Surrogate Surface Research Articles

Hybrid Pore-Scale Adsorption Model for CO2 and CH4 Storage in Shale

Making reliable estimates of gas adsorption in shale remains a challenge because the variability in their mineralogy and thermal maturity results in a broad distribution of pore-scale properties, including size, morphology and surface chemistry. Here, we demonstrate the development and application of a hybrid pore-scale model that uses surrogate surfaces to describe supercritical gas adsorption in shale. The model is based on the lattice density functional theory (DFT) and considers both slits and cylindrical pores to mimic the texture of shale. Inorganic and organic surfaces associated with these pores are accounted for by using two distinct adsorbate–adsorbent interaction energies. The model is parametrized upon calibration against experimental adsorption data acquired on adsorbents featuring either pure clay or pure carbon surfaces. Therefore, in its application to shale, the hybrid lattice DFT model only requires knowledge of the shale-specific organic and clay content. We verify the reliability of the model predictions by comparison against high-pressure CO2 and CH4 adsorption isotherms measured at 40 °C in the pressure range 0.01–30 MPa on four samples from three distinct plays, namely, the Bowland (UK), Longmaxi (China), and Marcellus shale (USA). Because it uses only the relevant pore-scale properties, the proposed model can be applied to the analysis of other shales, minimizing the heavy experimental burden associated with high-pressure experiments. Moreover, the proposed development has general applicability meaning that the hybrid lattice DFT can be used for the characterization of any adsorbent featuring morphologically and chemically heterogeneous surfaces.

Global optimization on non-convex two-way interaction truncated linear multivariate adaptive regression splines using mixed integer quadratic programming

Multivariate adaptive regression splines (MARS) has been adopted as a popular surrogate function to model the unknown relationships between input and output variables in complex systems. Searching optimal solutions on the complicated surrogate response surface of MARS serves as important tasks in various applications. In this paper, we present an efficient and effective approach to find a global optimal value of MARS models that incorporate two-way interaction terms which are products of truncated linear univariate functions (TITL-MARS). Specifically, with a MARS model consisting of linear and quadratic structures, we can reformulate the optimization problem on TITL-MARS into a mixed integer quadratic programming problem (TITL-MARS-OPT), which can be further solved in a more principled way. To illustrate the effectiveness of our proposed approach TITL-MARS-OPT, we come up with a genetic algorithm and a gradient descent algorithm to solve the original version of TITL-MARS, and we compared the performance of the proposed approach with the benchmark algorithms. Numerical experiments are conducted on a spectrum of examples with different levels of complexity, including 6 existing models and a real world application in the wind farm optimization. Our proposed approach can find a global optimal successfully and efficiently while the comparison algorithms fail to find a global optimal solution. In the end, Python code for TITL-MARS and TITL-MARS-OPT is made available on GitHub ( https://github.com/JuXinglong/TITL-MARS-OPT).

Optimisation of heat recovery from low-enthalpy aquifers with geological uncertainty using surrogate response surfaces and simple search algorithms

Optimisation of doublet well spacing in low-enthalpy geothermal systems is addressed by defining a novel objective function that is based on the Coefficient of Performance (CoP) and energy sweep efficiency. The definition of objective function that separates performance-based criteria from economic factors, allows us to better observe the effects of heterogeneity on optimisation. A checkerboard pattern of two doublets (two injection wells diagonally placed and two production wells diagonally placed over corners of a rectangle) is considered for a range of homogeneous to heterogeneous (spatially correlated and fluvial) synthetic low enthalpy reservoirs. Optimal length and width of this rectangle are sought in order to (a) maximise heat recovery from a conventionally-chosen licence area around the rectangular domain, (b) minimise heat recovery from outside this licence area, and (c) maximise CoP. We define fixed (15 years and 30 years) and varying life times of operation (between 15 and 30 years). For optimisation, in addition to a simple-search procedure of optimisation across a mesh of simulation nodes, we also utilise a surrogate response surface model to computationally solve the optimisation problem. Our results consistently show that for a fixed life time of 15 years and a discharge rate of 250 m3/hr, 400 m is the optimal well/doublet spacing. Increasing the life time and the discharge rate will increase the optimal well/doublet spacing. The results show while CoP is sensitive to the heterogeneity, adding energy sweep to the objective function makes the distances found for the homogeneous cases also consistent solutions for the heterogeneous cases.

