Approximate Propagation Research Articles

Phase Portraits and Abundant Soliton Solutions of a Hirota Equation with Higher-Order Dispersion

The Hirota equation, an advanced variant of the nonlinear Schrödinger equation with cubic nonlinearity, incorporates time-delay adjustments and higher-order dispersion terms, offering an enhanced approximation for wave propagation in optical fibers and oceanic systems. By utilizing the traveling wave transformation generated from Lie point symmetry analysis with the combination of generalized exponential differential rational function and modified Bernoulli sub-ODE techniques, several traveling wave solutions, such as periodic, singular-periodic, and kink solitons, emerge. To examine the solutions visually, parametric values are adjusted to create 3D, contour, and 2D illustrations. Additionally, the dynamic properties of the model are explored through bifurcation analysis. The exact results demonstrate that both techniques are practical and robust.

What Remote PPG Oximetry Tells Us about Pulsatile Volume?

While pulse oximetry using remote photoplethysmography (rPPG) is used in medicine and consumer health, sound theoretical foundations for this methodology are not established. Similarly to traditional pulse oximetry, rPPG oximetry uses two wavelengths to calculate the tissue oxygenation using the so-called ratio-of-ratios, R. However, the relationship between R and tissue oxygenation has not been derived analytically. As such, rPPG oximetry relies mostly on empirical methods. This article aimed to build theoretical foundations for pulse oximetry in rPPG geometry. Using the perturbation approach in diffuse approximation for light propagation in tissues, we obtained an explicit expression of the AC/DC ratio for the rPPG signal. Based on this ratio, the explicit expression for "ratio-of-ratios" was obtained. We have simulated the dependence of "ratio-of-ratios" on arterial blood saturation across a wide range (SaO2 = 70-100%) for several commonly used R/IR light sources (660/780, 660/840, 660/880, and 660/940 nm) and found that the obtained relationship can be modeled by linear functions with an extremely good fit (R2 = 0.98-0.99) for all considered R/IR pairs. Moreover, the location of the pulsatile volume can be extracted from rPPG data. From experimental data, we found that the depth of blood pulsations in the human forehead can be estimated as 0.6 mm on the arterial side, which points to the papillary dermis/subpapillary vascular plexus origin of the pulsatile volume.

3D Hybrid modeling for the ultrasonic phased array inspection of porosity in heavy plates: Simulation and experimental validation

Quantitative NDT modeling and simulation tools are important in design and fabrication of ultrasonic phased array probes with optimum characteristics and reduce the number of experimental efforts and development time. In order to achieve high accuracy (Amplitude deviation < 1 dB) with a low computational time in the simulation, the modeling tools require not only the high-frequency approximation of the ultrasonic wave propagation but also the high-precision electro-acoustic coupling model. This paper describes a computationally efficient 3D time-dependent hybrid model for the simulation of ultrasonic phased array inspection of porosity in heavy plates. This method combines the results of both 3D finite element analysis (PZFLex-FEA) and the semi-analytical ultrasonic simulation “Ray-Tracing”. As a first step, we demonstrate the FEA modeling of an ultrasonic phased array probe, which includes the full 1-3 piezocomposite design with the arrangement of array elements, conducting electrodes, multi-phase backing material, acoustical matching layer and electrical circuit of a 50 Ohm coaxial cable. In the current FEA simulation, we considered the acoustical and electrical cross-talk between the adjacent elements of an ultrasonic array sensor. In the second step, the FEA-simulated ultrasonic pulse-echo signal and its frequency spectrum of an array element in the farfield region are further used as the reference input signal in the CIVA-UT software to simulate the imaging of porosity in heavy plates, quantitatively. The design of the 3D wedge-geometry of the application specific array sensor is modeled and optimized using the CIVA raytracing and multi-phase backing models. The real excitation pulse of the industrial ultrasonic testing equipment “ROMIS (ROSEN Modular Inspection System)” is also included in the simulation to achieve high accuracy in the current hybrid modeling. The dependency of the detectability of the porosity on the frequency of the array sensor and the active element group so-called “Virtual Probe” are quantitatively analyzed. Finally, the 3D hybrid model simulated Time Corrected Gain (TCG) curves of the porosity like defects in the reference steel plates are compared with the real experiments and a good quantitative agreement is achieved.

Performance of range-separated long-range SOPPA short-range density functional theory method for vertical excitation energies.

