Annals of Applied Mathematics (AAM) was established in 1985, with the original title Annals of Differential Equations. AAM is an international journal devoted to the publication of original research papers and review articles in all areas of applied mathematics, with an emphasis on differential equations, probability and statistics, computational mathematics and discrete mathematics. The journal is published in one volume annually consisting of four issues. AAM is co-sponsored by Fuzhou University and China Society for Industrial and Applied Mathematics (CSIAM).
A Survey on the Performance of Krylov Subspace Methods in High Order Compact Schemes for Solving Poisson’s Equation for Application in Incompressible Fluid Flow Solvers
The efficiency of three Krylov subspace methods with their ILU0-preconditioned version in solving the systems with the nondiagonal sparse matrix is examined. The systems have arisen from the discretization of Poisson’s equation using the 4th and 6th-order compact schemes. Four matrix-vector multiplication techniques based on four sparse matrix storage schemes are considered in the algorithm of the Krylov subspace methods and their effects are explored. The convergence history, error reduction, iteration-resolution relation and CPU-time are addressed. The efficacy of various methods is evaluated against a benchmark scenario in which the conventional second-order central difference scheme is employed to discretize Poisson’s equation. The Krylov subspace methods, paired with four distinct matrix-vector multiplication strategies across three discretization approaches, are tested and implemented within an incompressible fluid flow solver to solve the elliptic segment of the equations. The resulting solution process CPU-time surface gives a new vision regarding speeding up a CFD code with proper selection of discretization stencil and matrix-vector multiplication technique.
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The Method of Fundamental Solutions for Optical Fluorescence
In this paper, we develop the method of fundamental solutions (MFS) for solving boundary value problems in the field of optical fluorescence. The governing system of diffusion-absorption equations for the excitation and emission fluences is transformed into a single fourth-order partial differential equation whose fundamental solution can be expressed as the difference of two fundamental solutions of the complex Helmholtz equation. The numerically obtained results confirm the accuracy of the MFS when compared with an available analytical solution. Numerical results are also provided for a physical application in optical fluorescence. Furthermore, extensions to three dimensions along with numerical verification are performed.
Stability and Convergence Analysis of a Linear Energy Stable Scheme for a Cahn-Hilliard Model with Smooth or Weakly Singular Non-Local Term
We consider a Cahn-Hilliard gradient flow model with a free energy functional, which contains a non-local term in addition to linear and non-linear local terms. The non-local terms can be based on smooth and weakly singular kernel operators. We establish the well-posedness of this problem, construct an unconditional energy stable scheme, and carry out a stability and convergence analysis. Several numerical results are presented to illustrate the efficiency and robustness of the proposed scheme.
A Hybrid Conjugate Gradient Method with Trust Region for Large-Scale Unconstrained Optimization Problems
In this work, we modify a conjugate gradient (CG) method recently proposed in the literature, where a PRP conjugate gradient method is modified using trust region. Particularly, we propose a hybrid CG method that incorporates the parameters $β^{PRP},$ $β^{FR}$ and $β^{CD},$ and this new search direction satisfies both the trust region feature and the sufficient descent conditions. Furthermore, under suitable conditions the developed method is proved to be globally convergent. The method is tested on some benchmark problems from the literature and numerical results show that it is quite efficient in solving large scale problems.
Numerical Approaches to Compute Spectra of Non-Self Adjoint Operators in Two and Three Dimensions
In this article we are interested in the numerical computation of spectra of non-self adjoint quadratic operators, in two and three spatial dimensions. Indeed, in the multidimensional case very few results are known on the location of the eigenvalues. This leads to solve nonlinear eigenvalue problems. In introduction we begin with a review of theoretical results and numerical results obtained for the one dimensional case. Then we present the numerical methods developed to compute the spectra (finite difference discretization) for the two and three dimensional cases. The numerical results obtained are presented and analyzed. One difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are unstable. This work is in continuity of a previous work in one spatial dimension [3].