Computational modeling of seizure spread on a cortical surface

In the field of computational epilepsy, neural field models helped to understand some large-scale features of seizure dynamics. These insights however remain on general levels, without translation to the clinical settings via personalization of the model with the patient-specific structure. In particular, a link was suggested between epileptic seizures spreading across the cortical surface and the so-called theta-alpha activity (TAA) pattern seen on intracranial electrographic signals, yet this link was not demonstrated on a patient-specific level. Here we present a single patient computational study linking the seizure spreading across the patient-specific cortical surface with a specific instance of the TAA pattern recorded in the patient. Using the realistic geometry of the cortical surface we perform the simulations of seizure dynamics in The Virtual Brain platform, and we show that the simulated electrographic signals qualitatively agree with the recorded signals. Furthermore, the comparison with the simulations performed on surrogate surfaces reveals that the best quantitative fit is obtained for the real surface. The work illustrates how the patient-specific cortical geometry can be utilized in The Virtual Brain for personalized model building, and the importance of such approach.

An efficient hybrid reliability analysis method based on active learning Kriging model and multimodal‐optimization‐based importance sampling

AbstractA novel method is proposed, which aims to solve rare‐event hybrid reliability problems with random and interval variables, where the performance function has various failure zones. It combines the active learning Kriging (ALK) model with importance sampling (IS) and evolutionary multimodal‐based multiobjective optimization (EMO‐MMO). The surrogate limit state surfaces (LSS) for the upper and lower failure probability bounds are respectively defined considering the Kriging variance. Failure candidate solutions located in different failure regions are generated by the EMO‐MMO method. Subsequently, all the most probable failure points (MPPs) are identified from those candidate solutions. The IS samples are simulated around the MPPs using the MPP‐based IS method. The IS samples located in unimportant regions are removed in order to improve the efficiency of approximating the surrogate LSSs. The optimal training points are selected from the truncated IS samples to update the Kriging model. After several training iterations, the surrogate LSSs are convergent. Ultimately, a reliable and unbiased estimation of the upper and lower failure probability bounds is provided. The performance of the ALK‐EMO‐IS‐HRA approach is verified through five application examples.

Evaluating the Design and Repeatability of a Novel Device to Measure Friction of Mechanical Surrogate Skins in Contact with Cotton Textiles

The ability to measure the level of friction between the human skin and a given textile is critical across fashion and textiles sectors, not least for the development of sporting and protective clothing. A portable custom-made device capable of measuring friction during the skin-textile interaction across often difficult or impossible to investigate body regions with objective repeatability has been established. The friction between a pre-shrunk 100% cotton textile and a quantity of four control surfaces (transparent and patterned polycarbonate plastic, and silicon and lorica surrogate skin) was measured three times per day across five consecutive days. The results clearly demonstrated that the novel friction test device had an excellent repeatability of 0.94 and 0.93 intraclass corelation coefficient for static and dynamic friction coefficient measurement, respectively. The silicon surrogate skin control surface produced the highest friction coefficient, while the pattered polycarbonate plate demonstrated the lowest friction coefficient, suggesting that the physical features of the control surface material influenced the recorded coefficient of friction. It was also revealed that the relationship between the static and dynamic friction coefficient is dependent on the surface material.

Geometry Optimization in Internal Coordinates Based on Gaussian Process Regression: Comparison of Two Approaches.

Geometry optimization based on Gaussian process regression (GPR) was extended to internal coordinates. We used delocalized internal coordinates composed of distances and several types of angles and compared two methods of including them. In both cases, the GPR surrogate surface is trained on geometries in internal coordinates. In one case, it predicts the gradient in Cartesian coordinates and in the other, in internal coordinates. We tested both methods on a set of 30 small molecules and one larger Rh complex taken from the study of a catalytic mechanism. The former method is slightly more efficient, while the latter method is somewhat more robust. Both methods reduce the number of required optimization steps compared to GPR in Cartesian coordinates or the standard L-BFGS optimizer. We found it advantageous to use automatically adjusted hyperparameters to optimize them.

