06 eV for the cases with and without He, respectively.

Up to now, blister formation on rough or technical tungsten surfaces exposed to hydrogen isotope plasma was believed to be completely suppressed. The few dedicated experiments on this issue that can be found in literature appear to support that claim. Using a novel technique of 3D difference imaging of tungsten surfaces, we now demonstrate that roughness introduced by chemical etching, i.e. without the associated mechanical deformation layer introduced by grinding, only moderately reduces blistering. A technical surface with comparable roughness produced by precision grinding (R a ≤ 1.6 µm) led to a strong reduction in blister size and density, but blisters were found nevertheless. In this article we give a detailed description of the investigated rough W surfaces and present a statistical evaluation of blistering on these surfaces after exposure to a low-temperature deuterium plasma.

From the basics to the forefront of modern research, this book presents all aspects of probability theory, statistics and data analysis from a Bayesian perspective for physicists and engineers. The book presents the roots, applications and numerical implementation of probability theory, and covers advanced topics such as maximum entropy distributions, stochastic processes, parameter estimation, model selection, hypothesis testing and experimental design. In addition, it explores state-of-the art numerical techniques required to solve demanding real-world problems. The book is ideal for students and researchers in physical sciences and engineering.

Integrated Data Analysis (IDA) offers a unified way of combining information relevant to fusion experiments. Thereby, IDA meets with typical issues arising in fusion data analysis. In IDA, all information is consistently formulated as probability density functions quantifying uncertainties in the analysis within the Bayesian probability theory. For a single diagnostic, IDA allows the identification of faulty measurements and improvements in the setup. For a set of diagnostics, IDA gives joint error distributions allowing the comparison and integration of different diagnostics results. Validation of physics models can be performed by model comparison techniques. Typical data analysis applications benefit from IDA capabilities of nonlinear error propagation, the inclusion of systematic effects and the comparison of different physics models. Applications range from outlier detection, background discrimination, model assessment and design of diagnostics. In order to cope with next step fusion device requirements, appropriate techniques are explored for fast analysis applications.

Ellipsometry is a unique technique of great sensitivity for in situ non‐destructive characterization of surfaces utilizing the change in the state of polarization of a light‐wave. It is extensively used in the semi‐conductor industry. To relate ellipsometric measurements to surface properties (as eg layer thickness changes in the range of nm or chemical composition), Bayesian probability theory is used. The parameter estimation process is complicated by the incomplete phase information of the measured data. Examples of 3‐D surface reconstructions of samples after ion bombardment demonstrate the tremendous information gain due to the Bayesian analysis.

Tungsten sputtering and carbon layer growth by carbon ion bombardment has been investigated by experiments with thin W layers. Preparation of tungsten layers by magnetron deposition allows one to control the surface roughness by choosing appropriate substrates. The fluence dependent elemental composition of the bombarded surface has been studied in situ with ion beam analysis allowing separate measurements of the areal densities of tungsten and carbon atoms. In contrast to weight-loss measurements, this approach is much less affected by nonuniformities of the incident flux and therefore allows one to determine the dependency of the principal physics processes on input parameters with much higher accuracy. After bombardment, ex situ scanning electron microscopy has been used for the qualitative understanding of the evolution of the surface topography with increasing fluence. The experiments clearly show the influence of surface roughness, leading to increased tungsten sputter yields and strongly reduced ca...

We present a new method using Bayesian probability theory and neural networks for the evaluation of speckle interference patterns for an automated analysis of deformation and erosion measurements. The method is applied to the fringe pattern reconstruction of speckle measurements with a Twyman-Green interferometer. Given a binary speckle image, the method returns the fringe pattern without noise, thus removing the need for smoothing and allowing a straightforward unwrapping procedure and determination of the surface shape. Because no parameters have to be adjusted, the method is especially suited for continuous and automated monitoring of surface changes.

Neural Networks are famous for their advantageous flexibility for problems when there is insufficient knowledge to set up a proper model. On the other hand this flexibility can cause over‐fitting and can hamper the generalization properties of neural networks. Many approaches to regularize NN have been suggested but most of them based on ad‐hoc arguments. Employing the principle of transformation invariance we derive a general prior in accordance with the Bayesian probability theory for a class of feedforward networks. Optimal networks are determined by Bayesian model comparison verifying the applicability of this approach.