An Extension of the $r^p$ Method for Wave Equations with Scale-Critical Potentials and First-Order Terms
The $r^p$ method, first introduced in [9], has become a robust strategy to prove decay for wave equations in the context of black holes and beyond. In this note, we propose an extension of this method, which is particularly suitable for proving decay for a general class of wave equations featuring a scale-critical time-dependent potential and/or first-order terms of small amplitude. Our approach consists of absorbing error terms in the $r^p$-weighted energy using a novel Grönwall argument, which allows a larger range of $p$ than the standard method. A spherically symmetric version of our strategy first appeared in [22] in the context of a weakly charged scalar field on a black hole whose equations also involve a scale-critical potential.
Deep Neural Network Approaches for Computing the Defocusing Action Ground State of Nonlinear Schrödinger Equation
The defocusing action ground state of the nonlinear Schrödinger equation can be characterized via three different but equivalent minimization formulations. In this work, we propose some deep neural network (DNN) approaches to compute the action ground state through the three formulations. We first consider the unconstrained formulation, where we propose the DNN with a shift layer and demonstrate its necessity towards finding the correct ground state. The other two formulations involve the $L^{p+1}$-normalization or the Nehari manifold constraint. We enforce them as hard constraints into the networks by further proposing a normalization layer or a projection layer to the DNN. Our DNNs can then be trained in an unconstrained and unsupervised manner. Systematical numerical experiments are conducted to demonstrate the effectiveness and superiority of the approaches.
The $H^p-H^q$ Estimates for a Class of Dispersive Equations with Finite Type Geometry
This paper studies the $H^p-H^q$ estimates of a class of oscillatory integrals related to dispersive equations under the assumption that the level hypersurfaces are convex and of finite type. As applications, we obtain the decay estimates for the solutions of higher order homogeneous and inhomogeneous Schrödinger equations.
Time Decay Estimates for Fourth-Order Schrödinger Operators in Dimension Three
This paper is concerned with the time decay estimates of the fourth order Schrödinger operator $H = ∆^2+V (x)$ in dimension three, where $V (x)$ is a real valued decaying potential. Assume that zero is a regular point or the first kind resonance of $H,$ and $H$ has no positive eigenvalues, we established the following time optimal decay estimates of $e^{−it H}$ with a regular term $H^{α/4}:$ When zero is the second or third kind resonance of $H,$ their decay will be significantly changed. We remark that such improved time decay estimates with the extra regular term $H^{α/4}$ will be interesting in the well-posedness and scattering of nonlinear fourth order Schrödinger equations with potentials.
Preface: Special Issue Dedicated to Professor Avy Soffer’s 70th Birthday (Part II)
Professor Avraham (Avy) Soffer was born in Israel on December 28, 1955. He earned his Bachelor’s, Master’s, and Doctor’s degrees in Science from Tel-Aviv University in 1976, 1980, and 1984, respectively. Between 1978 and 1983, he worked as an assistant at Tel-Aviv University.
Oscillation-Eliminating Discontinuous Galerkin Methods for Multicomponent Chemically Reacting Flows
This paper proposes a robust and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method for solving multicomponent chemically reacting flows, which is an extension and application of the recent work [M. Peng, Z. Sun, and K. Wu, Math. Comput., 2024, doi.org/10.1090/mcom/3998]. Following recently developed high-order bound-preserving discontinuous Galerkin method in [J. Du and Y. Yang, J. Comput. Phys., 469 (2022), 111548], we incorporate an OE procedure after each Runge-Kutta time stage to suppress spurious oscillations. The OE procedure is defined by the solution operator of a damping equation, which can be analytically solved without requiring discretization, making its implementation straightforward, non-intrusive, and efficient. Through careful design of the damping coefficients, the proposed OEDG method not only achieves the essentially non-oscillatory (ENO) property without compromising accuracy but also preserves the conservative property — an indispensable aspect of the bound-preserving technique introduced in [J. Du and Y. Yang, J. Comput. Phys., 469 (2022), 111548]. The effectiveness and robustness of the OEDG method are demonstrated through a series of one- and two-dimensional numerical tests on the compressible Euler and Navier–Stokes equations for chemically reacting flows. These results highlight the method’s capability to handle complex flow dynamics while maintaining stability and high-order accuracy.