Application of an automated machine learning-genetic algorithm (AutoML-GA) coupled with computational fluid dynamics simulations for rapid engine design optimization

In recent years, the use of machine learning-based surrogate models for computational fluid dynamics (CFD) simulations has emerged as a promising technique for reducing the computational cost associated with engine design optimization. However, such methods still suffer from drawbacks. One main disadvantage is that the default machine learning (ML) hyperparameters are often severely suboptimal for a given problem. This has often been addressed by manually trying out different hyperparameter settings, but this solution is ineffective in case of a high-dimensional hyperparameter space. Besides this problem, the amount of data needed for training is also not known a priori. In response to these issues that need to be addressed, the present work describes and validates an automated active learning approach, AutoML-GA, for surrogate-based optimization of internal combustion engines. In this approach, a Bayesian optimization technique is used to find the best machine learning hyperparameters based on an initial dataset obtained from a small number of CFD simulations. Subsequently, a genetic algorithm is employed to locate the design optimum on the ML surrogate surface. In the vicinity of the design optimum, the solution is refined by repeatedly running CFD simulations at the projected optima and adding the newly obtained data to the training dataset. It is demonstrated that AutoML-GA leads to a better optimum with a lower number of CFD simulations, compared to the use of default hyperparameters. The proposed framework offers the advantage of being a more hands-off approach that can be readily utilized by researchers and engineers in industry who do not have extensive machine learning expertise.

Heterogeneity in Human Induced Pluripotent Stem Cell–derived Alveolar Epithelial Type II Cells Revealed with ABCA3/SFTPC Reporters

Alveolar epithelial type 2 cells (AEC2s), the facultative progenitors of lung alveoli, are typically identified through the use of the canonical markers, SFTPC and ABCA3. Self-renewing AEC2-like cells have been generated from human induced pluripotent stem cells (iPSCs) through the use of knock-in SFTPC fluorochrome reporters. However, developmentally, SFTPC expression onset begins in the fetal distal lung bud tip and thus is not specific to mature AEC2s. Furthermore, SFTPC reporters appear to identify only those iPSC-derived AEC2s (iAEC2s) expressing the highest SFTPC levels. Here, we generate an ABCA3 knock-in GFP fusion reporter (ABCA3:GFP) that enables the purification of iAEC2s while allowing visualization of lamellar bodies, organelles associated with AEC2 maturation. Using an SFTPCtdTomato and ABCA3:GFP bifluorescent line for invitro distal lung-directed differentiation, we observe later onset of ABCA3:GFP expression and broader identification of the subsequently emerging iAEC2 population based on ABCA3:GFP expression compared with SFTPCtdTomato expression. Comparing ABCA3:GFP/SFTPCtdTomato double-positive with ABCA3:GFP single-positive (SP) cells by RNA sequencing and functional studies reveals iAEC2 cellular heterogeneity with both populations functionally processing surfactant proteins but the SP cells exhibiting faster growth kinetics, increased clonogenicity, increased expression of progenitor markers, lower levels of SFTPC expression, and lower levels of AEC2 maturation markers. Over time, we observe that each population (double-positive and SP) gives rise to the other and each can serve as the parents of indefinitely self-renewing iAEC2 progeny. Our results indicate that iAEC2s are a heterogeneous population of cells with differing proliferation versus maturation properties, the majority of which can be tracked and purified using the ABCA3:GFP reporter or surrogate cell surface proteins, such as SLC34A2 and CPM.

Deformation of an airfoil-shaped brain surrogate under shock wave loading

Improvised explosive devices (IEDs), during military operations, has increased the incidence of blast-induced traumatic brain injuries (bTBI). The shock wave is created following detonation of the IED. This shock wave propagates through the atmosphere and may cause bTBI. As a result, bTBI research has gained increased attention since this injury's mechanism is not thoroughly understood. To develop better protection and treatment against bTBI, further studies of soft material (e.g. brain and brain surrogate) deformation due to shock wave exposure are essential. However, the dynamic mechanical behavior of soft materials, subjected to high strain rates from shock wave exposure, remains unknown. Thus, an experimental approach was applied to study the interaction between the shock wave and an unconfined brain surrogate fabricated from a biomaterial (i.e. polydimethylsiloxane (PDMS)). The 1:70 ratio of curing agent-to-base determined the stiffness of the PDMS (Sylgard 184, Dow Corning Corporation). A stretched NACA 2414 (upper airfoil surface) geometry was utilized to resemble the shape of a porcine brain. Digital image correlation (DIC) technique was applied to measure the deformation on the brain surrogate's surface following shock wave exposure. A shock tube was utilized to create the shock wave and pressure transducers measured the pressure in the vicinity of the brain surrogate. A transient structural analysis using ANSYS Workbench was performed to predict the elastic modulus of 1:70 airfoil-shaped PDMS, at a strain rate on the order of 6 × 103 s−1. Both compression and protrusion of the PDMS surface were found due to the shock wave exposure. Negative pressure was found in a semi-ring area, which was the cause of protrusion. Oscillation of the brain surrogate, due to the shock wave loading, was found. The frequency of oscillation does not depend on the geometry. This work will add to the limited data describing the dynamic behavior of soft materials due to shock wave loading.

