Simpson's Rule Research Articles

Some new bounds for Simpson's rule involving special functions via harmonic h-convexity

Some new bounds for Simpson's rule involving special functions via harmonic h-convexity

An accurate vortex feature extraction method for Lagrangian vortex visualization on high-order flow field data

As vortex is of great significance to structure analysis and mechanism research in flow field, vortex feature extraction has always been a hot research topic in flow field visualization. We investigate the data precision and numerical algorithm accuracy impact on vortex feature area extraction of Lagrangian Averaged Vorticity Deviation (LAVD). Then, an LAVD-based high-order accurate vortex extraction algorithm is proposed, which incorporates vorticity computation of cell-vertexed data by Weighted Essentially Non-Oscillatory (WENO) scheme, vorticity and velocity computation of off-grid point by candidate stencil weight-based high-order polynomial interpolation method, flow map computation by 4th-order Runge–Kutta (RK) method and integration by compound Simpson rule. We perform vortex feature extraction and visualization on both analytical flow field and high-order unsteady flow field. The experiment results demonstrate that the proposed algorithm can practically reflect the vortex feature of high-order flow field and accurately describe small scale vortex structure, which is unavailable by low-order method.

How Much Is the Gap? Efficient Jump Risk-Adjusted Valuation of Leveraged Certificates

This paper develops a novel and highly efficient numerical algorithm for the gap risk-adjusted valuation of leveraged certificates. The existing literature relies on Monte Carlo simulations, which are not fast enough to be used in a market making environment. This is because issuers need to compute thousands of price updates per second. By valuing leveraged certificates as multi-window barrier options, we explicitly model random jumps that occur at known times, such as between the exchange closing and re-opening. Our algorithm combines the one-day transition probability with Simpson's numerical integration rule. This yields a backward induction scheme which requires a significantly coarser spacial and time grid than finite difference methods. We confirm its robustness and accuracy through Monte Carlo simulations.

Improving Quantitation of Left Ventricular Ejection Fraction in a Tertiary Echocardiography Lab- Merging Guidelines with Practice

Background: The American Society of Echocardiography strongly recommend quantitative assessment of LVEF on transthoracic echocardiographic (TTE) by 2DE; Modified Simpson's rule or 3D echocardiography (3DE) where available. Purpose: To assess whether education and implementation of the guidelines results can improve the performance and reporting of 2DE/3DE quantitative LVEF and quality improvement. To validate whether the use of automated 3D technology (Philips HeartModelA.I.(HM)) can produce accurate and reproducible online 3DE measurements of LVEF. Methods: Consecutive TTE studies performed over a month prior to and after education intervention and guideline implementation were included. Baseline TTE studies were assessed whether quantitative LVEF was performed by the sonographer and included in the final report by the cardiologist. HM was then utilised prospectively and correlated with conventional 3DE LVEF as the reference. Results: 1422 TTEs (664 before and 758 after intervention) were included. Pre-intervention, the quantitative LVEF was calculated in 27% and 33% were included in the final report. Following educational intervention, these increased to 45% and 52% respectively. 3DE LVEF (by HM) in 48 patients demonstrated good correlation and agreement with conventional 3D LVEF (intraclass correlation coefficient 0.89; p < 0.0001)and good test-retest reproducibility (intra-observer variability 0.2 + -1.0% and inter-observer variability 2.7 + - 2.8%). Conclusions: In a busy clinical echocardiography laboratory, over-reliance on visual estimation of LVEF remains common. Education and implementation of guidelines improves confidence, performance, and reporting of quantitative 2DE LVEF. Incorporation of new automated technology 3DE LVEF (using HM) is feasible, accurate and reproducible.

Pressure-volume Relationship in the Stress-echocardiography Laboratory: Does (Left Ventricular End-diastolic) Size Matter?