Oscillation elimination discontinuous Galerkin (OEDG) method
multicomponent chemically reacting flows.
Reconfiguration Graphs for Vertex Colorings of $P_5$-Free Graphs
For any positive integer $k,$ the reconfiguration graph for all $k$-colorings of a graph $G,$ denoted by $\mathcal{R}_k(G),$ is the graph where vertices represent the $k$-colorings of $G,$ and two $k$-colorings are joined by an edge if they differ in color on exactly one vertex. Bonamy et al. established that for any 2-chromatic $P_5$-free graph $G,$ $\mathcal{R}_k(G)$ is connected for each $k≥3.$ On the other hand, Feghali and Merkel proved the existence of a $7p$-chromatic $P_5$-free graph $G$ for every positive integer $p,$ such that $\mathcal{R}_{8p}(G)$ is disconnected. In this paper, we offer a detailed classification of the connectivity of $\mathcal{R}_k(G)$ concerning $t$-chromatic $P_5$-free graphs $G$ for cases $t = 3,$ and $t ≥ 4$ with $t+1 ≤ k ≤(\begin{matrix} t \\ 2 \end{matrix}).$ We demonstrate that $\mathcal{R}_k(G)$ remains connected for each 3-chromatic $P_5$-free graph $G$ and each $k ≥4.$ Furthermore, for each $t≥4$ and $t+1\le k\le (\begin{matrix} t \\ 2 \end{matrix}),$ we provide a construction of a $t$-chromatic $P_5$-free graph $G$ with $\mathcal{R}_k(G)$ being disconnected. This resolves a question posed by Feghali and Merkel.
An Enriched Petrov-Galerkin Method for Darcy Flow in Fractured Porous Media
We develop a locally mass-conservative enriched Petrov-Galerkin (EPG) method without any penalty term for the simulation of Darcy flow in fractured porous media. The discrete fracture model is applied to model the fractures as the lower dimensional fracture interfaces. The new method enriches the approximation trial space of the conforming continuous Galerkin (CG) method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions in the fractures and the surrounding porous media. We propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG method. The solution of the pressure can be decoupled into two steps with a standard CG method and a post-processing correction. The post-processing correction based on the bubble functions in the matrix and the fractures can be solved separately, which is useful for parallel computing. We derive a priori and a posteriori error estimates for the problem. Numerical examples are presented to illustrate the performance of the proposed method.
Existence of Global Attractor for Weakly Damped FDS Nonlinear Wave Equations
The paper investigates the well-posedness of global solutions and the existence of global attractors for weakly damped FDS nonlinear wave equations. It establishes the well-posedness of weak solutions using Galerkin approximation and a priori estimate. Subsequently, a dynamical system is constructed based on the well-posedness of the solution. The existence of a bounded absorbing set for the equations and the smooth properties of the operator semigroup are presented, leading to the existence of a global attractor.
The Characteristics of Solutions to Semilinear Wave Equation with Logarithmic Plus Polynomial Nonlinearities
The semilinear wave equation with logarithmic and polynomial nonlinearities is considered in this paper. By adjusting and using potential well method, we attain the global-in-time existence and infinite time blowup solutions at subcritical initial energy level $E(0)<d.$ Then using additional conditions on initial data, these results are enlarged at critical case $E(0)=d$ and arbitrarily positive case $E(0)>0.$
logarithmic and polynomial combined nonlinearity
Global Classical Solution and Asymptotic Behavior to a Kind of Linearly Degenerate Quasilinear Hyperbolic System
This paper is concerned with the global classical solution and the asymptotic behavior to a kind of linearly degenerate quasilinear hyperbolic system in several space variables. When the semilinear terms contain at least two waves with different propagation speeds, we can prove that the system considered admits a global classical solution by the weighted energy estimate under the small and suitable decay assumptions on the initial data. Furthermore, we can show that the solution converges to a solution of the linearized system based on the decay property of the nonlinearities.