Direct Measurement of Mercury Deposition at Rural and Suburban Sites in Washington State, USA

Because of mercury’s (Hg) capacity for long-range transport in the atmosphere, and its tendency to bioaccumulate in aquatic biota, there is a critical need to measure spatial and temporal patterns of Hg atmospheric deposition. Dry deposition of Hg is commonly calculated as the product of a measured atmospheric concentration and an assumed deposition velocity. An alternative is to directly assess Hg deposition via accumulation on surrogate surfaces. Using a direct measurement approach, this study quantified Hg deposition at a rural site (Pullman) and suburban site (Puyallup) in Washington State using simple, low-cost equipment. Dry deposition was measured using an aerodynamic “wet sampler” consisting of a Teflon plate, 35 cm in diameter, holding a thin layer (2.5 mm) of recirculating acidic aqueous receiving solution. In addition, wet Hg deposition was measured using a borosilicate glass funnel with a 20-cm-diameter opening and a 1 L Teflon sampling bottle. Hg deposition was estimated based on changes in total Hg in the aqueous phase of the samplers. Dry Hg deposition was 2.4 ± 1.4 ng/m2·h (average plus/minus standard deviation; n = 4) in Pullman and 1.3 ± 0.3 ng/m2·h (n = 6) in Puyallup. Wet Hg deposition was 7.0 ± 4.8 ng/m2·h (n = 4) in Pullman and 1.1 ± 0.2 ng/m2·h (n = 3) in Puyallup. Relatively high rates of Hg deposition in Pullman were attributed to regional agricultural activities that enhance mercury re-emission and deposition including agricultural harvesting and field burning. Hg concentration in precipitation negatively correlated with precipitation depth, indicating that Hg was scavenged from the atmosphere during the beginning of storm events. Because of their relative simplicity and robustness, direct measurement approaches such as those described in this study are useful in assessing Hg deposition, and for comparing results to less direct estimates and model estimates of Hg deposition.

Optimization of the Performance of Marine Diesel Engines to Minimize the Formation of SOx Emissions

Optimization procedures are required to minimize the amount of fuel consumption and exhaust emissions from marine engines. This study discusses the procedures to optimize the performance of any marine engine implemented in a 0D/1D numerical model in order to achieve lower values of exhaust emissions. From that point, an extension of previous simulation researches is presented to calculate the amount of SOx emissions from two marine diesel engines along their load diagrams based on the percentage of sulfur in the marine fuel used. The variations of SOx emissions are computed in g/kW·h and in parts per million (ppm) as functions of the optimized parameters: brake specific fuel consumption and the amount of air-fuel ratio respectively. Then, a surrogate model-based response surface methodology is used to generate polynomial equations to estimate the amount of SOx emissions as functions of engine speed and load. These developed non-dimensional equations can be further used directly to assess the value of SOx emissions for different percentages of sulfur of the selected or similar engines to be used in different marine applications.

ANN Based Design Parameter Estimation for Structural Systems

Estimation of the probability of failure of multi-dimensional structural systems is expensive from the computation perspective. To decrease the burden of computation, one can use simple approximation methods like Surrogate models, Kriging model, Support vector machine, Artificial neural network, and more based on the suitability for the problems. In Surrogate or Response surface modeling, the limit state function of any system is suitably approximated by making use of known mathematical models like polynomials, exponentials, etc. During the construction of surrogates, variables in the model should be well known prior to the approximation. In practical consideration, the design parameters are the unknowns that need to be evaluated before reliability-based design. Inverse Response surface procedure is proposed in the paper to address the above-mentioned issue. The procedure developed is the combination of adaptive Response surface method with appropriate experimental design i.e. Halton low discrepancy sequence sampling technique for evaluating the probability of failure or reliability index and an Artificial neural network is utilised as an inverse reliability procedure for design optimisation. The method gives an accurate result and the efficiency is increased for the same number of iterations in comparison to the work of David Lehky and Martina Somodikova [1] with Latin hypercube sampling as experimental design.