Introduction and objectivesThe variation between rest and peak stress end-systolic pressure-volume relation is an afterload-independent index of left ventricular contractility. Whether and to what extent it depends on end-diastolic volume remains unclear. The aim of this study was to assess the dependence of the delta rest-stress end-systolic pressure-volume relation on end-diastolic volume in patients with negative stress echo and all ranges of resting left ventricular function. MethodsWe analyzed interpretable data obtained in 891 patients (593 men, age 63 ± 12 years) with ejection fraction 47% ± 12%: 338 were normal or near-normal or hypertensive; 229 patients had coronary artery disease; and 324 patients had ischemic or nonischemic dilated cardiomyopathy. They were studied with exercise (n = 172), dipyridamole (n = 482) or dobutamine (n = 237) stress echocardiography. The end-systolic pressure-volume relation was evaluated at rest and peak stress from raw measurement of systolic arterial pressure by cuff sphygmomanometer and end-systolic volume by biplane Simpson rule 2-dimensional echocardiography. ResultsAbsolute values of delta rest-stress end-systolic pressure-volume relation were higher for exercise and dobutamine than for dipyridamole. In the overall population, an inverse relationship between end-systolic pressure-volume relation and end-diastolic volume was present at rest (r2 = 0.69, P < .001) and peak stress (r2 = 0.56, P < .001), but was absent if the delta rest-stress end-systolic pressure-volume relation was considered (r2 = 0.13). ConclusionsLeft ventricular end-diastolic volume does not affect the rest-stress changes in end-systolic pressure-volume relation in either normal or abnormal left ventricles during physical or pharmacological stress.

Relación presión/volumen en el laboratorio de ecocardiografía de estrés. ¿Cómo influye el tamaño del ventrículo izquierdo (dimensión diastólica del ventrículo izquierdo)?

Introduction and objectivesThe variation between rest and peak stress end-systolic pressure-volume relation is an afterload-independent index of left ventricular contractility. Whether and to what extent it depends on end-diastolic volume remains unclear. The aim of this study was to assess the dependence of the delta rest-stress end-systolic pressure-volume relation on end-diastolic volume in patients with negative stress echo and all ranges of resting left ventricular function. MethodsWe analyzed interpretable data obtained in 891 patients (593 men, age 63 ± 12 years) with ejection fraction 47% ± 12%: 338 were normal or near-normal or hypertensive; 229 patients had coronary artery disease; and 324 patients had ischemic or nonischemic dilated cardiomyopathy. They were studied with exercise (n = 172), dipyridamole (n = 482) or dobutamine (n = 237) stress echocardiography. The end-systolic pressure-volume relation was evaluated at rest and peak stress from raw measurement of systolic arterial pressure by cuff sphygmomanometer and end-systolic volume by biplane Simpson rule 2-dimensional echocardiography. ResultsAbsolute values of delta rest-stress end-systolic pressure-volume relation were higher for exercise and dobutamine than for dipyridamole. In the overall population, an inverse relationship between end-systolic pressure-volume relation and end-diastolic volume was present at rest (r2 = 0.69, P < .001) and peak stress (r2 = 0.56, P < .001), but was absent if the delta rest-stress end-systolic pressure-volume relation was considered (r2 = 0.13). ConclusionsLeft ventricular end-diastolic volume does not affect the rest-stress changes in end-systolic pressure-volume relation in either normal or abnormal left ventricles during physical or pharmacological stress.Full English text available from:www.revespcardiol.org/en

Membrane fouling in a membrane bioreactor: A novel method for membrane surface morphology construction and its application in interaction energy assessment

A novel method was developed to construct membrane surface morphology in this study. The method was based on fractal geometry theory, and involved a modified Weierstrass-Mandelbrot (WM) function for modeling surface topography. With MATLAB programming, membrane surface could be graphically modeled. It was found that the fractal dimension (Df) and cutoff frequency (Ls) were the two key parameters affecting the topography, and the membrane surface constructed by this method could be very close to the real situation. The method was then applied in interaction energy assessment regarding membrane fouling in a membrane bioreactor. The procedure which combines membrane surface topography construction, surface element integration (SEI) method and the composite Simpson's rule was theoretically proposed. The surface properties of foulant and membrane were experimentally measured. Accordingly, a membrane surface was reconstructed for interaction energy assessment. With these data, quantitatively assessing interfacial interactions between foulant particles and a nearly-real rough membrane surface is expected to be firstly realized. This study provided a complete solution for quantitative assessment of interaction energies between a foulant particle and rough membrane surface.

Determination of Right Ventricular Volume by Combining Echocardiographic Distance Measurements.

