A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials
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97 MATHEMATICS AND COMPUTING↗
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Scalable Bayesian Physics-Informed Kolmogorov-Arnold Networks
Uncertainty quantification (UQ) plays a pivotal role in scientific machine learning, especially when surrogate models are used to approximate complex systems. Although multilayer perceptions (MLPs) are commonly employed as surrogates, they often suffer from overfitting due to their large number of parameters. Kolmogorov-Arnold networks (KANs) offer an alternative solution with fewer parameters. However, gradient-based inference methods, such as Hamiltonian Monte Carlo (HMC), may result in computational inefficiency when applied to KANs, especially for large-scale datasets, due to the high cost of back-propagation. To address these challenges, we propose a novel approach, combining the dropout Tikhonov ensemble Kalman inversion (DTEKI) with Chebyshev KANs. This gradient-free method effectively mitigates overfitting and enhances numerical stability. In addition, we incorporate the active subspace method to reduce the parameter-space dimensionality, allowing us to improve the accuracy of predictions and obtain more reliable uncertainty estimates. Extensive experiments demonstrate the efficacy of our approach in various test cases, including scenarios with large datasets and high noise levels. Our results show that the new method achieves comparable or better accuracy, much higher efficiency as well as stability compared to HMC, in addition to scalability. Moreover, by leveraging the low-dimensional parameter subspace, our method preserves prediction accuracy while substantially reducing further the computational cost.
97 MATHEMATICS AND COMPUTING↗
Utilization of nuclear methods for materials analysis and the determination of concentration gradients
The various types of nuclear chemical analysis methods are discussed. The possibilities of analysis through activation and direct observation of nuclear reactions are described. Such methods make it possible to analyze trace elements and impurities with selectivity, accuracy, and a high degree of sensitivity. Such methods are used in measuring major elements present in materials which are available for analysis only in small quantities. These methods are well suited to superficial analyses and to determination of concentration gradients; provided the nature and energy of the incident particles are chosen judiciously. Typical examples of steels, pure iron and refractory metals are illustrated.
Darras, R.↗
Color optimization of red OLEDs via periodic and gradient deposition rate of fluorescent dopants
We have investigated a method to enhance the light emission efficiency and color tuning for mixture of dyes with varying the growth parameters, during Alq{sub 3}:Nile Red deposition. We used solvent bath and green laser (532 nm) for improvement of the performance of devices. This paper presents two kind of evaporation method, periodic and gradient. We observed that periodic emission layers can greatly improve the device performances such as electroluminescence, chromaticity coordinates, luminance, and reduced the turn on voltage. The periodic 3 structure with evaporation rate of (0.1 → 3 → 2 → 3 → 2 → 0.1 A°/s) has a luminance of 3000 cd/m{sup 2} and maximum efficiency of 1.8 cd/A at 18 V which are the highest values among devices.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Measurement of Seebeck coefficient using a large thermal gradient
The integral method of measuring the Seebeck voltage, V(T), has been applied to short rod-shaped samples. In the present method, one end of the sample is held at a fixed temperature and the other end is automatically varied through a temperature T range of interest, up to a maximum temperature of 1000 C. The Seebeck coefficient is then obtained from the slope of the V(T) vs T curve. The method offers simplicity of sample handling and minimal operator involvement.
Wood, C.↗
Charge gradient microscopy
A method for rapid imaging of a material specimen includes positioning a tip to contact the material specimen, and applying a force to a surface of the material specimen via the tip. In addition, the method includes moving the tip across the surface of the material specimen while removing electrical charge therefrom, generating a signal produced by contact between the tip and the surface, and detecting, based on the data, the removed electrical charge induced through the tip during movement of the tip across the surface. The method further includes measuring the detected electrical charge.