Soiling Quantification Using an Image-Based Method: Effects of Imaging Conditions

Solar power generation continues to grow as a dominant renewable energy source. Soiling because of dust accumulation is one of the important challenges facing large-scale solar photovoltaic (PV) plant operations, particularly, in the arid regions. The assessment and mitigation of soiling effects on large arrays of PV panels are often limited by the area coverage and how often data can be collected. To overcome this limitation, in this article, we propose a method to directly use images to quantify soiling on PV panels. A prototype lab system and image processing algorithm are developed to test the feasibility of the method. Specifically, a metric called black/white ratio is calculated from images of a surrogate surface with controlled dust loadings. Effects of different camera settings and view angles are investigated using lab experiments. The experimental results show that dust loading can be quantified under a range of proper imaging conditions, which indicates the potential to use aerial images for PV panel soiling quantification in the future.

Gaseous Oxidized mercury dry deposition measurements in the Four Corners area, U.S.A., after large power plant mercury emission reductions

Gaseous oxidized mercury (GOM) dry deposition measurements using surrogate surface passive samplers were collected at six sites in the Four Corners area, U.S.A., for the two-year period August 2017–August 2019, after the implementation of large power plant mercury emission reductions across the U.S.A. Two-year baseline GOM dry deposition measurements at the same six sites in the Four Corners area, taken before the implementation of U.S.A. power plant mercury control regulations, were conducted earlier from August 2009–August 2011. The GOM dry deposition rate estimate decreased at the Four Corners area high elevation remote mountain site of Molas Pass, Colorado (3249 m asl) from 0.4 ng/m2h for August 2009–August 2011 to 0.3 ng/m2h for August 2017–August 2019. In contrast, GOM dry deposition rate estimates for the remaining five sites increased for August 2017–August 2019, ranging from 0.8 to 1.3 ng/m2h, up from the August 2009–August 2011 range of 0.6–1.0 ng/m2h. Comparisons of median GOM dry deposition values showed a statistically significant decrease of 17 ng/m2 at the Molas Pass site between August 2009–August 2011 and August 2017–August 2019, and a statistically significant increase of 66 ng/m2 and 64 ng/m2, respectively, at the Mesa Verde National Park and Farmington Substation sites between August 2009–August 2011 and August 2017–August 2019. For the four years of GOM dry deposition data collected in the Four Corners area annual GOM dry deposition levels ranged from 2237 ng/m2yr (at the Molas Pass high elevation remote mountain site) to 11542 ng/m2yr (at the Mesa Verde National Park site), and the estimates were generally higher in magnitude in the spring and summer compared to the fall and winter. In light of the unexpected increases in GOM dry deposition rates at the non-remote sites, it is suggested that large regional wildfires and local anthropogenic mercury emission sources from cities and oil/gas production areas are possible notable contributors to the GOM dry deposition measurements collected in the Four Corners area.

Active learning method combining Kriging model and multimodal‐optimization‐based importance sampling for the estimation of small failure probability

AbstractA novel method which combines the active learning Kriging (ALK) model with important sampling is proposed in this paper. The main aim of the proposed method is to solve problems with very small failure probability and multiple failure regions. A surrogate limit state surface (LSS) which strikes a balance between the Kriging mean and variance is proposed. In each iteration, important samples of the surrogate LSS are generated, optimal training points are chosen, the Kriging model is updated and the surrogate LSS is refined. After several iterations, the surrogate LSS will converge to the true LSS. To obtain all the local and global most probable points (MPPs) on the surrogate LSS in each iteration, a recently proposed evolutionary algorithm from the field of multimodal optimization is introduced. In this way, none of the potential failure regions is missed and the unbiasedness of the proposed method is guaranteed. The contribution factor of each MPP is defined and a weighted multimodal instrumental sampling density is formulated. In this way, more attention is paid to the more important failure regions and training points are further saved. The performance of the proposed method is verified by six case studies.

