Efficient Numerical Algorithm Research Articles

Simulation-based, Finite-sample Inference for Privatized Data

Privacy protection methods, such as differentially private mechanisms, introduce noise into resulting statistics which often produces complex and intractable sampling distributions. In this paper, we propose a simulation-based “repro sample” approach to produce statistically valid confidence intervals and hypothesis tests, which builds on the work of Xie and Wang (2022). We show that this methodology is applicable to a wide variety of private inference problems, appropriately accounts for biases introduced by privacy mechanisms (such as by clamping), and improves over other state-of-the-art inference methods such as the parametric bootstrap in terms of the coverage and type I error of the private inference. We also develop significant improvements and extensions for the repro sample methodology for general models (not necessarily related to privacy), including 1) modifying the procedure to ensure guaranteed coverage and type I errors, even accounting for Monte Carlo error, and 2) proposing efficient numerical algorithms to implement the confidence intervals and p-values.

General numerical framework to derive structure preserving reduced order models for thermodynamically consistent reversible-irreversible PDEs

In this paper, we propose a general numerical framework to derive structure-preserving reduced-order models for thermodynamically consistent PDEs. Our numerical framework has two primary features: (a) a systematic way to extract reduced-order models for thermodynamically consistent PDE systems while maintaining their inherent thermodynamic principles and (b) a general process to derive accurate, efficient, and structure-preserving numerical algorithms to solve these reduced-order models. The platform's generality extends to various PDE systems governed by embedded thermodynamic laws, offering a unique approach from several perspectives. First, it utilizes the generalized Onsager principle to transform the thermodynamically consistent PDE system into an equivalent form, where the free energy of the transformed system takes a quadratic form in terms of the state variables. This transformation is known as energy quadratization (EQ). Through EQ, we gain a novel perspective on deriving reduced-order models that continue to respect the energy dissipation law. Secondly, our proposed numerical approach automatically provides algorithms to discretize these reduced-order models. The proposed algorithms are always linear, easy to implement and solve, and uniquely solvable. Furthermore, these algorithms inherently ensure the thermodynamic laws. Our platform offers a distinctive approach for deriving structure-preserving reduced-order models for a wide range of PDE systems with underlying thermodynamic principles.

Uniqueness, stability and algorithm for an inverse wave-number-dependent source problem

Abstract This paper is concerned with an inverse wave-number-/frequency-dependent source problem for the Helmholtz equation. In two and three dimensions, the unknown source term is supposed to be compactly supported in spatial variables but independent on one spatial variable. The dependence of the source function on wave-number/frequency is supposed to be unknown. Based on the Fourier-transform and Dirichlet-Laplacian methods, we develop two efficient non-iterative numerical algorithms to recover the wave-number-dependent source. Uniqueness proof and increasing stability analysis are carried out if the boundary measurement data of Dirichlet kind are available. Numerical experiments are conducted to illustrate the effectiveness and efficiency of the proposed methods.

Axisymmetric Indentation of Circular Rigid Plate on Layered Elastic Halfspace with Transverse Isotropy

This paper investigates the contact problem of a layered elastic halfspace with transverse isotropy under the axisymmetric indentation of a circular rigid plate. Fourier integral transforms and a backward transfer matrix method are used to obtain the analytical solution of the contact problem. The interaction between the rigid plate and the layered halfspace can be expressed with the standard Fredholm integral equations of the second kind. The induced elastic field in the layered halfspace can be expressed as the semi-infinite integrals of four known kernel functions. The convergence and singularity of the semi-infinite integrals near or at the surface of the layered halfspace are resolved using an isolating technique. The efficient numerical algorithms are used and developed for accurately calculating the Fredholm integral equations and the semi-infinite integrals. Numerical results show the correctness of the proposed method and the effect of layering non-homogeneity on the elastic fields in layered transversely isotropic halfspace induced by the axisymmetric indentation of a circular rigid plate.

