Analytical Convolution Research Articles

Analytic ray tracer on GPU for central receiver systems

An accurate ray tracer is essential to calculate the efficiency of central receiver systems. Furthermore, its runtime has a direct influence on the applicable optimization method for finding optimal layouts of heliostat fields. Developing a ray tracer that achieves both objectives is the main focus of this study. There exist several different ray tracing techniques, starting from the classical Monte Carlo method to analytical convolution methods. Within this work, we introduce a new analytical ray tracer with high accuracy at low runtime. This is achieved by integrating over the bivariate Gaussian distributions used in our convolution ray tracer. In such a way the raytracer is capable of including the effect of multiple solar rays without the need of simulating each ray. Besides the overall efficiency, the new ray tracer can also compute accurate flux maps with thousand times fewer rays than the Monte Carlo method. To further improve the overall model accuracy, a new equation to convolute the sun, slope and tracking error is derived. We show that it perfectly matches the results of a separate accounting for these errors. The newly integrated convolution ray tracer is developed on the same platform as a bidirectional Monte Carlo ray tracer and a convolution method. Besides a large cross-validation of the results, this allows for a reasonable direct run time comparison. With extensive case studies, the quality of the solution, the run time, and various other aspects are investigated. Furthermore, all ray tracers are also implemented on the GPU improving the run time compared to the CPU version by a factor of 50. We show that the integrated convolution ray tracer running on the GPU achieves an accuracy of 99.95% for an annual simulation in 0.7 s for the PS10 and in 1.8 s for the Gemasolar.

On the Evolution of Different Types of Green Water Events—Part II: Applicability of a Convolution Approach

Recent research related to the evolution of different types of green water events, generated in wave flume experiments, has shown that some events, such as plunging-dam-break (PDB) and hammer-fist (HF) types, can present multiple-valued water surface elevations during formation at the bow of the structure. However, the applicability of analytical models to capture the evolution (i.e., the spatio-temporal variation of water elevations) of these events has not been tested thoroughly. This could be useful when estimating green water loads in the preliminary design stage of marine structures. The present work extends the research by Fontes et al. (On the evolution of different types of green water events, Water, 13, 1148, 2021) to examine the applicability of an analytical convolution approach to represent the variation in time of single-valued water elevations of different types of green water events generated by incident wave trains, particularly PDB and HF types. Detailed experimental measurements using high-speed video in wave flume experiments were used to verify the applicability of the model for single and consecutive green water events of type PDB and HF. The present work is a tentative attempt to compare an analytical approach for HF evolution. Results were also compared with the classic analytical dam-break approach. It was found that the convolution model allows the variation of water elevations in time to be captured better in comparison with the dam-break approach. The convolution model described the trend of water elevations well, particularly at the bow of the structure. The model captured the peak times well in single and consecutive events with multiple-valued water surfaces. Results suggest that this conservative and simplified approach could be a useful engineering tool, if improved and extended, to include the evolution of green water events in time domain simulations. This could be useful in the design stages of marine structures subject to green water events.

Notes on Fitting and Analysis Frameworks for QENS Spectra of (Soft) Colloid Suspensions

With continuously improving signal-to-noise ratios, a statistically sound analysis of quasi-elastic neutron scattering (QENS) spectra requires to fit increasingly complex models which poses several challenges. Simultaneous fits of the spectra for all recorded values of the momentum transfer become a standard approach. Spectrometers at spallation sources can have a complicated non-Gaussian resolution function which has to be described most accurately. At the same time, to speed up the fitting, an analytical convolution with this resolution function is of interest. Here, we discuss basic concepts to efficient approaches for fits of QENS spectra based on standard MATLAB and Python fit algorithms. We illustrate the fits with example data from IN16B, BASIS, and BATS.