In this paper, benchmark results are presented on the calculation of vertical electronic excitation energies using a long-range second-order polarization propagator approximation (SOPPA) description with a short-range density functional theory description based on the Perdew-Burke-Ernzerhof (PBE) functional. The excitation energies are investigated for 132 singlet states and 71 triplet states across 28 medium-sized organic molecules. The results show that overall SOPPA-srPBE always performs better than PBE and that SOPPA-srPBE performs better than SOPPA for singlet states, but slightly worse than SOPPA for triplet states when CC3 results are the reference values.

Predicting second-harmonic generation in shear wave beams in tissue-mimicking phantoms

Planar nonlinear shear waves in isotropic media are subject to only cubic nonlinear effects and generate only odd harmonics during propagation. However, wavefront curvature in shear wave beams breaks the symmetry and yields quadratic nonlinear effects, and therefore, a second harmonic may be present in shear wave beams depending on the polarization of the wave. Focused radially polarized shear wave beams have been generated in tissue-mimicking phantoms [Cormack et al., IEEE TBME (2024)], and it is postulated that the second harmonic may be used to estimate the quadratic nonlinearity in such a medium as an additional biomarker for diseased tissue. Here, an analytical solution obtained in the paraxial approximation for nonlinear propagation of a shear wave beam in an absorbing medium [Spratt et al., AIP Conf. Proc. 1685, 080007 (2015)] is used to characterize the strength of second-harmonic generation. Feasibility of measuring the second harmonic experimentally in a weakly nonlinear, radially polarized focused shear wave beam propagating in a tissue-like medium is explored. Second-harmonic generation in focused shear wave beams with other polarizations is discussed. [P.G.K. was supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

Single-stage approach for estimating optical parameters in spectral quantitative photoacoustictomography.

In quantitative photoacoustic tomography, the optical parameters of a target, most importantly the concentrations of chromophores such as deoxygenated and oxygenated hemoglobin, are estimated from photoacoustic data measured on the boundary of the target. In this work, a numerical approximation of a forward model for spectral quantitative photoacoustic tomography is constructed by utilizing the diffusion approximation for light propagation, the acoustic wave equation for ultrasound propagation, and spectral models of optical absorption and scattering to describe the wavelength dependence of the optical parameters. The related inverse problem is approached in the framework of Bayesian inverse problems. Concentrations of four chromophores (deoxygenated and oxygenated hemoglobin, water, and lipid), two scattering parameters (reference scattering and scattering power), and the Grüneisen parameter are estimated in a single-stage from photoacoustic data. The methodology is evaluated using numerical simulations in different full-view and limited-view imaging settings. The results show that, utilizing spectral data and models, the spectral optical parameters and the Grüneisen parameter can be simultaneously estimated. Furthermore, the approach can also be utilized in limited-view imaging situations.

On the performance of HRPA(D) for NMR spin–spin coupling constants: Smaller molecules, aromatic and fluoroaromatic compounds

In this study, the performance of the doubles-corrected higher random-phase approximation [HRPA(D)] has been investigated in calculations of nuclear magnetic resonance spin–spin coupling constants (SSCCs) for 58 molecules with the experimental values used as the reference values. HRPA(D) is an approximation to the second-order polarization propagator approximation (SOPPA) and is, therefore, computationally less expensive than SOPPA. HRPA(D) performs comparable and sometimes even better than SOPPA, and therefore, when calculating SSCCs, it should be considered as an alternative to SOPPA. Furthermore, it was investigated whether a coupled-cluster singles, doubles and perturbative triples [CCSD(T)] or Møller-Plesset second order (MP2) geometry optimization was optimal for a SOPPA and a HRPA(D) SSCC calculation for eight smaller molecules. CCSD(T) is the optimal geometry optimization for the SOPPA calculation, and MP2 was optimal for HRPA(D) SSCC calculations.

Exploring Alternate Methods for the Calculation of High-Level Vibrational Corrections of NMR Spin-Spin Coupling Constants.