An Efficient Numerical Algorithm for Abnormal Integrals with Arbitrary Order Double Bessel Functions of the First Type

Abstract The integrals of the first type of double Bessel functions have a wide range of applications in geological exploration, mechanical and electromagnetic responses, signal processing, scattering, and wetting. In this paper, we develop a linear transformation accelerated convergence algorithm (LTACA) that combines the large argument approximate expression of the Bessel function (LAAEBF) and the integral accumulation to provide an efficient numerical algorithm for abnormal integrals with arbitrary order double Bessel functions of the first type. The effectiveness and high efficiency of the algorithm are verified by numerical examples, and its high accuracy is demonstrated by comparison with the Gaver-Stehfest inverse Laplace transform method (GSILTM). This offers a reliable and efficient computational method for the study of signal processing and mechanical problems.

A novel semi-explicit numerical algorithm for efficient 3D phase field modelling of quasi-brittle fracture

Phase field models have become an effective tool for predicting complex crack configurations including initiation, propagation, branching, intersecting and merging. However, several computational issues have hindered their utilisation in engineering practice, such as the convergence challenge in implicit algorithms, numerical stability issues in explicit methods and significant computational costs. Aiming to providing a more efficient numerical algorithm, this work integrates the explicit integral operator with the recently developed neighbored element method, for the first time, to solve the coupled governing equations in phase field models. In addition, the damage irreversibility can be ensured automatically, avoiding the need to introduce extra history variable for the maximum driving force in traditional algorithms. Six representative fracture benchmarks with different failure modes are simulated to verify the effectiveness of the proposed method, including the multiple cracks in heterogeneous concrete at mesoscale. It is found that this semi-explicit numerical algorithm yields consistent crack profiles and load capacities for all examples to the available experimental data and literature. In particular, the computational cost is significantly reduced when compared to the traditional explicit modelling. Therefore, the presented numerical algorithm is highly attractive and promising for phase-field simulations of complicated 3D solid fractures in structural-level engineering practices.

Standby and inspection policy optimization in systems exposed to common and operational shock processes

Motivated by practical applications like data storage, product defect detection, medical imagining, and sensing, this paper puts forward a new inspected standby system model where only one element can be online operating and only one element can stay in the standby mode at any time. Both operating and standby elements are exposed to random common shocks, causing their deterioration and even failures. The operating element may also be deteriorated by random operational shocks. The system undergoes periodic inspections to determine the refill of the operating or standby element. A new optimization problem is formulated and solved to determine the inspection and standby element addition policy with the objective to minimize the expected mission cost (EMC) attributed to factors including system downtime, number of inspections, element modes and failures, element activation and mode transitions. A new and efficient system state transition-based numerical algorithm is proposed to evaluate the EMC. A case study of a standby sensor system is provided to demonstrate the proposed model and impacts of several cost parameters as well as shock rates on the EMC and the optimal inspection and standby element addition policy, leading to insightful managerial guidelines for the system design and operation.

Revisit Liu & Katz (2006) And Zigunov & Charonko (2024): On The Equivalency Of Omni-Directional Integration And Pressure Poisson Equation

In this paper we demonstrate the equivalency of Omni-Directional Integration (ODI) and the Pressure Poisson Equation (PPE) for pressure field reconstruction from corrupted image velocimetry data. Over the years, it has been long debated which of the two families of methods is better for pressure reconstruction, direct pressure gradient integration (particularly ODI) versus PPE. Some have claimed that ODI is fundamentally different, and far more accurate than PPE; while other studies observed similar reconstruction accuracy between ODI and PPE (McClure & Yarusevych, 2017). This debate has been filled with confusion and conflicting results until a recent breakthrough by Zigunov & Charonko (2023, 2024) while trying to improve the computational efficiency of ODI. In a series of works, Zigunov & Charonko (2023, 2024) reformulated the iterative integration process of ODI into a system of linear equations resembling the discretized PPE, alluding to a deep connection between ODI and PPE. With careful numerical treatment, we show that ODI can be viewed as pursuing the minimal norm solution to a Poisson equation with pure Neumann boundary conditions. We provide a detailed and physical explanation for why some have reported poor robustness of the PPE, highlighting critical nuances in its numerical implementation, and explain why the ODI is more robust to random noise in the data. We hope to put an end to the PPE versus ODI debate and clear up the confusion surrounding how these and when these methods perform well. With these new comprehensions, we can leverage the established regularization techniques and efficient numerical algorithms of elliptic equations to improve PPE/ODI-based pressure field reconstruction.