On the Product of Planar Harmonic Mappings

Given two harmonic functions $$f_{1}=h_{1}+{\bar{g}}_{1}$$ and $$f_{2}=h_{2}+{\bar{g}}_{2}$$ defined on the open unit disk of the complex plane, the geometric properties of the product $$f_{1}\otimes f_{2}$$ defined by $$\begin{aligned} f_{1}\otimes f_{2}=h_{1}*h_{2}-\overline{g_{1}*g_{2}}+ i{\text {Im}}(h_{1}*g_{2}+h_{2}*g_{1}) \end{aligned}$$ are discussed. Here $$*$$ denotes the analytic convolution. Sufficient conditions are obtained for the product to be univalent and convex in the direction of the real axis. In addition, a convolution theorem, coefficient inequalities and closure properties for the product $$\otimes $$ are proved.

Transient Electromagnetic Analysis of Multilayer Graphene with Dielectric Substrate Using Marching-on-in-Degree Method

The marching-on-in-degree (MOD) method is applied in this paper to analyze the transient electromagnetic scattering of multilayer graphene and a dielectric substrate. The time domain resistive boundary condition (TD-RBC) integral equation and time domain Poggio–Miller–Chang–Harrington–Wu (PMCHW) integral equation of electric and magnetic currents are employed to model graphene and the dielectric substrate, respectively. These two sets of equations are coupled and solved with the MOD method. The dispersion of multilayer graphene’s surface conductivity/resistivity in the frequency domain is taken into account in the analytical convolution of temporal surface conductivity/resistivity and magnetic/electric current densities. The Rao-Wilton-Glisson (RWG) basis function over triangle patches and weighted Laguerre polynomial (WLP) are used as the spatial and temporal basis/testing functions, respectively. The orthogonal WLPs are defined from zero to +∞ and are convergent to zero with time passing. These advantages ensure late time stability of the transient solution. A stable electric/magnetic current is achieved. A radar cross section and extinction cross section in the frequency domain are also obtained and compared with commercial software results to verify the proposed method.

Region adaptive scheduling for time-dependent processes with optimal use of machine dynamics

In time-dependent processes, such as bonnet and fluid jet polishing, surface quality and accurate processing critically depend on careful planning of the tool feed. CNC feedrate commands are usually generated from a dwell time map calculated by deconvolution or numerical iteration. These methods are time-consuming, numerically unstable, and fail to consider dynamic stressing of the machine tool. In this research, Gaussian mixture model (GMM) is proposed to model experimental tool influence functions (TIF). This leads to a general analytical convolution model integrating processing depth, volumetric removal rate of TIF, path spacing and feedrate. Based on this model, a novel direct feedrate scheduling method is proposed, which is suitable for any kind of smooth time-dependent processing beam. Optimal feedrate scheduling within dynamic constraints of the machine tool is achieved by establishing acceptable path spacing and feed ranges, whilst dynamic stressing of the machine tool is optimized concurrently through adaptive path spacing. Simulations and experiments demonstrate the enhanced stability and usefulness of the proposed feedrate model in deterministic material removal. It also verifies that path adaptability allows for improved machine tool dynamics, without incurring a process accuracy penalty.

Analytical convolution model for shipping water evolution on a fixed structure

The shipping of water is a problem that affects naval and offshore structures. Estimating its propagation on the decks of these structures by using analytical methods has been a main concern of projects. However, classical approaches disregard the decay tendency of water elevation time series and tend to overestimate the resultant water on deck. This paper is concerned with estimating the evolution of water along the deck of a fixed structure due to shipping water events. An analytical convolution model is proposed to estimate water elevations. The model considers the freeboard exceedance time series and the mean shipping flow velocity as inputs and the frictional effects of the bottom by resistance coefficients, which enables an approximated representation of the water elevation time series over the deck. It was validated with experiments of isolated shipping water events that were generated with the wet dam-break approach. The results obtained with the proposed model captured the experimental results, approximating the peak values and the decay trend of time series. Improvement of the proposed approach over classical models to represent shipping water elevations was demonstrated by comparing the results obtained with those of the dam-break model of Stoker.