Traditional nuclear magnetic resonance (NMR) calculations typically treat systems with a Born-Oppenheimer-derived electronic wave function that is solved for a fixed nuclear geometry. One can numerically account for this neglected nuclear motion by averaging over property values for all nuclear geometries with a vibrational wave function and adding this expectation value as a correction to an equilibrium geometry property value. Presented are benchmark coupled-cluster singles and doubles (CCSD) vibrational corrections to spin-spin coupling constants (SSCCs) computed at the level of vibrational second-order perturbation theory (VPT2) using the vibrational averaging driver of the CFOUR program. As CCSD calculations of vibrational corrections are very costly, cheaper electronic structure methods are explored via a newly developed Python vibrational averaging program within the Dalton Project. Namely, results obtained with the second-order polarization propagator approximation (SOPPA) and density functional theory (DFT) with the B3LYP and PBE0 exchange-correlation functionals are compared to the benchmark CCSD//CCSD(T) and experimental values. CCSD//CCSD(T) corrections are also combined with literature CC3 equilibrium geometry values to form the highest-order vibrationally corrected values available (i.e., CC3//CCSD(T) + CCSD//CCSD(T)). CCSD//CCSD(T) statistics showed favorable statistics in comparison to experimental values, albeit at an unfavorably high computational cost. A cheaper CCSD//CCSD(T) + B3LYP method showed quite similar mean absolute deviation (MAD) values as CCSD//CCSD(T), concluding that CCSD//CCSD(T) + B3LYP is optimal in terms of cost and accuracy. With reference to experimental values, a vibrational correction was not worth the cost for all of the other methods tested. Finally, deviation statistics showed that CC3//CCSD(T) + CCSD//CCSD(T) vibrational-corrected equilibrium values deteriorated in comparison to CCSD//CCSD(T) attributed to the use of a smaller basis set or lack of solvation effects for the CC3 equilibrium calculations.

3D Hybrid Modeling for the Ultrasonic Phased Array Inspection of Porosity in Heavy Plates: Simulation and Experimental Validation

Quantitative non-destructive testing (NDT) modelling and simulation tools are important in design and fabrication of ultrasonic phased array (UTPA) sensors with optimum characteristics and reduce the number of experimental efforts and development time. In order to achieve high accuracy with a low computational time in the simulation, the modeling tools require not only the high-frequency approximation of the ultrasonic wave propagation but also the high-precision electro-acoustic coupling model. In this work, an application specific 4 MHz, 32-element linear phased array probe with an optimized angle beam wedge for the inspection of porosity in heavy plates was developed based on the new 3D hybrid modeling approach. As a first step, we performed the FEA modeling of an UTPA sensor, which includes the full 1-3 piezo-composite design with the arrangement of array elements, conducting electrodes, multi-phase damping material, acoustical matching layer and the electrical circuit of a 50 Ω coaxial cable. In addition, the real excitation pulse of an industrial UT equipment was included in the current modelling to achieve the high accuracy in the simulated result. In the second step, the FEA simulated pulse-echo characteristics of an array element were further used as the reference signal in the CIVA-UT Software to simulate the imaging of porosity in heavy plates, quantitatively. Furthermore, the 3D wedge-geometry of the array sensor was modeled and optimized to suppress the unwanted wedge-boundary reflected echoes and to improve the signal-to-noise ratio (SNR) in the detected signals. The 3D hybrid model simulated Time Corrected Gain (TCG) curves for the 3 mm SDH at different depths in the reference steel plate were compared with the real experiments and a good quantitative agreement was achieved.

An Enhanced Method to Achieve Exact DC Values for Frequency-dependent Transmission lines

This paper proposes an improved method to enhance the dc response of a frequency-dependent transmission line model used in EMT studies. A modification to the rational function approximation of propagation and characteristic admittance matrices of a transmission line is introduced to enforce exact dc values at 0 Hz. Furthermore, weighting factors are applied to improve accuracy at low frequencies. Finally, the order of the propagation function is reduced to decrease the computational effort.The validity of the proposed approach is demonstrated using examples involving underground cables and overhead lines. First, the effect of dc correction is demonstrated by comparing transmission line frequency domain characteristics. In addition, time domain simulations via open and short circuit conditions show a more accurate simulation of HVDC transmission lines with the proposed method.

Approximation of High-Frequency Wave Propagation in Dispersive Media

Approximation of High-Frequency Wave Propagation in Dispersive Media

New pecJ-n (n = 1, 2) Basis Sets for Selenium Atom Purposed for the Calculations of NMR Spin-Spin Coupling Constants Involving Selenium.