The position of the right ventricle (RV), often partly behind the sternum, implies difficulties to image the RV free wall using transthoracic echocardiography (TTE) and consequently limits the possibilities of stroke volume calculations. The aim of this study was to evaluate whether the volume of the right ventricle (RV) can be determined by combining TTE distance measurements that do not need the RV free wall to be fully visualized. The RV volume was approximated by an ellipsoid composed of three distances. Distance measurements, modeled RV stroke volumes (RVSV), and RV ejection fraction (RVEF) were compared to reference values obtained from cardiac magnetic resonance (CMR) imaging for 12 healthy volunteers. Inter-modality comparisons showed that distance measurements were significantly underestimated in TTE compared to CMR. The modeled RV volumes using TTE distance measurements were underestimated compared to reference CMR volumes. There was, however, for TTE an agreement between modeled RVSV and left ventricular stroke volumes determined by biplane Simpson's rule. Similar agreement was shown between modeled RVSV based on CMR distance measurements and the CMR reference. Regarding RVEF, further studies including patients with a wider range of RVEF are needed to evaluate the method. In conclusion, the ellipsoid model of the RV provides good estimates of RVSVs, but volumes based on distance measurements from different modalities cannot be used interchangeably.

A semi-numerical model for near-critical angle scattering.

Numerous phenomena in the fields of physics and mathematics as seemingly different as seismology, ultrasonics, crystallography, photonics, relativistic quantum mechanics, and analytical number theory are described by integrals with oscillating integrands that contain three coalescing criticalities, a branch point, stationary phase point, and pole as well as accumulation points at which the speed of integrand oscillation is infinite. Evaluating such integrals is a challenge addressed in this paper. A fast and efficient numerical scheme based on the regularized composite Simpson's rule is proposed, and its efficacy is demonstrated by revisiting the scattering of an elastic plane wave by a stress-free half-plane crack embedded in an isotropic and homogeneous solid. In this canonical problem, the head wave, edge diffracted wave, and reflected (or compensating) wave each can be viewed as a respective contribution of a branch point, stationary phase point, and pole. The proposed scheme allows for a description of the non-classical diffraction effects near the "critical" rays (rays that separate regions irradiated by the head waves from their respective shadow zones). The effects include the spikes present in diffraction coefficients at the critical angles in the far field as well as related interference ripples in the near field.

Evaluation of Long Term Effect of RV Apical Pacing on Global LV Function by Echocardiography.

We very often face pacemaker implanted patients during follow-up with shortness of breath and effort intolerance inspite of normal clinical parameters. The aim of our study is to evaluate the cause of effort intolerance and probable cause of sub-clinical Congestive Cardiac Failure (CCF) in a case of long term Right Ventricular (RV) apical pacing on global Left Ventricular (LV) function non- invasively by echocardiography. We studied 54 patients (Male 42, Female 12) of complete heart block (CHB) with RV apical pacing (40 VVI and 14 DCP). Mean duration of pacing was 58+4 months. All patients underwent 24 hours Holter monitoring to determine the percentage of ventricular pacing beats. 2-D Echocardiography was done to assess the regional wall motion of abnormality and global LV ejection fraction by modified Simpson's rule. These methods were coupled with the Doppler derived Myocardial Performance Index (MPI), tissue Doppler imaging, and mechanical regional dyssynchrony with 3-D Echocardiography. Data were analysed from 54 RV- apical paced patients and compared with age and body surface area of 60 controlled subjects (Male 46, Female 14). Evaluation of LV function in 54 patients demonstrated regional wall motion abnormality and Doppler study revealed both LV systolic and diastolic dysfunction compare with control subjects (regional wall motion abnormality 80±6% vs 30±3% with p-value<0.0001) which is proportional to the percentage of ventricular pacing beats (mean paced beat 78%). Global LVEF 50±4% vs 60±2% (p-valve <0.0001) and MPI 0.46 ±0.12 v/s 0.36±0.09 (p-value <0.0001). RV-apical pacing induces iatrogenic electrical dyssynchrony which leads to remodeling of LV and produces mechanical dyssynchrony which is responsible for LV dysfunction. Alternate site of RV pacing and/or biventricular pacing should be done to maintain biventricular electrical synchrony which will preserve the LV function.

Noninvasive model including right ventricular speckle tracking for the evaluation of pulmonary hypertension.