Phase field modeling and numerical algorithm for two-phase dielectric fluid flows

We develop a method for modeling and simulating a class of two-phase flows consisting of two immiscible incompressible dielectric fluids and their interactions with imposed external electric fields in two and three dimensions. We first present a thermodynamically-consistent and reduction-consistent phase field model for two-phase dielectric fluids. The model honors the conservation laws and thermodynamic principles, and has the property that, if only one fluid component is present in the system, the two-phase formulation will exactly reduce to that of the corresponding single-phase system. In particular, this model accommodates an equilibrium solution that is compatible with the zero-velocity requirement based on physics. This property leads to a simpler method for simulating the equilibrium state of two-phase dielectric systems. We further present an efficient numerical algorithm, together with a spectral-element (for two dimensions) or a hybrid Fourier-spectral/spectral-element (for three dimensions) discretization in space, for simulating this class of problems. This algorithm computes different dynamic variables successively in an un-coupled fashion, and involves only coefficient matrices that are time-independent in the resultant linear algebraic systems upon discretization, even when the physical properties (e.g. permittivity, density, viscosity) of the two dielectric fluids are different. This property is crucial and enables us to employ fast Fourier transforms for three-dimensional problems. Ample numerical simulations of two-phase dielectric flows under imposed voltage are presented to demonstrate the performance of the method herein and to compare the simulation results with theoretical models and experimental data.

Matrix equation representation of the convolution equation and its unique solvability

Abstract We consider the convolution equation F * X = B F* X=B , where F ∈ R 3 × 3 F\in {{\mathbb{R}}}^{3\times 3} and B ∈ R m × n B\in {{\mathbb{R}}}^{m\times n} are given and X ∈ R m × n X\in {{\mathbb{R}}}^{m\times n} is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and we analyze the unique solvability of the convolution equation.

CFD analysis of paraffin-based hybrid (Co–Au) and trihybrid (Co–Au–ZrO2) nanofluid flow through a porous medium

Abstract Ternary hybrid nanofluids possess improved thermal characteristics, enhanced stability, better physical strength, and multi-functionality as compared to hybrid or usual nanofluids. The aim of the ongoing study is to explore the novel thermal attributes of hybrid and trihybrid nanofluids through a porous medium. Whereas the nano-composition of cobalt (Co), gold (Au), and zirconium oxide (ZrO2) make amalgamation in the paraffin (Pfin) which is a base fluid. This nano-composition of the proposed nanoparticles, specifically, subject to the base fluid Pfin has not been interpreted before. The analysis not only covers the features of trihybrid nanofluids (Co–Au–ZrO2–Pfin) but it also describes the characteristics of hybrid (Co–Au–Pfin) as well as pure nanofluids (Co–Pfin). An efficient numerical algorithm is developed for which the numerical simulations are carried out. The approximations are performed in MATLAB software using “Successive under Relaxation (SUR)” technique. A comparison, under certain limiting conditions, with the established results appraises the efficiency of the numerical code. The outcomes evidently designate that temperature raises with the change in thermal radiation and volume fraction of gold and zirconium oxide in either case of pure, hybrid, or ternary nanofluids. The concentration ϕ 3 {\phi }_{3} of ZrO2 has a significant impact on Nusselt number rather than the concentration ϕ 1 {\phi }_{1} of cobalt and ϕ 2 {\phi }_{2} of gold. It has been comparatively noticed that the ternary nanofluids (Co–Au–ZrO2–Pfin) portray embellished and improvised thermal characteristics as compared to the other two cases.

