Random Batch Particle Methods for the Homogeneous Landau Equation

We consider in this paper random batch particle methods for efficiently solving the homogeneous Landau equation in plasma physics. The methods are stochastic variations of the particle methods proposed by Carrillo et al. [J. Comput. Phys.: X 7: 100066, 2020] using the random batch strategy. The collisions only take place inside the small but randomly selected batches so that the computational cost is reduced to $\mathcal{O}(N)$ per time step. Meanwhile, our methods can preserve the conservation of mass, momentum, energy and the decay of entropy. Several numerical examples are performed to validate our methods.     

Examining Saddle Point Searches in the Context of Off-Lattice Kinetic Monte Carlo

In calculating the time evolution of an atomic system on diffusive timescales,off-lattice kinetic Monte Carlo (OLKMC) can sometimes be used to overcome the limitations of Molecular Dynamics. OLKMC relies on the harmonic approximation toTransition State Theory, in which the rate of rare transitions from one energy minimum to a neighboring minimum scales exponentially with an energy barrier on thepotential energy surface. This requires locating the index-1 saddle point, commonlyreferred to as a transition state, that separates two neighboring energy minima. Inmodeling the evolution of an atomic system, it is desirable to find all the relevant transitions surrounding the current minimum. Due to the large number of minima on thepotential energy surface, exhaustively searching the landscape for these saddle pointsis a challenging task. In examining the particular case of isolated Lennard-Jones clusters of around 50 particles, we observe very slow convergence of the total number ofsaddle points found as a function of successful searches. We seek to understand thisbehavior by modeling the distribution of successful searches and sampling this distribution to create a stochastic process that mimics this behavior. Finally, we will discussan improvement to a rejection scheme for OLKMC where we terminate searches thatappear to be failing early in the search process.     

Monte Carlo Simulations of Light Distributions in an Embedded Tumour Model: Studies of Selectivity in Photodynamic Therapy

. It is well known that photosensitisers in photodynamic therapy (PDT) tend to localise in greater concentrations in tumours. This attractive feature may help confer on PDT the potential to selectivity destroy tumours while sparing the surrounding normal tissue. In this paper Monte Carlo simulations were used to study light distributions in a simple model consisting of tumour embedded in surrounding normal tissue subjected to superficial irradiance. The Monte Carlo model was coded to allow modelling of arbitrary geometries and multiple tissue types. This permitted the use of different optical properties for tumour and normal tissue. Two simulations were run using optical coefficients appropriate to breast carcinoma in adipose tissue and liver tumour in liver. Contours of equal fluence were plotted against depth for both simulations. Contours of equal photodynamic dose (fluence×drug concentration) were plotted for various tumour/normal drug ratios. By assuming a threshold for necrosis it was possible to estimate the depth of damage in the normal tissue and tumour simultaneously. A greater depth of selective tumour damage was observed in the breast tissue simulation for a given drug ratio due to the higher penetration of light compared to the liver. For a tumour to normal ratio of 4:1 selective damage to a depth greater than 4 mm was observed in the breast simulation compared to almost 3 mm in the case of the liver model. Paper received 1 December 1998; accepted after revision 13 July 1999.     

DECAY: A Monte Carlo Library for the Decay of a Particle with ROOT Compatibility

Recently, there is a need for a general-purpose event generator for decaysof an elementary particle or a hadron to a state of higher multiplicity $(N ≥ 2)$ that issimple to use and universal. We present the structure of such a library for preparationof generators that generate kinematics of decay processes and can be used to integrateany matrix element squared over phase space of this decay. Some test examples arepresented, and results are compared with results known from the literature. As one ofexamples we consider the Standard Model Higgs boson decay into four leptons. Thegenerators discussed here are compatible with the ROOT interface.     

Some Random Batch Particle Methods for the Poisson-Nernst-Planck and Poisson-Boltzmann Equations

We consider in this paper random batch interacting particle methods forsolving the Poisson-Nernst-Planck (PNP) equations, and thus the Poisson-Boltzmann(PB) equation as the equilibrium, in the external unbounded domain. To justify thesimulation in a truncated domain, an error estimate of the truncation is proved inthe symmetric cases for the PB equation. Then, the random batch interacting particle methods are introduced which are $mathcal{O}(N)$ per time step. The particle methods cannot only be considered as a numerical method for solving the PNP and PB equations,but also can be used as a direct simulation approach for the dynamics of the chargedparticles in solution. The particle methods are preferable due to their simplicity andadaptivity to complicated geometry, and may be interesting in describing the dynamics of the physical process. Moreover, it is feasible to incorporate more physical effectsand interactions in the particle methods and to describe phenomena beyond the scopeof the mean-field equations.     