Energy Equality for the Isentropic Compressible Navier-Stokes Equations Without Upper Bound of the Density
In this paper, we are concerned with the minimal regularity of both the density and the velocity for the weak solutions keeping energy equality in the isentropic compressible Navier-Stokes equations. The energy equality criteria without upper bound of the density are established for the first time. Our results imply that the lower integrability of the density $\rho$ means that more integrability of the velocity $v$ or the gradient of the velocity $∇v$ are necessary for energy conservation of the isentropic compressible fluid and the inverse is also true.
Non-Global Existence of Regular Solution to Initial Value Problem of Relativistic Euler Equations in $\mathbb{R}^N$
In fluid mechanics and astrophysics, relativistic Euler equations can be used to describe the effects of special relativity which are an extension of the classical Euler equations. In this paper, we will consider the initial value problem of relativistic Euler equations in an initial bounded region of $\mathbb{R}^N.$ If the initial velocity satisfies $$\max\limits_{\vec{x_0}\in ∂Ω(0)}\sum\limits_{i=1}^N v^2_i(0,\vec{x_0})<\frac{c^2A_1}{2},$$ where $A_1$ is a positive constant depend on some sufficiently large $T^∗,$ then we can get the non-global existence of the regular solution for the $N$-dimensional relativistic Euler equations.
Entire Sign-Changing Solutions to the Fractional Critical Schrödinger Equation
In this paper, we consider the fractional critical Schrödinger equation (FCSE) $$(-\Delta)^su-|u|^{2^*_s-2}u=0,$$ where $u∈\dot{H}^s(\mathbb{R}^N),$ $N≥4,$ $0<s<1$ and $2^∗_s=\frac{2N}{N−2s}$ is the critical Sobolev exponent of order $s.$ By virtue of the variational method and the concentration compactness principle with the equivariant group action, we obtain some new type of nonradial, sign-changing solutions of (FCSE) in the energy space $\dot{H}^s(\mathbb{R}^N)$. The key component is that we take the equivariant group action to construct several subspace of $\dot{H}^s(\mathbb{R}^N)$ with trivial intersection, then combine the concentration compactness argument in the Sobolev space with fractional order to show the compactness property of Palais-Smale sequences in each subspace and obtain the multiple solutions of (FCSE) in $\dot{H}^s(\mathbb{R}^N).$
Fractional critical Schrödinger equation
Preface: Special Issue Dedicated to Professor Avy Soffer’s 70th Birthday (Part I)
Numerical Study of Semidiscrete Penalty Approach for Stabilizing Boussinesq System with Localized Feedback Control
We investigate the numerical approximation for stabilizing the semidiscrete linearized Boussinesq system around an unstable stationary state. Stabilization is attained through internal feedback controls applied to the velocity and temperature equations, localized within an arbitrary open subset. This study follows the framework presented in [14], considering the continuous linearized Boussinesq system. The primary objective is to explore the penalization-based approximation of the free divergence condition in the semidiscrete case and provide a numerical validation of these results in a two-dimensional setting.