We present new compact pecJ-n (n = 1, 2) basis sets for the selenium atom developed for the quantum-chemical calculations of NMR spin-spin coupling constants (SSCCs) involving selenium nuclei. These basis sets were obtained at the second order polarization propagator approximation with coupled cluster singles and doubles amplitudes (SOPPA(CCSD)) level with the property-energy consistent (PEC) method, which was introduced in our previous papers. The existing SSCC-oriented selenium basis sets are rather large in size, while the PEC method gives more compact basis sets that are capable of providing accuracy comparable to that reached using the property-oriented basis sets of larger sizes generated with a standard even-tempered technique. This is due to the fact that the PEC method is very different in its essence from the even-tempered approaches. It generates new exponents through the total optimization of angular spaces of trial basis sets with respect to the property under consideration and the total molecular energy. New basis sets were tested on the coupled cluster singles and doubles (CCSD) calculations of SSCCs involving selenium in the representative series of molecules, taking into account relativistic, solvent, and vibrational corrections. The comparison with the experiment showed that the accuracy of the results obtained with the pecJ-2 basis set is almost the same as that provided by a significantly larger basis set, aug-cc-pVTZ-J, while that achieved with a very compact pecJ-1 basis set is only slightly inferior to the accuracy provided by the former.

Indirect nuclear spin-spin couplings with third-order contributions added to the SOPPA method.

In this article, a modification of the second-order polarization propagator approximation (SOPPA) method is introduced and illustrated for the calculation of the indirect nuclear spin-spin couplings. The standard SOPPA method, although cheaper in terms of computational cost, offers less accurate results than the ones obtained with coupled cluster methods. A new method, named SOPPA+A3-3, was therefore developed by adding the terms of the third-order A matrix that rely on the second-order double amplitudes. The performance of this third-order contribution was studied using the coupled cluster singles and doubles method as a reference, calculating the spin-spin couplings of molecules of diverse sizes and compositions, and comparing them to the SOPPA method. The results show that inclusion of this third-order contribution gives more accurate results than the standard SOPPA method with a level of accuracy close to that of the coupled cluster method with only a small increase in the computational cost of the response calculation that dominates the computational cost for small- to medium-sized molecules. The implementation of the first contributions to the third-order polarization propagator approximation in the Dalton program, thus, already shows a significant change in these molecular properties over those obtained with the standard SOPPA method.

An Efficient Approximate Expectation Propagation Detector With Block-Diagonal Neumann-Series

Expectation propagation (EP) achieves near-optimal performance for large-scale multiple-input multiple-output (L-MIMO) detection, however, at the expense of unaffordable matrix inversions. To tackle the issue, several low-complexity EP detectors have been proposed. However, they all fail to exploit the properties of channel matrices, thus resulting in unsatisfactory performance in non-ideal scenarios. To this end, in this paper, a block-diagonal Neumann-series-based expectation propagation approximation (BD-NS-EPA) algorithm is proposed, which is applicable for both ideal uncorrelated channels and the correlated channels with multiple-antenna user equipment system. First, a block-diagonal-based Neumann iteration is employed, which skillfully exerts the main information of the channels while reducing computational cost. An adjustable sorting message updating scheme then is introduced to reduce the update of redundant nodes during iterations. Numerical results show that, for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$128\times 32$ </tex-math></inline-formula> MIMO with the non-ideal channel, the proposed algorithm exhibits 0.3 dB away from the original EP when bit error-rate (BER) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$=10^{-3}$ </tex-math></inline-formula> , at the cost of mere 3% normalized complexity. The implementation results on SMIC 65-nm CMOS technology suggest that the proposed detector can achieve 1.252 Gbps/W and 0.275 Mbps/kGE hardware efficiency, further demonstrating that the proposed detectors can achieve a good trade-off between error-rate performance and hardware efficiency.

The Importance of the Density Functional Theory Exchange-Correlation Hartree-Fock Term in Magnetic Resonance: Application to an Aqueous Environment.

Within the framework of Density Functional Theory (DFT), the relevance of the term Hartree-Fock exchange (HFE) for a variety of molecular properties is a critical point. For this reason, we spend efforts to understand these relationships in nuclear magnetic resonance (NMR) parameters in a water solvent. This work takes advantage of the appropriate aug-cc-pVTZ-J basis set and the Minnesota family of DFT methods, which consider different portions of HFE contributions. With regard to solvent participation, the results are based on a sequential Monte Carlo/Quantum Mechanics procedure, which builds the structures of the liquid under realistic thermodynamic conditions. Compared to the accurate results of second-order polarization propagator approximation (SOPPA) and experimental data, all NMR parameters show a huge dependence on the size of the HFE contribution. For instance, the inclusion of this term in 1JOH and 2JHH indirect spin-spin couplings does vary with 49.661 and 25.459 Hz, respectively. The M06-HF method accounts for 100% of HFE and better matches the σO and σH shielding constants. On the other hand, 1JOH and 2JHH demand a medium contribution (54% of HFE), the best description being associated with the M06-2X method. Thus, the dependence varies regarding the phenomenology of the property in focus and the order for independent treatments. For elements that participate in hydrogen bonds simultaneously as donor and acceptor actors, the results indicate that explicit solvent molecules must be considered in the quantum mechanical calculations for better modeling of paramagnetic shielding constants.

