True Solution Research Articles

Solutions to elliptic and parabolic problems via finite difference based unsupervised small linear convolutional neural networks

In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods like PINNs rely on auto differentiation and sampling collocation points, leading to a lack of interpretability and lower accuracy than traditional numerical methods. As a result, we propose a fully unsupervised approach, requiring no training data, to estimate finite difference solutions for PDEs directly via small linear convolutional neural networks. Our proposed approach uses substantially fewer parameters than similar finite difference-based approaches while also demonstrating comparable accuracy to the true solution for several selected elliptic and parabolic problems compared to the finite difference method.

Convergence analysis of non-matching finite elements for a linear monotone additive Schwarz scheme for semi-linear elliptic problems

Abstract In this article, we are interested in the standard finite element approximation method of linear additive Schwarz iterations for a class of semi-linear elliptic problems, for two subdomains, in the context of non-matching grids. More precisely, by means of a uniform convergence result from the study by Lui and a fundamental lemma consisting of estimating, at each iteration, the gap between the continuous and the finite element Schwarz iterates, we prove that the discrete Schwarz sequences converge, in the maximum norm, to the true solution. Moreover, we also give numerical results to support the theoretical findings.

Using H‐Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady‐State Conditions

ABSTRACTUnsaturated zones are important for geotechnical design, geochemical reactions, and microbial reactions. The numerical analysis of unsaturated seepage is complex because it involves highly nonlinear partial differential equations. The permeability can vary by orders of magnitude over short vertical distances. This article defines and uses H‐convergence tests to quantify numerical errors made by uniform meshes with element size (ES) for 1D steady‐state conditions. The quantitative H‐convergence should not be confused with a qualitative mesh sensitivity study. The difference between numerical and mathematical convergences is stated. A detailed affordable method for an H‐convergence test is presented. The true but unknown solution is defined as the asymptote of the numerical solutions for all solution components when ES decreases to zero. The numerical errors versus ES are then assessed with respect to the true solution, and using a log–log plot, which indicates whether a code is correct or incorrect. If a code is correct, its results follow the rules of mathematical convergence in a mathematical convergence domain (MCD) which is smaller than the numerical convergence domain (NCD). If a code is incorrect, it has an NCD but no MCD. Incorrect algorithms of incorrect codes need to be modified and repaired. Existing codes are shown to converge numerically within large NCDs but generate large errors, up to 500%, in the NCDs, a dangerous situation for designers.

Computationally efficient and error aware surrogate construction for numerical solutions of subsurface flow through porous media

Limiting the injection rate to restrict the pressure below a threshold at a critical location can be an important goal of simulations that model the subsurface pressure between injection and extraction wells. The pressure is approximated by the solution of Darcy’s partial differential equation for a given permeability field. The subsurface permeability is modeled as a random field since it is known only up to statistical properties. This induces uncertainty in the computed pressure. Solving the partial differential equation for an ensemble of random permeability simulations enables estimating a probability distribution for the pressure at the critical location. These simulations are computationally expensive, and practitioners often need rapid online guidance for real-time pressure management. An ensemble of numerical partial differential equation solutions is used to construct a Gaussian process regression model that can quickly predict the pressure at the critical location as a function of the extraction rate and permeability realization. The Gaussian process surrogate analyzes the ensemble of numerical pressure solutions at the critical location as noisy observations of the true pressure solution, enabling robust inference using the conditional Gaussian process distribution.Our first novel contribution is to identify a sampling methodology for the random environment and matching kernel technology for which fitting the Gaussian process regression model scales as O(nlogn) instead of the typical O(n3) rate in the number of samples n used to fit the surrogate. The surrogate model allows almost instantaneous predictions for the pressure at the critical location as a function of the extraction rate and permeability realization. Our second contribution is a novel algorithm to calibrate the uncertainty in the surrogate model to the discrepancy between the true pressure solution of Darcy’s equation and the numerical solution. Although our method is derived for building a surrogate for the solution of Darcy’s equation with a random permeability field, the framework broadly applies to solutions of other partial differential equations with random coefficients.