In vitro fluorescence measurements and Monte Carlo simulation of laser irradiation propagation in porcine skin tissue

In dermatology, the in vivo spectral fluorescence measurements of human skin can serve as a valuable supplement to standard non-invasive techniques for diagnosing various skin diseases. However, quantitative analysis of the fluorescence spectra is complicated by the fact that skin is a complex multi-layered and inhomogeneous organ, with varied optical properties and biophysical characteristics. In this work, we recorded, in vitro, the laser-induced fluorescence emission signals of healthy porcine skin, one of the animals, which is considered as one of the most common models for investigations related to medical diagnostics of human cutaneous tissues. Differences were observed in the form and intensity of the fluorescence signal of the porcine skin, which can be attributed to the different concentrations of the native fluorophores and the variable physical and biological conditions of the skin tissue. As the light transport in the tissue target is directly influencing the absorption and the fluorescence emission signals, we performed Monte Carlo simulation of the light distribution in a five-layer model of human skin tissue, with a pulsed ultraviolet laser beam.     

A Symplectic Based Neural Network Algorithm for Quantum Controls under Uncertainty

Robust quantum control with uncertainty plays a crucial role in practicalquantum technologies. This paper presents a method for solving a quantum controlproblem by combining neural network and symplectic finite difference methods. Theneural network approach provides a framework that is easy to establish and train. Atthe same time, the symplectic methods possess the norm-preserving property for thequantum system to produce a realistic solution in physics. We construct a generalhigh dimensional quantum optimal control problem to evaluate the proposed methodand an approach that combines a neural network with forward Euler’s method. Ouranalysis and numerical experiments confirm that the neural network-based symplecticmethod achieves significantly better accuracy and robustness against noises.     

Analysis and Application of Single Level,Multi-Level Monte Carlo and Quasi-Monte Carlo Finite Element Methods for Time-Dependent Maxwell's Equations with Random Inputs

This article is devoted to three quadrature methods for the rapid solutionof stochastic time-dependent Maxwell's equations with uncertain permittivity, permeability and initial conditions. We develop the mathematical analysis of the error estimate for single level Monte Carlo method, multi-level Monte Carlo method, and thequasi-Monte Carlo method. The theoretical results are supplemented by numericalexperiments.     