Improved Algorithms of Quantum Imaginary Time Evolution for Ground and Excited States of Molecular Systems.

Quantum imaginary time evolution (QITE) is a recently proposed quantum-classical hybrid algorithm that is guaranteed to reach the lowest state of systems. In this study, we present several improvements on QITE, mainly focusing on molecular applications. We analyze the derivation of the underlying QITE equation order-by-order and suggest a modification that is theoretically well founded. Our results clearly indicate the soundness of the here-derived equation, enabling a better approximation of the imaginary time propagation by a unitary. We also discuss how to accurately estimate the norm of an imaginary-time-evolved state and applied it to excited-state calculations using the quantum Lanczos algorithm. Finally, we propose the folded-spectrum QITE scheme as a straightforward extension of QITE for general excited-state simulations. The effectiveness of all these developments is illustrated by simulations with or without noise effect, offering further insights into quantum algorithms for imaginary time evolution.

Exact solutions for the time-evolution of quantum spin systems under arbitrary waveforms using algebraic graph theory

A general approach is presented that offers exact analytical solutions for the time-evolution of quantum spin systems during parametric waveforms of arbitrary functions of time. The proposed method utilizes the path-sum method that relies on the algebraic and combinatorial properties of walks on graphs. A full mathematical treatment of the proposed formalism is presented, accompanied by an implementation in Matlab. Using computation of the spin dynamics of monopartite, bipartite, and tripartite quantum spin systems under chirped pulses as exemplar parametric waveforms, it is demonstrated that the proposed method consistently outperforms conventional numerical methods, including ODE integrators and piecewise-constant propagator approximations.

Orbital stability of smooth solitary waves for the Degasperis-Procesi equation

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L 3 L^3 , but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L 2 L^2 -norm, resulting in L 3 L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L 2 L^2 norm for DP equation, which cannot be used to control the L 3 L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L ∞ L^\infty orbital norm of the perturbation is bounded above by a function of its L 2 L^2 orbital norm, yielding the higher order control and the orbital stability in the L 2 ∩ L ∞ L^2\cap L^\infty space.

Design of the high-dimensional SPKF algorithm and application

In the framework of Kalman filtering, the key points is the estimation of the covariance of prediction error which is used to calculate the Kalman filter gain. For the traditional algorithm, e.g. extended Kalman filter, the priori information is obtained on the basis of the Taylor expansion approximation of the nonlinear system. Because the linear propagation approximation of prior information cannot truly reflect the evolution of the probability distribution of the prediction errors in nonlinear systems, however, the traditional method affects the estimation accuracy of the extended Kalman filter. To solve these problems and be easy to apply to high-dimensional nonlinear systems, this paper improves the SPKF algorithm. Firstly, the prediction error propagation model is re-established to eliminate the approximation error of Taylor expansion in the derivation of the probability density function of the prediction error. Then, a Fisher matrix calculation method that is easier for high-dimensional system is obtained by defining the calculation method of Hilbert inner product of vector and matrix function. To illustrate the merits of the novel algorithm, this paper took the oxygen generation system by using electrolytic water in space station as a case study and conducted the state estimation by respectively using the extended Kalman filter, unscented Kalman filter and high-dimensional stochastic projection Kalman filter. The simulation results shown that the new algorithm can obtain higher estimation precision than the traditional algorithm.

Orbital stability of the sum of smooth solitons in the Degasperis-Procesi equation

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model as an asymptotic approximation for the unidirectional propagation of shallow water waves. This work is to establish the L2∩L∞ orbital stability of a wave train containing N smooth solitons which are well separated. The main difficulties stem from the subtle nonlocal structure of the DP equation. One consequence is that the energy space of the DE equation based on the conserved quantity induced by the translation symmetry is only equivalent to the L2-norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. Our remedy is to introduce a priori estimates based on certain smooth initial conditions. Moreover, another consequence is that the nonlocal structure of the DP equation significantly complicates the verification of the monotonicity of local momentum and the positive definiteness of a refined quadratic form of the orthogonalized perturbation.