Improvements and reproducibility of an optimal grazing-incidence position method to L-shell x-ray fluorescence measurements of lead in bone and soft tissue phantoms

L-shell x-ray fluorescence (LXRF) is a non-invasive approach to lead (Pb) concentration measurements in human bone. The method is based on the detection of the characteristic x-ray photons of Pb at 10.5 and 12.6 keV and experimental studies were designed to perform in vivo human bone Pb measurements. In later studies, however, the initial LXRF methodology was shown to have poor accuracy and precision. In a recent publication, we investigated an optimal grazing-incidence position (OGIP) approach using a sub-millimeter x-ray beam from an integrated x-ray tube and polycapillary x-ray lens table-top system. The OGIP method effectively reduced the x-ray scatter and produced a Pb detection limit of ∼5 μg/g for a 2 mm soft tissue phantom thickness. In this study, the OGIP methodology was improved by using 10 s x-ray spectra acquisitions at sequential positions 0.5 mm apart. The measured Sr Kα peak height versus position data was used to spectroscopically identify the bone phantom and the OGIP. The data was fitted with the analytical convolution between a Gaussian and an exponential decay. The position corresponding to the maximum of the fitted convolution function was then selected as the OGIP. Four phantom sets were used. A cylindrical plaster-of-Paris bone phantom doped with Pb in a concentration of 74 μg/g was used as a bare bone phantom or with one of the three overlying polyoxymethylene cylindrical shell soft tissue phantoms of 1, 2, and 3 mm thickness. The reproducibility of the OGIP method was assessed in five independent trials using each of the four phantom sets. The coefficient of variation (COV) percentage values of the Sr Kα peak height measurements were below 5%. The new procedure decreased by more than threefold the duration and radiation dose of the earlier approach.

Effective X-ray beam size measurements of an X-ray tube and polycapillary X-ray lens system using a scanning X-ray fluorescence method

Size measurements of an X-ray beam produced by an integrated polycapillary X-ray lens (PXL) and X-ray tube system were performed by means of a scanning X-ray fluorescence (SXRF) method using three different metallic wires. The beam size was obtained by fitting the SXRF data with the analytical convolution between a Gaussian and a constant functions. For each chemical element in the wire an effective energy was calculated based on the incident X-ray spectrum and its photoelectric cross section. The proposed method can be used to measure the effective X-ray beam size in XRF microscopy studies.

A heuristic methodology to economic dispatch problem incorporating renewable power forecasting error and system reliability

With the constant increment of wind power generation driven by economic and environmental factors, the optimal utilization of generation resources has become a critical problem discussed by many authors. Within this topic, determination of optimal spinning reserve (SR) requirements is a key and complex issue due to the variable and unpredictable nature of renewable generation besides of generation unit reliability. Cost/benefit relationship has been suggested as a way to determine the optimal amount of power generation to be committed by taking into account renewable power forecasting error and system reliability. In this paper, a technique that combines an analytical convolution process with Monte Carlo Simulation (MCS) approach is proposed to efficiently build cost/benefit relationship. The proposed method uses discrete probability theory and identifies those cases at which convolution analysis can be used by recognizing those situations at which SR does not have any effect; while in the other cases MCS is applied. This approach allows improving significantly the computational efficiency. The proposed technique is illustrated by means of two case studies of 10 and 140 units, demonstrating the capabilities and flexibility of the proposed methodology.

The behavior of conditional Wiener integrals on product Wiener space

AbstractIn this paper we first investigate the behavior of conditional Wiener integrals on product Wiener spaces and then proceed to establish a Fubini theorem for conditional Wiener integrals. We then define a very general multiple conditional analytic Fourier‐Feynman transform and a multiple conditional analytic convolution product for functionals on Wiener space and then establish a relationship between them. Next we obtain a basic formula for the conditional Wiener integral involving the first variation and the n‐dimensional Wiener directional derivative. Finally we establish several evaluation formulas for conditional Wiener and Feynman integrals involving the n‐dimensional Wiener directional derivative of functionals on Wiener space.