Radially transverse isotropic inclusions in isotropic elastic media: Local fields, neutral inclusions, effective elastic properties

Radially transverse isotropic inclusions in homogeneous isotropic elastic host media are considered. Mellin transform method is used for solution of the volume integral equation of the problem for an isolated inclusion subjected to a constant external stress (strain) field. The tensor structure of the solution is revealed with precision to three scalar functions of the radial coordinate, and the system of ordinary differential equations for these functions is derived. For multilayered radially transverse isotropic inclusions with constant elastic coefficients inside layers, explicit solution of these equations is obtained. An efficient numerical algorithm of solution for inclusions with an arbitrary number of the layers is proposed. Neutral inclusions that do not disturb homogeneous external fields applied to the medium are considered. It is shown that an inclusion with an isotropic core and radially transverse isotropic external layer can be weak neutral by appropriate choice of the layer elastic constants. The effective field method is used for determination of the effective elastic stiffness tensor of a homogeneous isotropic medium containing a random set of radially transverse isotropic inclusions.

Efficient Multistep Algorithms for First-Order IVPs with Oscillating Solutions: II Implicit and Predictor–Corrector Algorithms

This research introduces a fresh methodology for creating efficient numerical algorithms to solve first-order Initial Value Problems (IVPs). The study delves into the theoretical foundations of these methods and demonstrates their application to the Adams–Moulton technique in a five-step process. We focus on developing amplification-fitted algorithms with minimal phase-lagor phase-lag equal to zero (phase-fitted). The request of amplification-fitted (zero dissipation) is to ensure behavior like symmetric multistep methods (symmetric multistep methods are methods with zero dissipation). Additionally, the stability of the innovative algorithms is examined. Comparisons between our new algorithm and traditional methods reveal its superior performance. Numerical tests corroborate that our approach is considerably more effective than standard methods for solving IVPs, especially those with oscillatory solutions.

Numerical algorithm for a general fractional diffusion equation

In this manuscript, the analytical solution to the general time fractional diffusion equation followed by the regularity analysis of its solution is presented. Further, a hybrid efficient numerical algorithm for a general fractional derivative of order α∈(0,1) is designed. The physical application of the developed approximation formula is employed to design a hybrid numerical algorithm to determine the numerical solution of the corresponding general time fractional diffusion equation. The proposed numerical techniques are subjected to solvability, numerical stability, and convergence analysis. Numerical example is utilized to verify that (3−α)-th order convergence is attained in time which is higher than the existing scheme (Xu et al., 2013). In the spatial direction, the fourth order convergence is obtained utilizing the compact finite difference method. Besides, such a hybrid numerical algorithm is also used in the stochastic model.

Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media

Abstract The objective of this paper is to develop efficient numerical algorithms for the linear advection-diffusion equation in fractured porous media. A reduced fracture model is considered where the fractures are treated as interfaces between subdomains and the interactions between the fractures and the surrounding porous medium are taken into account. The model is discretized by a backward Euler upwind-mixed hybrid finite element method in which the flux variable represents both the advective and diffusive fluxes. The existence, uniqueness, as well as optimal error estimates in both space and time for the fully discrete coupled problem are established. Moreover, to facilitate different time steps in the fracture-interface and the subdomains, global-in-time, nonoverlapping domain decomposition is utilized to derive two implicit iterative solvers for the discrete problem. The first method is based on the time-dependent Steklov–Poincaré operator, while the second one employs the optimized Schwarz waveform relaxation (OSWR) approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively with possibly variable time step sizes. The convergence of the OSWR-based method with conforming time grids is also proved. Finally, numerical results in two dimensions are presented to verify the optimal order of convergence of the monolithic solver and to illustrate the performance of the two decoupled schemes with local time-stepping on problems of high Péclet numbers.