Neural networks (NN) are famous for their advantageous flexibility for problems when there is insufficient knowledge to set up a proper model. On the other hand, this flexibility can cause overfitting and can hamper the generalization of neural networks. Many approaches to regularizing NN have been suggested but most of them are based on ad hoc arguments. Employing the principle of transformation invariance, we derive a general prior in accordance with the Bayesian probability theory for feed-forward networks. An optimal network is determined by Bayesian model comparison, verifying the applicability of this approach. Additionally the prior presented affords cell pruning.

Ion beam diagnostics are routinely used for quantitative analysis of the surface composition of mixture materials up to a depth of a few μm. Unfortunately, advantageous properties of the diagnostics, like high depth resolution in combination with a large penetration depth, no destruction of the surface, high sensitivity for large as well as for small atomic numbers, and high sensitivity are mutually exclusive. Among other things, this is due to the ill-conditioned inverse problem of reconstructing depth distributions of the composition elements. Robust results for depth distributions are obtained with adaptive methods in the framework of Bayesian probability theory. The method of adaptive kernels allows for distributions which contain only the significant information of the data while noise fitting is avoided. This is achieved by adaptively reducing the degrees of freedom supporting the distribution. As applications for ion beam diagnostics Rutherford backscattering spectroscopy and particle induced X-ray emission are shown.

A new simulation program for Rutherford backscattering spec- troscopy (RBS) together with an adaptive kernel method in the Bayesian proba- bility theory framework was applied to the analysis of RBS data. Reconstructed depth profiles are free from noise-induced ringing, even for strongly overlapping RBS peaks. This has been achieved by the use of the adaptive kernel method, which generates the least informative depth profile according to the data and in addition allows one to calculate the uncertainty of the obtained depth profiles. The method is applied to erosion measurements of carbon samples.

Scanning electron microscopy images and energy-dispersive x-ray spectra were recorded for a total of about 4×104 dust particles collected on the same position within the vacuum vessel via silicon wafers during four consecutive full-tungsten first wall campaigns of ASDEX Upgrade between 2007 and 2009. By careful analysis of the elemental composition and shape of the sampled particles, seven statistically relevant classes of dust were identified. The particle flux and area coverage of each class were normalized to the total plasma duration of each sampling period, revealing a high sensitivity of the dust composition to device conditioning. According to the present results, particles produced by arcing on divertor tiles with delaminated coatings were transported to the main chamber first wall.

Carbon removal in the so-called remote areas, where no energetic species can reach the surface, is investigated with low-temperature glow discharges in pure oxygen. Plasma-deposited amorphous hydrogenated carbon thin films are used as a model system for redeposited films. Erosion measurements are performed on flat substrates as well as on two different three-dimensional (3D) test structures. One design consists of 19 mm deep gaps with widths ranging from 0.5 to 4 mm to simulate ITER tile gaps. A second design consists of a flat box-like structure where particles can enter only through a narrow slit. Measurements are performed for substrate temperatures ranging from 290 to 580 K. Erosion rates follow an Arrhenius-type dependence on substrate temperature with an effective activation energy of 0.25 eV. While at room temperature the surface loss probability of the dominant eroding species is 50% it becomes smaller at elevated temperatures. At elevated temperatures, the films are not only removed faster but simultaneously erosion penetrates deeper into the gaps.

Within the Bayesian framework, we utilize Gaussian processes for parametric studies of long running computer codes. Since the simulations are expensive, it is necessary to exploit the computational budget in the best possible manner. Employing the sum over variances —being indicators for the quality of the fit—as the utility function, we establish an optimized and automated sequential parameter selection procedure. However, it is also often desirable to utilize the parallel running capabilities of present computer technology and abandon the sequential parameter selection for a faster overall turn-around time (wall-clock time). This paper proposes to achieve this by marginalizing over the expected outcomes at optimized test points in order to set up a pool of starting values for batch execution. For a one-dimensional test case, the numerical results are validated with the analytical solution. Eventually, a systematic convergence study demonstrates the advantage of the optimized approach over randomly chosen parameter settings.