To find parameters from transthorathic echocardiography (TTE) including speckle-tracking (ST) analysis of the right ventricle (RV) to identify precapillary pulmonary hypertension (PH). Forty-four patients with suspected PH undergoing right heart catheterization (RHC) were consecutively included (mean age 63.1 ± 14 years, 61% male gender). All patients underwent standardized TTE including ST analysis of the RV. Based on the subsequent TTE-derived measurements, the presence of PH was assessed: Left ventricular ejection fraction (LVEF) was calculated by Simpsons rule from 4Ch. Systolic pulmonary artery pressure (sPAP) was assessed with continuous wave Doppler of systolic tricuspid regurgitant velocity and regarded raised with values ≥ 30 mmHg as a surrogate parameter for RA pressure. A concomitantly elevated PCWP was considered a means to discriminate between the precapillary and postcapillary form of PH. PCWP was considered elevated when the E/e' ratio was > 12 as a surrogate for LV diastolic pressure. E/e' ratio was measured by gauging systolic and diastolic velocities of the lateral and septal mitral valve annulus using TDI mode. The results were then averaged with conventional measurement of mitral valve inflow. Furthermore, functional testing with six minutes walking distance (6MWD), ECG-RV stress signs, NT pro-BNP and other laboratory values were assessed. PH was confirmed in 34 patients (precapillary PH, n = 15, postcapillary PH, n = 19). TTE showed significant differences in E/e' ratio (precapillary PH: 12.3 ± 4.4, postcapillary PH: 17.3 ± 10.3, no PH: 12.1 ± 4.5, P = 0.02), LV volumes (ESV: 25.0 ± 15.0 mL, 49.9 ± 29.5 mL, 32.2 ± 13.6 mL, P = 0.027; EDV: 73.6 ± 24.0 mL, 110.6 ± 31.8 mL, 87.8 ± 33.0 mL, P = 0.021) and systolic pulmonary arterial pressure (sPAP: 61.2 ± 22.3 mmHg, 53.6 ± 20.1 mmHg, 31.2 ± 24.6 mmHg, P = 0.001). STRV analysis showed significant differences for apical RV longitudinal strain (RVAS: -7.5% ± 5.6%, -13.3% ± 4.3%, -14.3% ± 6.3%, P = 0.03). NT pro-BNP was higher in patients with postcapillary PH (4677.0 ± 7764.1 pg/mL, precapillary PH: 1980.3 ± 3432.1 pg/mL, no PH: 367.5 ± 420.4 pg/mL, P = 0.03). Patients with precapillary PH presented significantly more often with ECG RV-stress signs (P = 0.001). Receiver operating characteristics curve analyses displayed the most significant area under the curve (AUC) for RVAS (cut-off < -6.5%, AUC 0.91, P < 0.001), sPAP (cut-off > 33 mmHg, AUC 0.86, P < 0.001) and ECG RV stress signs (AUC 0.83, P < 0.001). The combination of these parameters had a sensitivity of 82.8% and a specificity of 17.2% to detect precapillary PH. The combination of non-invasive measurements allows feasible assessment of PH and seems beneficial for the differentiation between the pre- and postcapillary form of this disease.

Length Scales in Bayesian Automatic Adaptive Quadrature

Two conceptual developments in the Bayesian automatic adaptive quadrature approach to the numerical solution of one-dimensional Riemann integrals [Gh. Adam, S. Adam, Springer LNCS 7125, 1–16 (2012)] are reported. First, it is shown that the numerical quadrature which avoids the overcomputing and minimizes the hidden floating point loss of precision asks for the consideration of three classes of integration domain lengths endowed with specific quadrature sums: microscopic (trapezoidal rule), mesoscopic (Simpson rule), and macroscopic (quadrature sums of high algebraic degrees of precision). Second, sensitive diagnostic tools for the Bayesian inference on macroscopic ranges, coming from the use of Clenshaw-Curtis quadrature, are derived.