应用蒙特卡罗模拟优化肾功能亢进患者万古霉素给药方案的研究

目的根据万古霉素在肾功能亢进(ARC)和肾功能正常患者中的药物代谢动力学/药物效应动力学(PK/PD)原理,应用蒙特卡罗模拟优化ARC和肾功能正常患者万古霉素对常用革兰阳性菌的初始给药方案。方法使用文献公开发表的ARC及肾功能正常患者中的群体PK模型,收集2017至2019年南京医科大学附属苏州医院甲氧西林耐药金黄色葡萄球菌(MRSA)、甲氧西林耐药表皮葡萄球菌(MRSE)、粪肠球菌、屎肠球菌的最小抑菌浓度(MIC)分布。采用蒙特卡罗模拟计算不同万古霉素日剂量的目标获得概率(PTA)和累积反应分数(CFR),以预测达到与PK/PD参数指标AUC0~24 h/MIC的万古霉素给药剂量,推荐最佳给药方案。结果蒙特卡罗模拟结果显示,MIC=0.5时ARC患者对革兰阳性菌万古霉素日剂量2.0 g/d的PTA为97.30%,而肾功能正常患者只需要1.5 g/d,PTA即可达99.16%。MIC=1时,肾功能正常患者需要万古霉素2.5 g/d,PTA可达92.91%,而ARC患者需要万古霉素3.5 g/d,PTA才可达88.71%。当MIC≥2时,万古霉素常规推荐的给药剂量(2.0~4.0 g/d)很难达到理想的PTA值。对于常见革兰阳性菌,肾功能正常患者需要万古霉素3.0 g/d,CFR可达91.36%,而ARC患者需要万古霉素4.5 g/d,CFR可达90.60%。MRSA和屎肠球菌的MIC≤1的占比分别为97.25%和94.50%。肾功能正常患者MRSA感染给予万古霉素2.5 g/d,CFR可达93.95%,屎肠球菌给予万古霉素3.0 g/d,CFR可达94.46%;而ARC患者MRSA感染给予万古霉素3.5 g/d,CFR可达91.42%,屎肠球菌给予4.0 g/d,CFR可达93.07%。MRSE的MIC=1和2占比分别为64.64%和27.44%,粪肠球菌为77.09%和20.13%。肾功能正常患者粪肠球菌感染给予万古霉素4.0 g/d,CFR可达92.55%,MRSE需要万古霉素4.5 g/d,CFR可达92.79%,而ARC患者MRSE和粪肠球菌需要万古霉素6.0 g/d,CFR才可达88.35%和92.03%。当革兰阳性菌MIC≤0.5时ARC患者给予万古霉素2.0 g/d和肾功能正常者予1.5 g/d,PTA均可≥90%。MIC=1时肾功能正常患者推荐万古霉素3.0 g/d,ARC患者推荐万古霉素4.5 g/d,PTA和CFR可达90%以上。MIC≥2的革兰阳性菌万古霉素常规给药日剂量(2.0~4.0 g/d)对于ARC和肾功能正常患者均很难达到目标的PTA及CFR。结论与肾功能正常患者相比,ARC患者万古霉素需要更大的日剂量才可达目标PTA和CFR。MIC≥2的革兰阳性菌感染,建议改用其他敏感性更好的抗革兰阳性菌药物治疗。     

A Highly Scalable Boundary Integral Equation and Walk-on-Spheres (BIE-WOS) Method for the Laplace Equation with Dirichlet Data

In this paper, we study a highly scalable communication-free parallel domain boundary decomposition algorithm for the Laplace equation based on a hybrid method combining boundary integral equations and walk-on-spheres (BIE-WOS)method, which provides a numerical approximation of the Dirichlet-to-Neumann(DtN) mapping for the Laplace equation. The BIE-WOS is a local method on theboundary of the domain and does not require a structured mesh, and only needs acovering of the domain boundary by patches and a local mesh for each patch for a local BIE. A new version of the BIE-WOS method with second kind integral equations isintroduced for better error controls. The effect of errors from the Feynman-Kac formulabased path integral WOS method on the overall accuracy of the BIE-WOS method isanalyzed for the BIEs, especially in the calculation of the right hand sides of the BIEs.For the special case of flat patches, it is shown that the second kind integral equationof BIE-WOS method can be simplified where the local BIE solutions can be given inclosed forms. A key advantage of the parallel BIE-WOS method is the absence of communications during the computation of the DtN mapping on individual patches ofthe boundary, resulting in a complete independent computation using a large numberof cluster nodes. In addition, the BIE-WOS has an intrinsic capability of fault tolerance for exascale computations. The nearly linear scalability of the parallel BIE-WOSmethod on a large-scale cluster with 6400 CPU cores is verified for computing the DtNmapping of exterior Laplace problems with Dirichlet data for several domains.     

Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams

Finite-width light distributions in arterial tissue during Argon laser irradiation (476 nm) are simulated using the Monte Carlo method. Edge effects caused by radial diffusion of the light extend +/- 1.5 mm inward from the perimeter of a uniform incident beam. For beam diameters exceeding 3 mm the light distribution along the central axis can be described by the one-dimensional solution for an infinitely wide beam. The overlapping edge effects for beam diameters smaller than 3 mm reduce the penetration of the irradiance in the tissue. The beam profile influences the light distribution significantly. The fluence rates near the surface for a Gaussian beam are two times higher on the central axis and decrease faster radially than for a flat profile. The diverging light from a fiber penetrates tissue in a manner similar to collimated light.     