Supplementary optical solitonic expressions for Gerdjikov–Ivanov equations with three Kudryashov-based methods

The present manuscript uses three Kudryashov-based methods to analytically inspect the class of Gerdjikov–Ivanov equations, which comprises the standard Gerdjikov–Ivanov equation and the perturbed Gerdjikov–Ivanov equation. Various optical solitonic solutions have been constructed. Certainly, as the reported solitonic structures happened to be exponential functions, diverse true solitonic solutions can easily be resorted to upon suitably fixing the involving parameters, including mainly the bright and singular solitons. Lastly, the study graphically examined some of the constructed structures, which were then found to portray some interesting known shapes in the theory of solitary waves and nonlinear Schrödinger equations. Additionally, the Kudryashov-index d has been noted to play a significant role in the propagation of complex waves in the nonlinear media described by Gerdjikov–Ivanov equations.

Two Numerical Schemes Via an Auxiliary Function for Nonlocal Problems With Integrable Kernels

ABSTRACTIn this study, we explore volume constrained nonlocal problem with integrable kernels. A central challenge addressed is the discontinuity of true solutions across the boundary, posing difficulties for designing efficient numerical methods. To address this issue, we introduce an auxiliary function which serves as the solution of an auxiliary problem. This approach effectively transforms the original problem to a new nonlocal problem with a true solution which is continuous across the boundary. We propose two numerical schemes to solve the new nonlocal problem: one is exclusively applicable to one‐dimensional case, while the other is suitable for general dimensional cases. Experimental results show that both schemes exhibit optimal convergence order under mild condition.

Positivity and Boundedness Preserving Numerical Scheme for a Stochastic Multigroup Susceptible-Infected-Recovering Epidemic Model with Age Structure.

Since the stochastic age-structured multigroup susceptible-infected-recovering (SIR) epidemic model is nonlinear, the solution of this model is hard to be explicitly represented. It is necessary to construct effective numerical methods so as to predict the number of infections. In addition, the stochastic age-structured multigroup SIR model has features of positivity and boundedness of the solution. Therefore, in this article, in order to ensure that the numerical and analytical solutions must have the same properties, by modifying the classical Euler-Maruyama (EM) scheme, we generate a positivity and boundedness preserving EM (PBPEM) method on temporal space for stochastic age-structured multigroup SIR model, which is proved to have a strong convergence to the true solution over finite time intervals. Moreover, by combining the standard finite element method and the PBPEM method, we propose a full-discrete scheme to show the numerical solutions, as well as analyze the error estimations. Finally, the full-discrete scheme is applied to a general stochastic two-group SIR model and the Chlamydia epidemic model, which shows the superiority of the numerical method.

Phase Preparation of Xiao-chai-hu Decoction and its Pharmacodynamics of Acute Liver Injury.