Impaired Left Atrial Reservoir Function in Metabolic Syndrome Predicts Symptoms in HFpEF Patients

&lt;p&gt;&lt;strong&gt;Background and Aim.&lt;/strong&gt; The Metabolic Syndrome (MetS) has been shown to be independently associated with increased risk for incident heart failure (HF) and coronary artery disease. The aim of this study was to investigate the additional effect of MetS on LA dysfunction in a group of patients with HF and preserved ejection fraction (HFpEF) and its relationship with symptoms.&lt;/p&gt;&lt;p&gt;&lt;strong&gt;Methods. &lt;/strong&gt;This study included 194 consecutive patients (age 62 ± 9 years) with stable HFpEF. LV dimensions, ejection fraction (EF), mitral annulus peak systolic excursion (MAPSE), myocardial velocities (s’, e’ and a’), LA dimensions and volumes were measured. Total LA emptying fraction (LA EF) was measured by Simpson rule volumes. Based on the NCEP-ATP III criteria, patients were divided into two groups; MetS (n=95) and non-MetS (n=108) and were compared with 34 age and gender matched controls.&lt;/p&gt;&lt;p&gt;&lt;strong&gt;Results. &lt;/strong&gt;Age and gender were not different between patients and control neither between MetS and non-Met. LV dimensions, EF and longitudinal function indices were also not different. The&lt;strong&gt; &lt;/strong&gt;MetS patients had higher LV mass index (p=0.038), lower septal and lateral e’ (p=0.003 and p=0.001, respectively) velocities, larger LA minimal volume (p=0.007) and lower LA EF (p&amp;lt;0.001) compared with the non-MetS patients. Age, LA EF and MetS independently predicted the NYHA class.&lt;/p&gt;&lt;strong&gt;Conclusions. &lt;/strong&gt;Despite no difference in LV systolic function, patients with HFpEF and MetS have worse LA emptying fraction, compared with HFpEF and non-MetS patients. In addition, LA reservoir function impairment and MetS independently predict patients limiting symptoms, thus add to a better understanding of HFpEF.

Explicit numerical solutions of a microbial survival model under nonisothermal conditions.

Differential equations used to describe the original and modified Geeraerd models were, respectively, simplified into an explicit equation in which the integration of the specific inactivation rate with respect to time was numerically approximated using the Simpson's rule. The explicit numerical solutions were then used to simulate microbial survival curves and fit nonisothermal survival data for identifying model parameters in Microsoft Excel. The results showed that the explicit numerical solutions provided an easy way to accurately simulate microbial survival and estimate model parameters from nonisothermal survival data using the Geeraerd models.

Fast computation of seabed spherical-wave reflection coefficients in geoacoustic inversion.

This paper develops a fast numerical approach to computing spherical-wave reflection coefficients (SWRCs) for layered seabeds, which provides substantial savings in computation time when used as the forward model for geoacoustic inversion of broadband seabed reflectivity data. The approach exploits the Sommerfeld-integral representation of SWRCs as the Hankel transform of a function proportional to the plane-wave reflection coefficient (PWRC), and applies Levin integration to the rapidly oscillating integrand cast as the product of a (pre-computed) media-independent matrix and a vector involving PWRCs at a sparse sampling of integration angles. Compared to conventional Simpson's rule integration for computation of the SWRC, the Levin integration yields speed-up factors of an order of magnitude or more. Further, it results in reduced memory requirements for storage of pre-computed quantities, a desirable property when a graphics processing unit (GPU) is used for parallel computation of SWRCs. The paper applies trans-dimensional Bayesian inversion to investigate the impact of forward modeling in terms of PWRCs and SWRCs on the estimation of geoacoustic parameters and uncertainties. Model comparisons are quantified in simulated- and measured-data inversions by comparing the estimated geoacoustic parameters to the true parameters or core measurements, respectively, and by calculating the deviance information criterion for model selection.

Derivation of magnetic fields on a metal cylinder excited by longitudinal and transverse magnetic dipole transmitters: I. Cylinder in unbounded dissipative dielectric medium

AbstractWe derive new and exact analytical and convergent integral representations for the frequency‐dependent complex magnetic fields Hz(a, ϕ, z) and Hϕ(a, ϕ, z) excited by oscillating point magnetic dipole transmitters on the surface of an infinitely long metal cylinder of radius a in an unbounded dissipative dielectric medium. Hz(a, ϕ, z) is calculated for a longitudinally oriented magnetic dipole parallel to the cylinder axis and Hϕ(a, ϕ, z) for a transversely oriented magnetic dipole perpendicular to the axis. The solutions are relevant to the computation of phase shifts and attenuations measured by electromagnetic propagation logging tools, which have oscillating longitudinal and transverse magnetic dipole transmitters either on a metal drill collar or on a cylindrical antenna pad. The integral representations can be readily evaluated using simple numerical integration algorithms, e.g., Simpson's rule, to accurately compute the complex magnetic fields on the cylinder surface. A second paper will address the two‐layer cylindrical media problem.