Effect of insertion factors on dental implant insertion torque/energy-experimental results

Anchorage of dental implants is quantified with a mechanical engagement to insertion, for example maximum insertion torque (MIT) and insertion energy (IE). Good anchorage of dental implants highly correlates to positive clinical outcomes. However, it is still unclear how bone density, drill protocol, surface finish and cutting flute affect anchorage. In this study, effects of the insertion factors on both MIT and IE were investigated using a full-factorial experiment at two levels: bone surrogate density (0.32 g/cm3 versus 0.48 g/cm3), drill protocol (Ø2.4/2.8 versus Ø2.8/3.2 mm), implant surface finish (machined versus anodized surface) and cutting flute (with versus without). Osteotomies were prepared on rigid polyurethane foam blocks with dimensions of 40 × 40 × 8 mm. Screw shaped dental implants with variable tapered body were consecutively inserted into and removed from the polyurethane foam blocks three times under constant axial displacement and rotational speed. Axial force and torque were recorded synchronously. Insertion energy was calculated from the area under the torque-displacement curve. In this study, we found the main insertion mechanics were thread forming for the first insertion. For the second and third insertions, the main mechanics shifted to thread tightening. Maximum insertion torque (MIT) responded differently to the four insertion factors in comparison to IE. Bone surrogate density, drill protocol and surface finish had the largest main effects for first MIT. For the first IE, drill protocol, surface finish and cutting flute were significant contributors. These results suggest that MIT and IE are influenced by different mechanics: the first MIT and the first IE were sensitive to thread tighten and forming, respectively. Together MIT and IE provide a complete assessment of dental implant anchorage.

Crashworthiness optimization of hierarchical hexagonal honeycombs under out-of-plane impact

This paper presents a numerical study of regular and hierarchical honeycomb structures subjected to out-of-plane impact loading. The specific energy absorption capacity of honeycomb structures via nonlinear explicit finite element analysis is investigated. The constructed finite element models are validated using experimental data available in the literature. The honeycomb structures are optimized by using a surrogate-based optimization approach to achieve maximum specific energy absorption capacity. Three surrogate models polynomial response surface approximations, radial basis functions, and Kriging models are used; Kriging models are found to be the most accurate. The optimum specific energy absorption value obtained for hierarchical honeycomb structures is found to be 148% greater than that of regular honeycomb structures.

Wind farm macro-siting optimization with insightful bi-criteria identification and relocation mechanism in genetic algorithm

The existence of wake effect can affect the total power generation of a wind farm. To alleviate the impact of wake effect, numerous algorithms under the paradigm of evolutionary computation have been proposed to find the optimal layout of wind turbines. Previously, inspired by the self-adjustment capability among the individuals of a species in the natural world, we empowered the genetic algorithm (GA) with self-adaptivity and found that, by relocating the least efficient wind turbine to a new location with the help of a surrogate response surface of the power generation distribution, the performance of GA can be significantly improved. Following previous research, we discovered another major bottleneck that can cause the algorithm to be trapped into a suboptimal solution. A new bi-criteria identification and relocation (BCIR) mechanism is introduced to different versions of GA, including the conventional GA and our previous improved versions of GA. The introduction of BCIR does not require additional computation complexity. The effectiveness of this new mechanism is verified by conducting extensive experiments in two case studies, and both results show significant improvement over GA after adopting the new mechanism of BCIR.

Restricted-Variance Molecular Geometry Optimization Based on Gradient-Enhanced Kriging

Machine learning techniques, specifically gradient-enhanced Kriging (GEK), have been implemented for molecular geometry optimization. GEK-based optimization has many advantages compared to conventional—step-restricted second-order truncated expansion—molecular optimization methods. In particular, the surrogate model given by GEK can have multiple stationary points, will smoothly converge to the exact model as the number of sample points increases, and contains an explicit expression for the expected error of the model function at an arbitrary point. Machine learning is, however, associated with abundance of data, contrary to the situation desired for efficient geometry optimizations. In this paper, we demonstrate how the GEK procedure can be utilized in a fashion such that in the presence of few data points, the surrogate surface will in a robust way guide the optimization to a minimum of a potential energy surface. In this respect, the GEK procedure will be used to mimic the behavior of a conventional second-order scheme but retaining the flexibility of the superior machine learning approach. Moreover, the expected error will be used in the optimizations to facilitate restricted-variance optimizations. A procedure which relates the eigenvalues of the approximate guessed Hessian with the individual characteristic lengths, used in the GEK model, reduces the number of empirical parameters to optimize to two: the value of the trend function and the maximum allowed variance. These parameters are determined using the extended Baker (e-Baker) and part of the Baker transition-state (Baker-TS) test suites as a training set. The so-created optimization procedure is tested using the e-Baker, full Baker-TS, and S22 test suites, at the density functional theory and second-order Møller–Plesset levels of approximation. The results show that the new method is generally of similar or better performance than a state-of-the-art conventional method, even for cases where no significant improvement was expected.