Efficient numerical algorithm for multi-level ionization of high-atomic-number gases

An efficient numerical algorithm for laser driven multi-level ionization of high-atomic-number gases is proposed and implemented in an electromagnetic particle-in-cell code SPACE. The algorithm is based on analytical solutions to the system of differential equations describing ionization evolution. Using analytical solutions resolves the multiscale issue of ionization due to different characteristic time scales of ionization processes and the main code time step. Algorithm efficiency and memory requirements are significantly improved by using a locally reduced system of differential equations. The algorithm also assigns proper orbital quantum numbers and their projections to ionization states. The algorithm is verified and validated using experimental data.

Higher-order multi-scale computational approach and its convergence for nonlocal gradient elasticity problems of composite materials

A novel higher-order multi-scale computational approach for efficient nonlocal gradient elasticity simulation of composite materials is proposed in this paper. Based on the use of decoupling technique, raw fourth-order nonlocal gradient equations with periodically oscillatory coefficients are decoupled as the new second-order multi-scale equations for reducing the continuity requirements when implementing multi-scale simulation. Next, a novel macro-micro coupled multi-scale computational model with higher-order corrected terms is rigorously devised for high-resolution multi-scale and nonlocal analysis of composite materials via multi-scale asymptotic analysis. Then, the local error analysis of higher-order multi-scale solutions in the point-wise sense illustrates that establishing higher-order asymptotic corrected terms plays a vital role in investigating the micromechanical mechanisms. Moreover, a rigorous global error estimation with an explicit rate of higher-order multi-scale solutions is first derived in the energy norm sense. In addition, an efficient multi-scale numerical algorithm is presented to effectively simulate nonlocal gradient elasticity problems of composite materials based on finite element method (FEM). And we also derive the convergence estimation of the proposed numerical algorithm. Finally, numerical experiments are conducted to gauge the efficiency and accuracy of the proposed higher-order multi-scale method. This paper offers a unified higher-order two-scale computational framework for enabling nonlocal gradient elasticity behavior simulation of composite materials.

Partial Control and Beyond: Controlling Chaotic Transients with the Safety Function

Chaotic dynamical systems often exhibit transient chaos, where trajectories behave chaotically for a short amount of time before escaping to an external attractor. Sustaining transient chaotic dynamics under disturbances is challenging yet desirable for many applications. The partial control approach exploits the inherent symmetry and geometric structure of chaotic saddles, the topological object responsible of transient chaos, to enable surprising control with only small perturbations. Here, we review the latest findings in partial control techniques with the aim to sustain chaos or accelerate escapes by exploiting these intricate invariant sets. We introduce the fundamental concept of safe sets regions where orbits persist despite noise. This paper presents recent generalizations through safety functions and escape functions that automatically find the minimum control needed. Efficient numerical algorithms are presented and several examples of application are illustrated. Rather than eliminating chaos entirely, partial control techniques provide a framework to reliably control transient chaotic dynamics with minimal interventions. This approach has promising applications across diverse fields including physics, engineering, biology, and more.

Reconstruction techniques for complex potentials

An approach for solving a variety of inverse coefficient problems for the Sturm–Liouville equation −y″ + q(x)y = ρ2y with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations for solutions. With their aid the problem is reduced to a system of linear algebraic equations for the coefficients of the representations. The potential is recovered from an arithmetic combination of the first two coefficients. Special cases of the considered problems include the recovery of the potential from a Weyl function, inverse two-spectrum Sturm–Liouville problems, as well as the inverse scattering problem on a finite interval. The approach leads to efficient numerical algorithms for solving coefficient inverse problems. Numerical efficiency is illustrated by several examples.

Reconstruction techniques for quantum trees

The inverse problem of recovery of a potential on a quantum tree graph from the Weyl matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm–Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two‐spectrum inverse problems on separate edges and reduces them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests.