Approximation Of Differential Equations Research Articles

Pushing coarse-grained models beyond the continuum limit using equation learning

Mathematical modelling of biological population dynamics often involves proposing high-fidelity discrete agent-based models that capture stochasticity and individual-level processes. These models are often considered in conjunction with an approximate coarse-grained differential equation that captures population-level features only. These coarse-grained models are only accurate in certain asymptotic parameter regimes, such as enforcing that the time scale of individual motility far exceeds the time scale of birth/death processes. When these coarse-grained models are accurate, the discrete model still abides by conservation laws at the microscopic level, which implies that there is some macroscopic conservation law that can describe the macroscopic dynamics. In this work, we introduce an equation learning framework to find accurate coarse-grained models when standard continuum limit approaches are inaccurate. We demonstrate our approach using a discrete mechanical model of epithelial tissues, considering a series of four case studies that consider problems with and without free boundaries, and with and without proliferation, illustrating how we can learn macroscopic equations describing mechanical relaxation, cell proliferation, and the equation governing the dynamics of the free boundary of the tissue. While our presentation focuses on this biological application, our approach is more broadly applicable across a range of scenarios where discrete models are approximated by approximate continuum-limit descriptions.

Well-posedness and finite element approximation of mixed dimensional partial differential equations

In this article, a mixed dimensional elliptic partial differential equation is considered, posed in a bulk domain with a large number of embedded interfaces. In particular, well-posedness of the problem and regularity of the solution are studied. A fitted finite element approximation is also proposed and an a priori error bound is proved. For the solution of the arising linear system, an iterative method based on subspace decomposition is proposed and analyzed. Finally, numerical experiments are presented and rapid convergence using the proposed preconditioner is achieved, confirming the theoretical findings.

Regularity and numerical approximation of fractional elliptic differential equations on compact metric graphs

The fractional differential equation L β u = f L^\beta u = f posed on a compact metric graph is considered, where β > 0 \beta >0 and L = κ 2 − ∇ ( a ∇ ) L = \kappa ^2 - \nabla (a\nabla ) is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients κ , a \kappa ,a . We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when f f is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power L − β L^{-\beta } . For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the L 2 ( Γ × Γ ) L_2(\Gamma \times \Gamma ) -error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for L = κ 2 − Δ , κ > 0 {L = \kappa ^2 - \Delta , \kappa >0} are performed to illustrate the results.

An explicit approximation for super-linear stochastic functional differential equations

Since it is difficult to implement implicit schemes on the infinite-dimensional space, we aim to develop the explicit numerical method for approximating super-linear stochastic functional differential equations (SFDEs). Precisely, borrowing the truncation idea and linear interpolation we propose an explicit truncated Euler–Maruyama (EM) scheme for SFDEs, and obtain the boundedness and convergence in Lp(p≥2). We also prove the convergence rate with 1/2 order. Different from some previous works (Mao, 2003; Zhang et al., 2018), we release the global Lipschitz restriction on the diffusion coefficient. Furthermore, we reveal that numerical solutions preserve the underlying exponential stability. Moreover, we give several examples to support our theory.

On the layer crossing problem for a semi-infinite hydraulic fracture

This paper analyses the problem of a semi-infinite fluid-driven fracture propagating through multiple stress layers in a permeable elastic medium. Such a problem represents the tip region of a planar hydraulic fracture. When the hydraulic fracture crosses a stress layer, the use of a standard tip asymptotic solution may lead to a considerable reduction of accuracy, even for the simplest case of a height-contained fracture. In this study, we propose three approaches to incorporate the effect of stress layers into the tip asymptote: non-singular integral formulation, toughness-corrected asymptote, and an ordinary differential equation approximation of the non-singular integral formulation mentioned above. As illustrated in the paper, these approaches for stress-corrected asymptotes differ in computational complexity, the complexity of implementation, and the accuracy of the approximation. In addition, the size of the validity region of the stress-corrected asymptote is evaluated, and it is shown to be greatly reduced relative to the case without layers. In order to address the issue, the stress relaxation factor is introduced. This, in turn, allows for enhancing the accuracy of the layer-crossing computation on a relatively coarse mesh to utilize the stress-corrected asymptote in hydraulic fracturing simulators for the purpose of front tracking.