Self-emulsifying nano-phase of traditional Chinese medicine are a research hotspot. Xiao-Chai-Hu decoction is a commonly used compound decoction in clinical practice, which is of great research significance. The aim of this study was to isolate and characterize the self-emulsifying nano-phase and other phases of Xiao-Chai-Hu decoction, and to study the effects of each phase on acute liver injury. The liquid medicine was prepared employing centrifugation followed by dialysis. Single- factor investigation methodology was utilized to optimize the preparation parameters for both phases. Characterization of the formulated phase involved analyses such as surface morphology assessment, measurement of nanoparticle size and Zeta potential using an analyzer, observation of the Tyndall effect, conducting diffusion and dilution tests, examination under a microscope, and structural visualization via transmission electron microscopy (TEM). Furthermore, an acute liver injury model was established in rats through intraperitoneal injection of D-Galactosamine (D-Gal- N). To assess hepatic function and oxidative stress status, serum levels of alanine aminotransferase (ALT) and aspartate aminotransferase (AST), superoxide dismutase (SOD) activity, and malondialdehyde (MDA) content in liver tissue were quantified. The liver coefficients for each group were calculated as an additional parameter. For histopathological evaluation, liver tissue sections from the experimental group were stained with Hematoxylin and Eosin (H&E) and examined microscopically under light conditions. These revisions aim to enhance clarity, correct minor grammatical errors (such as capitalization of "HE" to "H&E"), and ensure a smoother flow of information without altering the scientific content of your original text. Successful establishment and separation of four distinct phases were achieved, including the self-emulsifying nano-phase, precipitation phase, suspension phase, and true solution phase. The self-emulsifying nano-phase was characterized as spherical particles with an average diameter of approximately 100 nm. Pharmacodynamic assessments revealed that both Xiao-Chai-Hu decoction and its self-emulsifying nano-phase significantly reduced liver coefficients and alanine aminotransferase (ALT) levels compared to controls (P<0.05). However, no statistically significant differences were observed in regards to aspartate aminotransferase (AST) concentrations, malondialdehyde (MDA) content, or superoxide dismutase (SOD) activity between the treatment groups and control (P>0.05). These findings indicate that both Xiao-Chai-Hu decoction and its self-emulsifying nano-formulation ameliorated D-GalN-induced acute liver injury, albeit without statistically distinguishable efficacy between them (P>0.05). The presence of a self-emulsifying nano-phase within Xiao-Chai-Hu decoction is confirmed, and this nano-phase emerges as a therapeutically efficacious component in mitigating acute liver injury.

Novel spectral methods for shock capturing and the removal of tygers in computational fluid dynamics

Spectral methods yield numerical solutions of the Galerkin-truncated versions of nonlinear partial differential equations (PDEs) involved especially in fluid dynamics. In the presence of discontinuities, such as shocks, spectral approximations develop Gibbs oscillations near the discontinuity. This causes the numerical solution to deviate quickly from the true solution. For spectral approximations of the inviscid Burgers equation in one-dimension (1D), nonlinear wave resonances lead to the formation of localised oscillatory structures called tygers in well-resolved areas of the flow, far from the shock. Recently, the author of [1] has proposed novel spectral relaxation (SR) and spectral purging (SP) schemes for the removal of tygers and Gibbs oscillations in spectral approximations of nonlinear conservation laws. For the 1D inviscid Burgers equation, it is shown in [1] that the novel SR and SP approximations of the solution converge strongly in L2 norm to the entropic weak solution, under an appropriate choice of kernels and related parameters. In this work, we carry out a detailed numerical investigation of SR and SP schemes when applied to the 1D inviscid Burgers equation and report the efficiency of shock capture and the removal of tygers. We then extend our study to systems of nonlinear hyperbolic conservation laws, such as the 2×2 system of the shallow water equations and the standard 3×3 system of 1D compressible Euler equations. For the latter, we generalise the implementation of SR methods to non-periodic problems using Chebyshev polynomials. We then turn to singular flow in the 1D wall approximation of the 3D-axisymmetric wall-bounded incompressible Euler equation [2]. Here, in order to determine the blowup time of the solution, we compare the decay of the width of the analyticity strip, obtained from the pure pseudospectral method (PPS), with the improved estimate obtained using the novel spectral relaxation scheme.

Correction: Integral representation bound of the true solution to the BVP of double-sided fractional diffusion advection reaction equation

Correction: Integral representation bound of the true solution to the BVP of double-sided fractional diffusion advection reaction equation

Assessing the influence of image capture frequency and asynchronicity on the accuracy of digital image correlation for fatigue analysis

Efficiency of Digital Image Correlation (DIC) technique for fatigue analysis can be significantly increased by capturing images at periodic intervals for correlation. However, the scarcity of images to capture the history of deformation can cause inaccuracies in full-field measurements. In this study, a comprehensive analysis is conducted to quantitatively assess this error by considering various scenarios. These include skipping of frames within every cycle and across cycles, as well as deviations between intended and captured frames. DIC measurements based on a high density of frames in every cycle is assumed to be true solution to evaluate error for all the cases. Analysis indicates that the skipping frames within every cycle introduces error in full-field strain, which increases with the propagation of crack. Interestingly, magnitude of error for periodic skipping of frames across cycles is observed to be more than skipping of frames within cycle. Furthermore, error shows an increasing trend as the gap between periodic frames becomes larger. Analysis also shows that random deviations disturbing periodicity of skipped frames for DIC can further increase the error in full-field strains. Overall, accuracy of the accelerated approach is seen to strongly depend on presence of discontinuous features and local gradients in response.