36 MATERIALS SCIENCE↗
Analytical energy gradient based on spin-free infinite-order Douglas-Kroll-Hess method with local unitary transformation
In this study, the analytical energy gradient for the spin-free infinite-order Douglas-Kroll-Hess (IODKH) method at the levels of the Hartree-Fock (HF), density functional theory (DFT), and second-order Møller-Plesset perturbation theory (MP2) is developed. Furthermore, adopting the local unitary transformation (LUT) scheme for the IODKH method improves the efficiency in computation of the analytical energy gradient. Numerical assessments of the present gradient method are performed at the HF, DFT, and MP2 levels for the IODKH with and without the LUT scheme. The accuracies are examined for diatomic molecules such as hydrogen halides, halogen dimers, coinage metal (Cu, Ag, and Au) halides, and coinage metal dimers, and 20 metal complexes, including the fourth–sixth row transition metals. In addition, the efficiencies are investigated for one-, two-, and three-dimensional silver clusters. The numerical results confirm the accuracy and efficiency of the present method.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Extending an Economical Third-Order Inviscid Nodal-Gradient Cell-Centered Finite-Volume Method to Mixed-Element Grids
In this paper, we extend an economical third-order nodal-gradient cell-centered finite-volume method, originally developed for tetrahedral grids, to mixed-element grids. It is shown that the efficient quadratic interpolation formula essential to eliminating second derivatives from a third-order accurate discretization can be easily extended to an arbitrary cell type. For the surface flux integration, we consider a split-face approach, where a quadrilateral face is split into two triangles, and the third-order method for tetrahedra is directly applied. Also, we derive a second-derivative-free, high-order, volume quadrature formula for an arbitrary cell. Numerical results are presented for accuracy verification and applications with three-dimensional nontetrahedral grids.
Computational Fluid Dynamics↗
Gradient Coding With Iterative Block Leverage Score Sampling
Gradient coding is a method for mitigating straggling servers in a centralized computing network that uses erasure-coding techniques to distributively carry out first-order optimization methods. Randomized numerical linear algebra uses randomization to develop improved algorithms for large-scale linear algebra computations. In this study, we propose a method for distributed optimization that combines gradient coding and randomized numerical linear algebra. The proposed method uses a randomized ℓ 2 -subspace embedding and a gradient coding technique to distribute blocks of data to the computational nodes of a centralized network, and at each iteration the central server only requires a small number of computations to obtain the steepest descent update. The novelty of our approach is that the data is replicated according to importance scores, called block leverage scores, in contrast to most gradient coding approaches that uniformly replicate the data blocks. Furthermore, we do not require a decoding step at each iteration, avoiding a bottleneck in previous gradient coding schemes. We show that our approach results in a valid ℓ 2 -subspace embedding, and that our resulting approximation converges to the optimal solution.
97 MATHEMATICS AND COMPUTING↗
Semiautomatic registration of 3D transabdominal ultrasound images for patient repositioning during postprostatectomy radiotherapy
Purpose: The aim of the present work is to propose and evaluate registration algorithms of three-dimensional (3D) transabdominal (TA) ultrasound (US) images to setup postprostatectomy patients during radiation therapy. Methods: Three registration methods have been developed and evaluated to register a reference 3D-TA-US image acquired during the planning CT session and a 3D-TA-US image acquired before each treatment session. The first method (method A) uses only gray value information, whereas the second one (method B) uses only gradient information. The third one (method C) combines both sets of information. All methods restrict the comparison to a region of interest computed from the dilated reference positioning volume drawn on the reference image and use mutual information as a similarity measure. The considered geometric transformations are translations and have been optimized by using the adaptive stochastic gradient descent algorithm. Validation has been carried out using manual registration by three operators of the same set of image pairs as the algorithms. Sixty-two treatment US images of seven patients irradiated after a prostatectomy have been registered to their corresponding reference US image. The reference registration has been defined as the average of the manual registration values. Registration error has been calculated by subtracting the reference registration from the algorithm result. For each session, the method has been considered a failure if the registration error was above both the interoperator variability of the session and a global threshold of 3.0 mm. Results: All proposed registration algorithms have no systematic bias. Method B leads to the best results with mean errors of −0.6, 0.7, and −0.2 mm in left–right (LR), superior–inferior (SI), and anterior–posterior (AP) directions, respectively. With this method, the standard deviations of the mean error are of 1.7, 2.4, and 2.6 mm in LR, SI, and AP directions, respectively. The latter are inferior to the interoperator registration variabilities which are of 2.5, 2.5, and 3.5 mm in LR, SI, and AP directions, respectively. Failures occur in 5%, 18%, and 10% of cases in LR, SI, and AP directions, respectively. 69% of the sessions have no failure. Conclusions: Results of the best proposed registration algorithm of 3D-TA-US images for postprostatectomy treatment have no bias and are in the same variability range as manual registration. As the algorithm requires a short computation time, it could be used in clinical practice provided that a visual review is performed.