Some Improvements to a Third Order Variant of Newton’s Method from Simpson’s Rule

In this paper, we present three improvements to a three-point third order variant of Newton’s method derived from the Simpson rule. The first one is a fifth order method using the same number of functional evaluations as the third order method, the second one is a four-point 10th order method and the last one is a five-point 20th order method. In terms of computational point of view, our methods require four evaluations (one function and three first derivatives) to get fifth order, five evaluations (two functions and three derivatives) to get 10th order and six evaluations (three functions and three derivatives) to get 20th order. Hence, these methods have efficiency indexes of 1.495, 1.585 and 1.648, respectively which are better than the efficiency index of 1.316 of the third order method. We test the methods through some numerical experiments which show that the 20th order method is very efficient.

Mitral Annular Plane Systolic Excursion-Derived Ejection Fraction: A Simple and Valid Tool in Adult Males With Left Ventricular Systolic Dysfunction.

Validation of a mitral annular plane systolic excursion (MAPSE)-derived formula to calculate the ejection fraction where EF=4.8×MAPSE (mm)+5.8 in adult males with left ventricular (LV) dysfunction. Echocardiographic assessment of LV function generally requires expert echocardiographer and is somewhat subjective and prone to reader discordance. MAPSE has been suggested as a surrogate measurement for LV function. Prospective analysis of 170 male patients with systolic dysfunction by two-dimensional transthoracic echocardiography was carried out. MAPSE and ejection fraction measured by qualitative visual inspection, M-mode, and biplane modified Simpson's rule were measured. MAPSE-derived EF was compared against other conventional methods to measure EF using Bland-Altman analysis and independent t-test. There was a significant positive correlation between average MAPSE and EF measured by M-mode (r=0.554, P<0.001), Simpson's rule (r=0.585, P<0.001), and visual inspection (r=0.611, P<0.001). An average MAPSE cutoff value <= 5 provided the best balanced sensitivity (67.1%) and specificity (76.5%) to predict EF < 30%. The mean difference between MAPSE-derived EF and EF measured by visual inspection and by Simpson's method was 3.86±5.24% and 3.57±5.97%, respectively. The least mean difference of 0.5±5.69% was present between MAPSE-derived EF and M-mode-measured EF (P value 0.2). MAPSE-derived EF using the equation EF=4.8×MAPSE (mm)+5.8 is a valid technique in adult males with severely impaired LV EF.

Development of Numerical Intergration Program

This researchaims at developing anumerical integrationprogram and comparing calculated tolerances obtained from individual numerical integrationmodels. In developing the program, PDLC (Program Development Life Cycle) was employed, from design problem analysis,coding, and to program testing. The program developed derives numerical integration by using the trapezoidal rule and Simpson's rule. The research results can be used can serve as a numerical integrationprogram for pedagogical purposes, and the obtained test values were used to determine errors.

A novel partial integrodifferential equation-based framework for pricing interest rate derivatives under jump-extended short-rate models

Interest rate derivatives under jump-extended short-rate models have commonly been valued using lattice methods. Unfortunately, lattice methods have pitfalls, mainly in terms of accuracy, efficiency and ease of programming. This paper proposes a much faster and more accurate valuation method based on partial integrodifferential equations (PIDEs). Spatial differential and integral terms are discretized by fourthorder finite differences and Simpson's rule, respectively. For the time-stepping, we investigate a Crank-Nicolson scheme, an implicit-explicit (IMEX) scheme and an exponential time integration scheme. We demonstrate our method by pricing bonds as well as European and American options on both zero-coupon and coupon bonds under jump-extended constant-elasticity-of-variance processes. Through a variety of test problems, we demonstrate that our PIDE-based approach is remarkably superior to lattice methods in terms of accuracy, speed and ease of implementation. By way of example, pricing a one-year zero-coupon bond under the Vasicek model extended with exponential jumps, our method attains an accuracy of 104 in 0:09 seconds with the IMEX scheme, whereas a truncated tree takes 78 seconds to reach the same accuracy.