Patterns of rogue waves in the sharp-line Maxwell–Bloch system

The Maxwell–Bloch system describes light-matter interactions in a semi-infinitely long one dimensional two-level optical medium. Rogue wave patterns in the Maxwell–Bloch system under sharp-line limit are analytically studied. It is shown that when single internal parameter in bilinear expressions of rogue waves gets large, these waves would exhibit clear geometric patterns, which comprise fundamental (Peregrine) rogue waves arranged in shapes such as triangle, pentagon, heptagon and nonagon structures, with a possible lower-order rogue wave at the center. These rogue wave patterns are analytically determined from the root structure of the Yablonskii–Vorob’ev polynomial hierarchy through dilation, rotation, stretch and shear. It is also shown that when multiple internal parameters in the rogue wave solutions get large, new rogue wave patterns would arise, including heart-shaped structures, fan-shaped structures, and many others. Analytically, these patterns are determined by the root structure of the Adler–Moser polynomials through a linear transformation. Comparison between analytical predictions of these rogue patterns and true solutions shows excellent agreement.

A Bayesian approach to modeling finite element discretization error

In this work, the uncertainty associated with the finite element discretization error is modeled following the Bayesian paradigm. First, a continuous formulation is derived, where a Gaussian process prior over the solution space is updated based on observations from a finite element discretization. To avoid the computation of intractable integrals, a second, finer, discretization is introduced that is assumed sufficiently dense to represent the true solution field. A prior distribution is assumed over the fine discretization, which is then updated based on observations from the coarse discretization. This yields a posterior distribution with a mean that serves as an estimate of the solution, and a covariance that models the uncertainty associated with this estimate. Two particular choices of prior are investigated: a prior defined implicitly by assigning a white noise distribution to the right-hand side term, and a prior whose covariance function is equal to the Green’s function of the partial differential equation. The former yields a posterior distribution with a mean close to the reference solution, but a covariance that contains little information regarding the finite element discretization error. The latter, on the other hand, yields posterior distribution with a mean equal to the coarse finite element solution, and a covariance with a close connection to the discretization error. For both choices of prior a contradiction arises, since the discretization error depends on the right-hand side term, but the posterior covariance does not. We demonstrate how, by rescaling the eigenvalues of the posterior covariance, this independence can be avoided.

Optimal control of variably distributed‐order time‐fractional diffusion equation: Analysis and computation

AbstractFractional diffusion equations exhibit competitive capabilities in modeling many challenging phenomena such as the anomalously diffusive transport and memory effects. We prove the well‐posedness and regularity of an optimal control of a variably distributed‐order fractional diffusion equation with pointwise constraints, where the distributed‐order operator accounts for, for example, the effect of uncertainties. We accordingly develop and analyze a fully‐discretized finite element approximation to the optimal control without any artificial regularity assumption of the true solution. Numerical experiments are also performed to substantiate the theoretical findings.

Efficient generation of barrier crossing trajectories using approximate Brownian bridges.