Diagnosis of breast cancer using diffuse reflectance spectroscopy: Comparison of a Monte Carlo versus partial least squares analysis based feature extraction technique

BACKGROUND AND OBJECTIVE: We explored the use of diffuse reflectance spectroscopy in the ultraviolet-visible (UV-VIS) spectrum for the diagnosis of breast cancer. A physical model (Monte Carlo inverse model) and an empirical model (partial least squares analysis) based approach, were compared for extracting diagnostic features from the diffuse reflectance spectra. STUDY DESIGN/METHODS: The physical model and the empirical model were employed to extract features from diffuse reflectance spectra measured from freshly excised breast tissues. A subset of extracted features obtained using each method showed statistically significant differences between malignant and non-malignant breast tissues. These features were separately input to a support vector machine (SVM) algorithm to classify each tissue sample as malignant or non-malignant. RESULTS AND CONCLUSIONS: The features extracted from the Monte Carlo based analysis were hemoglobin saturation, total hemoglobin concentration, beta-carotene concentration and the mean (wavelength averaged) reduced scattering coefficient. Beta-carotene concentration was positively correlated and the mean reduced scattering coefficient was negatively correlated with percent adipose tissue content in normal breast tissues. In addition, there was a statistically significant decrease in the beta-carotene concentration and hemoglobin saturation, and a statistically significant increase in the mean reduced scattering coefficient in malignant tissues compared to non-malignant tissues. The features extracted from the partial least squares analysis were a set of principal components. A subset of principal components showed that the diffuse reflectance spectra of malignant breast tissues displayed an increased intensity over wavelength range of 440-510 nm and a decreased intensity over wavelength range of 510-600 nm, relative to that of non-malignant breast tissues. The diagnostic performance of the classification algorithms based on both feature extraction techniques yielded similar sensitivities and specificities of approximately 80% for discriminating between malignant and non-malignant breast tissues. While both methods yielded similar classification accuracies, the model based approach provided insight into the physiological and structural features that discriminate between malignant and non-malignant breast tissues.     

Continuous-Variable Deep Quantum Neural Networks for Flexible Learning of Structured Classical Information

Quantum computation using optical modes has been well-established in itsability to construct deep neural networks. These networks have been shown to beflexible both architecturally as well as in terms of the type of data being processed.We leverage this property of the Continuous-Variable (CV) model to construct stackedsingle mode networks that are shown to learn structured classical information, whileplacing no restrictions on the size of the network, and at the same time maintaining itscomplexity. The hallmark of the CV model is its ability to forge non-linear functionsusing a set of gates that allows it to remain completely unitary. The proposed modelexemplifies that the appropriate photonic hardware can be integrated with presentday optical communication systems to meet our information processing requirements.In this paper, using the Strawberry Fields software library on the MNIST dataset ofhand-written digits, we demonstrate the adaptability of the network to learn classicalinformation in a multitude of machine learning tasks to very large fidelities.     

A Distributed Optimal Control Problem with Averaged Stochastic Gradient Descent

In this work, we study a distributed optimal control problem, in which the governing system is given by second-order elliptic equations with log-normal coefficients. To lessen the curse of dimensionality that originates from the representation of stochastic coefficients, the Monte Carlo finite element method is adopted for numerical discretization where a large number of sampled constraints are involved. For the solution of such a large-scale optimization problem, stochastic gradient descent method is widely used but has slow convergence asymptotically due to its inherent variance. To remedy this problem, we adopt an averaged stochastic gradient descent method which performs stably even with the use of relatively large step sizes and small batch sizes. Numerical experiments are carried out to validate our theoretical findings.     

Implicit Quadrature-Free Direct Reconstruction Method for Efficient Scale-Resolving Simulations

The present study develops implicit physical domain-based discontinuous Galerkin (DG) methods for efficient scale-resolving simulations on mixed-curvedmeshes. Implicit methods are essential to handle stiff systems in many scale-resolvingsimulations of interests in computational science and engineering. The physicaldomain-based DG method can achieve high-order accuracy using the optimal bases setand preserve the required accuracy on non-affine meshes. When using the quadrature-based DG method, these advantages are overshadowed by severe computational costson mixed-curved meshes, making implicit scale-resolving simulations unaffordable.To address this issue, the quadrature-free direct reconstruction method (DRM) is extended to the implicit DG method. In this approach, the generalized reconstructionapproximates non-linear flux functions directly in the physical domain, making thecomputing-intensive numerical integrations precomputable at a preprocessing step.The DRM operator is applied to the residual computation in the matrix-free method.The DRM operator can be further extended to the system matrix computation for thematrix-explicit Krylov subspace method and preconditioning. Finally, the A-stableRosenbrock-type Runge–Kutta methods are adopted to achieve high-order accuracyin time. Extensive verification and validation from the manufactured solution to implicit large eddy simulations are conducted. The computed results confirm that theproposed method significantly improves computational efficiency compared to thequadrature-based method while accurately resolving detailed unsteady flow featuresthat are hardly captured by scale-modeled simulations.     