On the Monotonicity of $Q^3$ Spectral Element Method for Laplacian
The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schemes on unstructured meshes. On the other hand, it is possible to construct high order accurate monotone schemes on structured meshes. All previously known high order accurate inverse positive schemes are or can be regarded as fourth order accurate finite difference schemes, which is either an M-matrix or a product of two M-matrices. For the $Q^3$ spectral element method for the two-dimensional Laplacian, we prove its stiffness matrix is a product of four M-matrices thus it is unconditionally monotone. Such a scheme can be regarded as a fifth order accurate finite difference scheme. Numerical tests suggest that the unconditional monotonicity of $Q^k$ spectral element methods will be lost for $k≥9$ in two dimensions, and for $k≥4$ in three dimensions. In other words, for obtaining a high order monotone scheme, only $Q^2$ and $Q^3$ spectral element methods can be unconditionally monotone in three dimensions.
On a Hybrid Method for Inverse Transmission Eigenvalue Problems
In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of a penetrable medium scatterer. The linear sampling method is employed to determine the transmission eigenvalues within a certain wavenumber interval based on far-field measurements. Based on a prior information given by the linear sampling method, the neural network approach is proposed for the reconstruction of the unknown scatterer. We divide the wavenumber intervals into several subintervals, ensuring that each transmission eigenvalue is located in its corresponding subinterval. In each such subinterval, the wavenumber that yields the maximum value of the indicator functional will be included in the input set during the generation of the training data. This technique for data generation effectively ensures the consistent dimensions of model input. The refractive index and shape are taken as the output of the network. Due to the fact that transmission eigenvalues considered in our method are relatively small, certain super-resolution effects can also be generated. Numerical experiments are presented to verify the effectiveness and promising features of the proposed method in two and three dimensions.
Generalized Isospectral-Nonisospectral Modified Korteweg-de Vries Integrable Hierarchies and Related Properties
In this article, a new technique for deriving integrable hierarchy is discussed, i.e., such that are derived by combining the Tu scheme with the vector product. Several classes of spectral problems are introduced by three-dimensional loop algebra and six-dimensional loop algebra whose commutators are vector product, and the six-dimensional loop algebra is derived from the enlargement of the three-dimensional loop algebra. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy is worked out. The derived integrable hierarchies are reduced to the modified Korteweg-de Vries (mKdV) equation, generalized coupled mKdV integrable system and non-isospectral mKdV equation under specific parameter selection. Starting from a 3×3 matrix spectral problem, we subsequently construct an explicit $N$-fold Darboux transformation for integrable system (2.8) with the help of a gauge transformation of the corresponding spectral problem. At the same time, the determining equations of nonclassical symmetries associated with mKdV equation are presented in this paper. It follows that we investigate the coverings and the nonlocal symmetries of the nonisospectral mKdV equation by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product.
Error Analysis of the Nonconforming $P_1$ Finite Element Method to the Sequential Regularization Formulation for Unsteady Navier-Stokes Equations
In this paper we investigate the nonconforming $P_1$ finite element approximation to the sequential regularization method for unsteady Navier-Stokes equations. We provide error estimates for a full discretization scheme. Typically, conforming $P_1$ finite element methods lead to error bounds that depend inversely on the penalty parameter $\epsilon.$ We obtain an $\epsilon$-uniform error bound by utilizing the nonconforming $P_1$ finite element method in this paper. Numerical examples are given to verify theoretical results.
A Convergent Numerical Algorithm for the Stochastic Growth-Fragmentation Problem
The stochastic growth-fragmentation model describes the temporal evolution of a structured cell population through a discrete-time and continuous-state Markov chain. The simulations of this stochastic process and its invariant measure are of interest. In this paper, we propose a numerical scheme for both the simulation of the process and the computation of the invariant measure, and show that under appropriate assumptions, the numerical chain converges to the continuous growth-fragmentation chain with an explicit error bound. With a triangle inequality argument, we are also able to quantitatively estimate the distance between the invariant measures of these two Markov chains.