We examine continuous random walks that are conditioned to reach one region before another. These conditioned processes are used to generate stochastic trajectories for barrier crossing events, which are generally rare and difficult to sample. The processes are generated using a Brownian bridge technique, resulting in near perfect sampling efficiency without accruing error in the conditional statistics of the process. The construction requires the hitting probability or committer function, which is a solution to the backward Fokker-Planck equation, a partial-differential equationthat can be difficult to solve through general means. Therefore, we derive a one-dimensional approximation through asymptotic methods for barrier crossing trajectories. We show that this approximation has a simple analytical representation and approaches the true solution as the barrier height increases. Brownian bridge trajectories generated with this approximate solution are then shown to result in accurate conditional statistics when used in conjunction with importance sampling, even in the case when potential energy barriers are not large. We show this idea's effectiveness by simulating rare events in a stochastic chemical reaction network (Schögl reaction) with multiple steady states. This methodology shows great promise for future implementation to simulate rare barrier crossing events for a wide variety of physical processes.

Non-convex sparse regularization via convex optimization for blade tip timing

Blade Tip Timing (BTT), an emerging technology poised to replace strain gauges, enables contactless measurement of rotor blade vibration. However, the blade vibration signals measured by BTT systems often suffer from significant undersampling. Sparse reconstruction methods are instrumental in addressing the challenge of undersampled signal reconstruction. However, traditional approaches grounded in ℓ1 regularization tend to underestimate the amplitude of the true solution. This underestimation is particularly pronounced in the resonance state of the rotor blade, hindering effective prediction of the blade’s operational state. To overcome this limitation, this paper introduces a novel non-convex regularized BTT model, employing a non-convex penalty term with convex-preserving properties to achieve nearly unbiased reconstruction accuracy. Additionally, we propose a new threshold iteration algorithm designed for the swift solution of this model. The accuracy and robustness of the proposed method in identifying the multimodal vibration parameters of rotor blades are validated through simulations and experiments. Comparatively, the proposed method closely aligns with the Orthogonal Matching Pursuit (OMP) method in recognizing blade multimodal vibration amplitude, showcasing significant improvement over the ℓ1 regularization method. Furthermore, it demonstrates lower sensitivity to changes in BTT probe layout when compared to the OMP and ℓ1 regularization methods.

Study on the bioactive components of Banxia Xiexin Decoction with different decocting methods and its effects on ulcerative colitis rats from the perspective of phase states

Ethnopharmacological relevanceBanxia Xiexin Decoction (BXD) is one of the seven classic prescriptions of the special decoction method (SDM) of “removing dregs and decocting again”, which has been widely used in inflammatory bowel diseases such as ulcerative colitis (UC). However, the impacts of SDM have not been fully investigated, either on the components or on the biological effects. Aim of the studyThis study aimed to investigate the rational of SDM traditionally recorded about BXD, re-decoction after dreg-removal, by comparing with the contemporary general decoction method (GDM) from the perspective of phase states, in the bioactive components from the perspective of phase states, and their corresponding pharmacodynamic effects on a particular UC rat model. MethodsThe BXD decoctions were respectively obtained by SDM and GDM, together with the different samples with different decocting time. The phase state samples (true solution, colloidal phase, and precipitated phase), were also obtained after a series of separation process and characterized. The multi-components in the in-process decoctions, original decoctions and phase state samples were quantitatively determined. HPLC fingerprint spectrum of the samples were also detected and compared with chemometrics analysis. A rat model of ulcerative colitis with cold-heat complex syndrome was established, on which the pharmacodynamic effects of different phases of SDM-made BXD were investigated. ResultsThe results showed that the contents of eight marker components in SDM-made decoction were significantly higher than those in GDM-made decoction. Compared with the precipitated phases and true solutions, the colloidal phase was confirmed to obtain absolutely higher contents of the components (except berberine). The analysis on HPLC fingerprints also revealed that the profiles of colloidal phase showed the majority of the characteristics of original decoctions, when compared with the other phases. The results showed the BXD group, precipitated phase group and colloidal phase group had certain therapeutic effects on the ulcerative colitis rats with cold-heat complex syndrome, among which the original decoction group showed optimal effects, followed by the colloidal phase. ConclusionThe study has provided the experimental evidence of the bioactive components and pharmacodynamic effects on the rational of SDM, as originally recorded about the classic prescription, which might provide useful idea for the interpretation on medicinal properties of TCM compound prescriptions, and contemporary TCM innovative drug developments.