Tolerance Analysis Method Using Improved Convolution Method

Convolution method is studied to analyze statistical tolerance for linear dimension chain and nonlinear dimension chain. Hybrid convolution method is proposed, which is the integration of analytical convolution and numerical convolution. In order to reduce the algorithm errors, improved convolution method is proposed. Comparing with other statistical tolerance analysis methods, this method is faster and accurate. At last, an example is used to demonstrate the method proposed in this paper.

Analytical solutions for sketch-based convolution surface modeling on the GPU

Convolution surfaces are attractive for modeling objects of complex evolving topology. This paper presents some novel analytical convolution solutions for planar polygon skeletons with both finite-support and infinite-support kernel functions. We convert the double integral over a planar polygon into a simple integral along the contour of the polygon based on Green’s theorem, which reduces the computational cost and allows for efficient parallel computation on the GPU. For finite support kernel functions, a skeleton clipping algorithm is presented to compute the valid skeletons. The analytical solutions are integrated into a prototype modeling system on the GPU (Graphics Processing Unit). Our modeling system supports point, polyline and planar polygon skeletons. Complex objects with arbitrary genus can be modeled easily in an interactive way. Resulting convolution surfaces with high quality are rendered with interactive ray casting.

Spectral components that form UV absorption spectrum of Ce3+ and Ce(IV) valence states in matrix of photothermorefractive glasses

We have measured the UV absorption spectra of photothermorefractive glasses of the system Na2O-ZnO-Al2O3-NaF-SiO2 doped by cerium oxide in the range of (2.8–5.0) × 104 cm−1 (360–200 nm). The spectra have been processed by the method of dispersion analysis based on the analytical convolution model for the complex dielectric function of glasses. We show that the absorption band centered at 3.3 × 104 cm−1 (∼303 nm) that is attributed to the transition 2F 5/2 → 5d in the Ce3+ ion, is an envelope of three spectral components. The broad absorption range (3.5–4.7) × 104 cm−1 (200–270 nm) that is commonly interpreted as a charge transfer band of the Ce(IV) valence state, is an envelope of at least three spectral components.

Liquidity-, Funding- and Market-Risk Measurement and Stress-Testing

We introduce a new framework to integrate liquidity risk, funding risk and market risk, which goes beyond the simple bid-ask spread overlay to a VaR number. In our approach, we overlay a whole distribution of liquidity uncertainty to each future market-risk scenario. Then we allow for the liquidity uncertainty to vary scenario by scenario, depending on what liquidation policy or funding policy is implemented in that scenario.The result is one easy-to-interpret and easy-to-implement formula for the total liquidity-plus-market-risk PL compute total risk and decompose it into a novel liquidity-plus-market risk formula; and define a liquidity score as a monetary measure of portfolio liquidity.Our approach relies on three pillars: first, the literature on optimal execution, to model liquidity risk as a function of the actual trading involved; second, an analytical conditional convolution, to blend market risk and liquidity/funding risk; third the Fully Flexible Probabilities framework, to model and stress-test market risk even in highly non-normal portfolios with complex derivatives. Our approach can be implemented efficiently with portfolios of thousand of securities. The code for the case study is available for download

An Analytical Convolution Method Combined With the Conformal Fourier Transform for Solving 1-D Integral Equations

A new solver, the analytical convolution method (ACM) combined with the conformal Fourier transform (CFT), is proposed for 1-D integral equations in electromagnetics. ACM directly obtains the convolution between the Green's function and the induced current density in a closed form. CFT provides a highly accurate Fourier transform of the discontinuous current density. Numerical results show that the ACM-CFT solver is several orders of magnitude more accurate than the traditional methods such as the conjugate-gradient fast Fourier transform (FFT) method, and its CPU time is significantly shorter.