Empirical Study on Option Pricing under Markov Regime Switching Economics
In this research, we summarize the results of a practical study of index options based on the option valuation model which was proposed by Siu and Yang (Acta Math. Appl. Sin. Engl. Ser., 25(3) (2009), pp. 339–388), where an EMM kernel is integrated which takes into account all risk components of a regime-switching model. Further, the regime-switching risk of an economy in the options is priced using a hidden Markov regime-switching model with the risky underlying asset being modulated by a discrete-time, finite-state, hidden Markov chain whose states represent the hidden states of an economy. We apply such a model to the pricing of Hang Seng Index options based on the real-world financial data from October 2009 to October 2010 (i.e., for the year in which the model was proposed). We employed the entropy martingale measure (EMM) approach proposed by Siu and Yang (Acta Math. Appl. Sin. Engl. Ser., 25(3) (2009), pp. 339–388) to determine the optimal martingale measure for the Markov-modulated GBM. In addition, we have proposed a numerical technique called the weighted difference method to compliment the EMM approach. We have also verified the extended jump-diffusion model under regime-switching that we proposed recently (Int. J. Finan. Eng., 6(4) (2019), 1950038) using the 50ETF options which are obtained from Shanghai Stock Exchange covering a time span from January 2018 to December 2022. Further, we have highlighted the challenges for the EMM kernel-based Markov regime-switching model for pricing the out-of-the-money index options in the real world.
Probabilistic Error Estimate for Numerical Discretization of High-Index Saddle Dynamics with Inaccurate Models
We prove probabilistic error estimates for high-index saddle dynamics with or without constraints to account for the inaccurate values of the model, which could be encountered in various scenarios such as model uncertainties or surrogate model algorithms via machine learning methods. The main contribution lies in incorporating the probabilistic error bound of the model values with the conventional error estimate methods for high-index saddle dynamics. The derived results generalize the error analysis of deterministic saddle dynamics and characterize the affect of the inaccuracy of the model on the convergence rate.
A Review of Process Optimization for Additive Manufacturing Based on Machine Learning
Additive manufacturing (AM), also known as 3D printing, has emerged as a groundbreaking technology that has transformed the manufacturing industry. Its ability to produce intricate and customized parts with remarkable speed and reduced material waste has revolutionized traditional manufacturing approaches. However, the AM process itself is a complex and multifaceted undertaking, with various parameters that can significantly influence the quality and efficiency of the printed parts. To address this challenge, researchers have explored the integration of machine learning (ML) techniques to optimize the AM process. This paper presents a comprehensive review of process optimization for additive manufacturing based on machine learning, highlighting the recent advancements, methodologies, and challenges in this field.
Fast High Order and Energy Dissipative Schemes with Variable Time Steps for Time-Fractional Molecular Beam Epitaxial Growth Model
In this paper, we propose and analyze high order energy dissipative time-stepping schemes for time-fractional molecular beam epitaxial (MBE) growth model on the nonuniform mesh. More precisely, $(2−α)$-order, second-order and $(3−α)$-order time-stepping schemes are developed for the time-fractional MBE model based on the well known L1, L2-$1_σ,$ and L2 formulations in discretization of the time-fractional derivative, which are all proved to be unconditional energy dissipation in the sense of a modified discrete nonlocal-energy on the nonuniform mesh. In order to reduce the computational storage, we apply the sum of exponential technique to approximate the history part of the time-fractional derivative. Moreover, the scalar auxiliary variable (SAV) approach is introduced to deal with the nonlinear potential function and the history part of the fractional derivative. Furthermore, only first order method is used to discretize the introduced SAV equation, which will not affect high order accuracy of the unknown thin film height function by using some proper auxiliary variable functions $V (ξ)$. To our knowledge, it is the first time to unconditionally establish the discrete nonlocal-energy dissipation law for the modified L1-, L2-$1_{σ^-},$ and L2-based high-order schemes on the nonuniform mesh, which is essentially important for such time-fractional MBE models with low regular solutions at initial time. Finally, a series of numerical experiments are carried out to verify the accuracy and efficiency of the proposed schemes.
Time-fractional molecular beam epitaxial growth