Joint sparse optimization: lower-order regularization method and application in cell fate conversion

Multiple measurement signals are commonly collected in practical applications, and joint sparse optimization adopts the synchronous effect within multiple measurement signals to improve model analysis and sparse recovery capability. In this paper, we investigate the joint sparse optimization problem via ℓp,q regularization ( 0⩽q⩽1⩽p ) in three aspects: theory, algorithm and application. In the theoretical aspect, we introduce a weak notion of joint restricted Frobenius norm condition associated with the ℓp,q regularization, and apply it to establish an oracle property and a recovery bound for the ℓp,q regularization of joint sparse optimization problem. In the algorithmic aspect, we apply the well-known proximal gradient algorithm to solve the ℓp,q regularization problems, provide analytical formulas for proximal subproblems of certain specific ℓp,q regularizations, and establish the global convergence and linear convergence rate of the proximal gradient algorithm under some mild conditions. More importantly, we propose two types of proximal gradient algorithms with the truncation technique and the continuation technique, respectively, and establish their convergence to the ground true joint sparse solution within a tolerance relevant to the noise level and the recovery bound under the assumption of restricted isometry property. In the aspect of application, we develop a novel method, based on joint sparse optimization with lower-order regularization and proximal gradient algorithm, to infer the master transcription factors for cell fate conversion, which is a powerful tool in developmental biology and regenerative medicine. Numerical results indicate that the novel method facilitates fast identification of master transcription factors, give raise to the possibility of higher successful conversion rate and in the hope of reducing biological experimental cost.

Potential Nano/Microcenters of Crystal Nucleation in Reagent-Grade Purity Solvents and Their Differentiation by Fluorescent-Tagged Antiscalant

A common issue in studies on liquid-phase chemical processes is that the natural solid nanoimpurities present in reagent-grade chemicals are ignored. Little is known about these impurities’ nature, sizes, concentrations, and behavior, yet they significantly affect the efficiency of antiscalants in municipal and laboratory solutions. Recent research has focused on: (i) estimating nanoimpurity concentrations in in-house deionized water and semiconductor-grade isopropanol using “light sheet” optical ultramicroscopy, and (ii) visualizing antiscalant sorption on these impurities. Using a fluorescent-tagged antiscalant aminobis(methylenephosphonic acid) (ADMP-F), we tracked its affinity to particulate matter in deionized water and reagent-grade KCl solutions. Our study showed that the total concentration of nanoparticles with a size larger than 20 nm is about 106 units/mL in deionized water and 105 units/mL in isopropanol. Extrapolation of these values to a size ≥1 nm resulted in concentrations of 1011 and 108 units/mL. The addition of KCl or ADMP-F significantly increased foreign nanoparticle populations. ADMP-F is selectively adsorbed by only some impurities, while most antiscalant molecules remain as true solution. To our knowledge, this is the first instance of fluorescently labeled aminoalkylphosphonates being able to differentiate particulate matter traces in reagent-grade purity solutions. Therefore, the role of nanoparticles as crystallization centers should be seriously reconsidered, especially in their important application in scale inhibition.

Steady Boussinesq convection: Parametric analyticity and computation

AbstractSteady solutions to the Navier–Stokes equations with internal temperature forcing are considered. The equations are solved in two dimensions using the Boussinesq approximation to couple temperature and density fluctuations. A perturbative Stokes expansion is used to prove that that steady flow variables are parametrically analytic in the size of the forcing. The Stokes expansion is complemented with analytic continuation, via functional Padé approximation. The zeros of the denominator polynomials in the Padé approximants are observed to agree with a numerical prediction for the location of singularities of the steady flow solutions. The Padé representations not only prove to be good approximations to the true flow solutions for moderate intensity forcing, but are also used to initialize a Newton solver to compute large amplitude solutions. The composite procedure is used to compute steady flow solutions with forcing several orders of magnitude larger than the fixed‐point method developed in previous work.