60 APPLIED LIFE SCIENCES↗
Efficient gradient estimation using finite differencing and likelihood ratios for kinetic Monte Carlo simulations
While many optimization and control methods for stochastic processes require gradient information from the process of interest, obtaining gradient information from experiments is prohibitively expensive and time-consuming. As a result, such information is often obtained from stochastic process simulations. Computing gradients efficiently and accurately from stochastic simulations is challenging, especially for simulations involving computationally expensive models with significant inherent noise. In this work, we analyze and characterize the applicability of two gradient estimation methods for kinetic Monte Carlo simulations: finite differencing and likelihood ratio. We developed a systematic method for choosing an optimal perturbation size for finite differencing and discuss, for both methods, important implementation issues such as scaling with respect to the number of elements in the gradient vector. Through a series of numerical experiments, the methods were compared across different time and size regimes to characterize the precision and accuracy associated with each method. We determined that the likelihood ratio method is appropriate for estimating gradients at short (transient) times or for systems with small population sizes, whereas finite differencing is better-suited for gradient estimation at long times (steady state) or for systems with large population sizes.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Programmable Chemical Gradient Patterns by Soft Grayscale Lithography
A method for fabricating chemical gradients on planar and nonplanar substrates using grayscale lithography is reported. Compliant grayscale amplitude masks are fabricated using a vacuum-assisted microfluidic filling protocol that employs dilutions of a carbon-black-containing polydimethylsiloxane emulsion (bPDMS) within traditional clear PDMS (cPDMS) to create planar, fully self-supporting mask elements. The mask is then placed over a surface functionalized with a hydrophobic coumarin-based photocleavable monolayer, which exposes a polar group upon irradiation. The mask serves to modulate the intensity of incident UV light, thereby controlling the density of molecules cleaved. The resulting molecular-level grayscale patterns are characterized by condensation microscopy and imaging mode time-of-flight secondary-ion mass spectrometry (ToF-SIMS). Due to the inherent flexibility of this technique, the photofuse as well as the gradient patterns can be designed for a wide range of applications; in this paper two proof-of-concept demonstrations are shown. The first utilizes the ability to control the resulting contact angle of the surface for the fabrication of a passive pressure-sensitive microfluidic gating system. The second is a model surface modification process that utilizes the functional groups deprotected during the photocleavage to pattern the deposition of moieties with complementary chemistry. The spatial layout, resolution, and concentration of these covalently linked molecules follow the gradient pattern created by the grayscale mask during exposure. Furthermore, the programmable chemical gradient fabrication scheme presented in this work allows explicit engineering of both surface properties that dictate nonspecific interactions (surface energy, charge, etc.) and functional chemistry necessary for covalent bonding.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Multithreaded parallelization of the energy and analytic gradient in the fragment molecular orbital method
The fragment molecular orbital method in GAMESS is parallelized in a multithreaded OpenMP implementation combined with the MPI version of the two-level generalized distributed data interface. The energy and analytic gradient in gas phase and the polarizable continuum model of solvation are parallelized in this hybrid three-level scheme, achieving a large memory footprint reduction and a high parallel efficiency on Intel Xeon Phi processors. The parallel efficiency is demonstrated on the Stampede2 and Theta supercomputers using up to 2048 nodes (262 144 threads).
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Land surface model parameter optimisation using in situ flux data: comparison of gradient-based versus random search algorithms (a case study using ORCHIDEE v1.9.5.2)
Land surface models (LSMs), which form the land component of earth system models, rely on numerous processes for describing carbon, water and energy budgets, often associated with highly uncertain parameters. Data assimilation (DA) is a useful approach for optimising the most critical parameters in order to improve model accuracy and refine future climate predictions. In this study, we compare two different DA methods for optimising the parameters of seven plant functional types (PFTs) of the ORCHIDEE LSM using daily averaged eddy-covariance observations of net ecosystem exchange and latent heat flux at 78 sites across the globe. We perform a technical investigation of two classes of minimisation methods – local gradient-based (the L-BFGS-B algorithm, limited memory Broyden–Fletcher–Goldfarb–Shanno algorithm with bound constraints) and global random search (the genetic algorithm) – by evaluating their relative performance in terms of the model–data fit and the difference in retrieved parameter values. We examine the performance of each method for two cases: when optimising parameters at each site independently (“single-site” approach) and when simultaneously optimising the model at all sites for a given PFT using a common set of parameters (“multi-site” approach). We find that for the single site case the random search algorithm results in lower values of the cost function (i.e. lower model–data root mean square differences) than the gradient-based method; the difference between the two methods is smaller for the multi-site optimisation due to a smoothing of the cost function shape with a greater number of observations. The spread of the cost function, when performing the same tests with 16 random first-guess parameters, is much larger with the gradient-based method, due to the higher likelihood of being trapped in local minima. When using pseudo-observation tests, the genetic algorithm results in a closer approximation of the true posterior parameter value in the L-BFGS-B algorithm. We demonstrate the advantages and challenges of different DA techniques and provide some advice on using it for the LSM parameter optimisation.