SU-GG-I-91: The Study of AAA Algorithm and PBC Algorithm of Intensity Modulated Radiation Therapy for Breast Cancer

Objective To compare AAA with PBC in the IMRT after breast‐conserving surgey in regard to the homogeneity of dose in the target area, ipsilateral lungdose and irradiation volume,and cardiac dose and irradiation volume.Metheods Ten breast cancer patients with inverse IMRT plans and ten breast cancer patients with forward IMRT plans were allotted.Prescription dose was 50Gy in breast volume with X‐ray, Their plans were separately calculated by Analytical Anisotropic Algorithm(AAA) and Pencial Beam Convolution(PBC).Plan Target Volume(PTV), ipsilateral lung and heart were evaluated with dose‐volume histograms(DVH)in two plans,with the Paired‐Sample T Test taken with SPSS 13.0.Results In inverse IMRT plan the above priscription dose of PTV with AAA were equal to the dose with PBC.The mean PTV dose with AAA were lower than the PBC by 73.91cGy,the uniformity higher than the PBC by 2.5%,V20 of ipsilateral lung higher than the PBC by 2.53%,the mean heartdose lower than the PBC by 53.39cGy,but there were no significance for all of them. The conformity index of PTV with AAA were significantly higher than the PBC by 5.71%. In forward IMRT plan the above priscription dose of PTV with AAA were higher than the PBC by 8.53%,the mean PTV dose higher than the PBC by 91.03cGy,the uniformity higher than the PBC by 10.08%,the heart mean dose higher than the PBC by 3.37cGy,but there were no significance for all of them.The conformity index of ung with AAA were significantly higher than the PBC by 0.91%. V20 of ipsilateral lung with AAA were significantly higher than the PBC by 0.91%. Conclusions The conformity of PTV with AAA is better than the PBC in breast cancer in IMRT plan,but the V20 of the ipsilateral lung is higher than the PBC.

Comparison of alternative probabilistic techniques for adequacy assessment of small isolated wind/diesel systems

This paper compares two alternative probabilistic techniques for the adequacy assessment of small isolated wind/diesel systems. These approaches are the analytical convolution method and the sequential Monte Carlo simulation. The results obtained from these techniques are illustrated by a series of comparative studies. The main purpose of the paper is to investigate the accuracy of the analytical technique assuming that the sequential Monte Carlo simulation method would provide the most precise results. The results show that there is no significant difference between the outcomes of the two techniques. The paper also shows that a multi-state wind energy conversion system model can be used to produce reasonably accurate results in small isolated wind/diesel system adequacy studies.

NON-NORMAL STATISTICAL TOLERANCE ANALYSIS USING ANALYTICAL CONVOLUTION METHOD

In statistical tolerance analysis, it is usually assumed that the statistical tolerance is normally distributed. But in practice, there are many non-normal distributions, such as uniform distribution, triangular distribution, etc. The simple way to analyze non-normal distributions is to approximately represent it with normal distribution, but the accuracy is low. Monte-Carlo simulation can analyze non-normal distributions with higher accuracy, but is time consuming. Convolution method is an accurate method to analyze statistical tolerance, but there are few reported works about it because of the difficulty. In this paper, analytical convolution is used to analyze non-normal distribution, and the probability density functions of closed loop component are obtained. Comparing with other methods, convolution method is accurate and faster.

Prototype Modeling from Sketched Silhouettes based on Convolution Surfaces

AbstractThis paper presents a hybrid method for creating three‐dimensional shapes by sketching silhouette curves. Given a silhouette curve, we approximate its medial axis as a set of line segments, and convolve a linearly weighted kernel along each segment. By summing the fields of all segments, an analytical convolution surface is obtained. The resulting generic shape has circular cross‐section, but can be conveniently modified via sketched profile or shape parameters of a spatial transform. New components can be similarly designed by sketching on different projection planes. The convolution surface model lends itself to smooth merging between the overlapping components. Our method overcomes several limitations of previous sketched‐based systems, including designing objects of arbitrary genus, objects with semi‐sharp features, and the ability to easily generate variants of shapes.