Determination of the concentration scaling law of the scattering coefficient of water solutions of Intralipid at 832 nm by comparison between collimated detection measurements and Monte Carlo simulations

BACKGROUND AND OBJECTIVES: Intralipid (IP) is a scatterer extensively used in the building of phantoms for Biomedical Optics measurements. Recently, deviations from the linearity have been shown for the concentration scaling law of the scattering coefficient of IP water solutions at visible wavelengths. In this work this scaling law was determined at 832 nm. STUDY DESIGN/MATERIALS AND METHODS: Space resolved transmittance measurements of a laser beam at 832 nm through water solutions of IP and ink were performed and compared with the corresponding results of Monte Carlo simulations. RESULTS: The comparison provides a quadratic dependence of mu'(s) on the volume-to-volume scatterer concentration, C(IP), in the range of C(IP) values (0.0024相似文献   

Quantum Implementation of Numerical Methods for Convection-Diffusion Equations: Toward Computational Fluid Dynamics

We present quantum numerical methods for the typical initial boundaryvalue problems (IBVPs) of convection-diffusion equations in fluid dynamics. The IBVPis discretized into a series of linear systems via finite difference methods and explicittime marching schemes. To solve these discrete systems in quantum computers, wedesign a series of quantum circuits, including four stages of encoding, amplification,adding source terms, and incorporating boundary conditions. In the encoding stage,the initial condition is encoded in the amplitudes of quantum registers as a state vectorto take advantage of quantum algorithms in space complexity. In the following threestages, the discrete differential operators in classical computing are converted into unitary evolutions to satisfy the postulate in quantum systems. The related arithmeticcalculations in quantum amplitudes are also realized to sum up the increments fromthese stages. The proposed quantum algorithm is implemented within the open-sourcequantum computing framework Qiskit [2]. By simulating one-dimensional transientproblems, including the Helmholtz equation, the Burgers’ equation, and Navier-Stokesequations, we demonstrate the capability of quantum computers in fluid dynamics.     

Recurrence Phenomenon for Vlasov-Poisson Simulations on Regular Finite Element Mesh

In this paper, we focus on one difficulty arising in the numerical simulation of the Vlasov-Poisson system: when using a regular grid-based solver with periodic boundary conditions, perturbations present at the initial time artificially reappear at a later time. For regular finite-element mesh in velocity, we show that thisrecurrence time is actually linked to the spectral accuracy of the velocity quadraturewhen computing the charge density. In particular, choosing trigonometric quadratureweights optimally defers the occurrence of the recurrence phenomenon. Numerical results using the Semi-Lagrangian Discontinuous Galerkin and the Finite Element/Semi-Lagrangian method confirm the analysis.     

Numerical Simulations for Full History Recursive Multilevel Picard Approximations for Systems of High-Dimensional Partial Differential Equations

One of the most challenging issues in applied mathematics is to developand analyze algorithms which are able to approximately compute solutions of high-dimensional nonlinear partial differential equations (PDEs). In particular, it is veryhard to develop approximation algorithms which do not suffer under the curse of dimensionality in the sense that the number of computational operations needed by thealgorithm to compute an approximation of accuracy $ε$>0 grows at most polynomiallyin both the reciprocal 1/$ε$ of the required accuracy and the dimension $d∈mathbb{N}$of the PDE.Recently, a new approximation method, the so-called full history recursive multilevel Picard (MLP) approximation method, has been introduced and, until today, this approximation scheme is the only approximation method in the scientific literature which hasbeen proven to overcome the curse of dimensionality in the numerical approximationof semilinear PDEs with general time horizons. It is a key contribution of this articleto extend the MLP approximation method to systems of semilinear PDEs and to numerically test it on several example PDEs. More specifically, we apply the proposedMLP approximation method in the case of Allen-Cahn PDEs, Sine-Gordon-type PDEs,systems of coupled semilinear heat PDEs, and semilinear Black-Scholes PDEs in up to1000 dimensions. We also compare the performance of the proposed MLP approximation algorithm with a deep learning based approximation method from the scientificliterature.