58 GEOSCIENCES↗
Friction stir gradient alloying: A high-throughput method to explore the influence of V in enabling HCP to BCC transformation in a γ-FCC dominated high entropy alloy
The compositional possibilities in high entropy alloys (HEAs) is very vast and effective strategies are needed to establish potential alloy chemistries. In this study, friction stir gradient alloying (FSGA), a recently-introduced high-throughput (HT) technique incorporating compositional and microstructural gradients, was used to explore the possibility of introducing a bcc transformation domain in γ-fcc dominated TRIP HEA Fe 38.5 Mn 20 Cr 15 Co 20 Si 5 Cu 1.5 (at.%) by vanadium addition. FSGA has accelerated the process of attaining the composition-microstructure library and the results suggest that vanadium addition of ~1 at.% results in nucleation of α-bcc. Additionally, this is the first observation of ε-hcp to α-bcc transformation in a TRIP HEA and supports the Olson-Cohen model of martensitic transformation (γ-fcc → ε-hcp → α-bcc). ε-hcp is nucleated with a pyramidal orientation by the formation of planar faults on the {111} close-packed planes of γ-fcc, and α-bcc is nucleated in the vicinity of ε-hcp along the {111} γ-fcc trace. There are indications for the formation of basal orientated ε-hcp from intersecting ε-hcp planes orientated for the pyramidal slip. Thus, the study shows the nucleation and growth of the α-bcc phase attributed to the dual effects of chemistry and strain and potential co-existence of all the three phases.
36 MATERIALS SCIENCE↗
The gradient discretization framework for virtual element and partition of unity methods for the Schrodinger equation
In this work we propose a Gradient Discretization framework for the conforming virtual element method and the partition of unity method for the Schrodinger equation with pseudopotentials. This formulation works for any order of accuracy on general polytopal ¨ meshes.
97 MATHEMATICS AND COMPUTING↗
Comparison of Two Approaches to Constructing Second- and Third-Order Nodal-Gradient Cell-Centered Finite-Volume Methods for Mixed-Element Grids
In this paper, we introduce a third-order nodal-gradient cell-centered finite volume scheme applicable to mixed-element grids based on a single numerical flux per face. Previously, we considered a split-quadrilateral-face approach, where we split each quadrilateral face of a non-tetrahedral cell into two triangles and then apply second- and third-order schemes designed for tetrahedral grids. This approach is relatively simple to implement but expensive because it requires two numerical flux evaluations per quadrilateral face. In this study, we consider another approach, where the surface flux quadrature is performed with a single numerical flux per face with flux corrections. It requires a 3×3 correction matrix to be stored at each face but is more economical than the split-quadrilateral-face approach. The two approaches are compared for three-dimensional flows on irregular hexahedral grids, and their relative merits are discussed.
Third-order scheme↗
Scaling up the Single Transducer Thickness-Independent Ultrasonic Imaging Method for Accurate Characterization of Microstructural Gradients in Monolithic and Composite Tubular Structures
Ultrasonic velocity/time-of-flight imaging that uses back surface reflections to gauge volumetric material quality is highly suited for quantitative characterization of microstructural gradients including those due to pore fraction, density, fiber fraction, and chemical composition variations. However, a weakness of conventional pulse-echo ultrasonic velocity/time-of-flight imaging is that the image shows the effects of thickness as well as microstructural variations unless the part is uniformly thick. This limits this imaging method's usefulness in practical applications. Prior studies have described a pulse-echo time-of-flight-based ultrasonic imaging method that requires using a single transducer in combination with a reflector plate placed behind samples that eliminates the effect of thickness variation in the image. In those studies, this method was successful at isolating ultrasonic variations due to material microstructure in plate-like samples of silicon nitride, metal matrix composite, and polymer matrix composite. In this study, the method is engineered for inspection of more complex-shaped structures-those having (hollow) tubular/curved geometry. The experimental inspection technique and results are described as applied to (1) monolithic mullite ceramic and polymer matrix composite 'proof-of-concept' tubular structures that contain machined patches of various depths and (2) as-manufactured monolithic silicon nitride ceramic and silicon carbide/silicon carbide composite tubular structures that might be used in 'real world' applications.
Roth, Don J.↗