Nine-point compact sixth-order approximation for two-dimensional nonlinear elliptic partial differential equations: Application to bi- and tri-harmonic boundary value problems

Nine point sixth order compact numerical approximations are suggested to solve 2D nonlinear elliptic partial differential equations (NLEPDEs) and for the estimation of normal derivatives on a uniform rectangular grid subject to Dirichlet boundary conditions. We deliberate error analysis and reveal that, under specific conditions, our method converges to the sixth order. In addition, we extend our technique to vector form in order to solve the system of NLEPDEs. In application, we discuss nine-point compact sixth order approximations for bi- and tri-harmonic elliptic boundary value problems. Numerical experiments are carried out on several benchmark problems including bi- and tri-harmonic equations, and verified the sixth order convergence of the proposed methods.

Approximation of Stochastic Volterra Equations with kernels of completely monotone type

In this work, we develop a multifactor approximation for d d -dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an L 2 L^2 -estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in ( 0 , 1 / 2 ) (0,1/2) , we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model.

Higher order approximations for fractional order integro-parabolic partial differential equations on an adaptive mesh with error analysis

Higher order approximations for fractional order integro-parabolic partial differential equations on an adaptive mesh with error analysis

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

In this work, first, a family of fourth‐order methods is proposed to solve nonlinear equations. The methods satisfy the Kung‐Traub optimality conjecture. By developing the methods into memory methods, their efficiency indices are increased. Then, the methods are extended to the multi‐step methods for finding the solutions to systems of problems. The formula for the order of convergence of the multi‐step iterative methods is , where is the step number of the methods. It is clear that computing the Jacobian matrix derivative evaluation and its inversion are expensive; therefore, we compute them only once in every cycle of the methods. The important feature of these multi‐step methods is their high‐efficiency index. Numerical examples that confirm the theoretical results are performed. In applications, some nonlinear problems related to the numerical approximation of fractional differential equations (FDEs) are constructed and solved by the proposed methods.

L 2 convergence of smooth approximations of stochastic differential equations with unbounded coefficients

The aim of this article is to obtain convergence in mean in the uniform topology of piecewise linear approximations of stochastic differential equations (SDEs) with C 1 drift and C 2 diffusion coefficients with uniformly bounded derivatives. Convergence analyses for such Wong-Zakai approximations most often assume that the coefficients of the SDE are uniformly bounded. Almost sure convergence in the unbounded case can be obtained using now standard rough path techniques, although L q convergence appears yet to be established and is of importance for several applications involving Monte-Carlo approximations. We consider L 2 convergence in the unbounded case using a combination of traditional stochastic analysis and rough path techniques. We expect our proof technique extend to more general piecewise smooth approximations.

A Broken FEEC Framework for Electromagnetic Problems on Mapped Multipatch Domains

We present a framework for the structure-preserving approximation of partial differential equations on mapped multipatch domains, extending the classical theory of finite element exterior calculus (FEEC) to discrete de Rham sequences which are broken, i.e., fully discontinuous across the patch interfaces. Following the Conforming/Nonconforming Galerkin (CONGA) schemes developed in Campos Pinto and Sonnendrücker (Math Comput 85:2651–2685, 2016) and Campos Pinto and Güçlü (Broken-FEEC discretizations and Hodge Laplace problems. arXiv:2109.02553, 2022), our approach is based on: (i) the identification of a conforming discrete de Rham sequence with stable commuting projection operators, (ii) the relaxation of the continuity constraints between patches, and (iii) the construction of conforming projections mapping back to the conforming subspaces, allowing to define discrete differentials on the broken sequence. This framework combines the advantages of conforming FEEC discretizations (e.g. commuting projections, discrete duality and Hodge–Helmholtz decompositions) with the data locality and implementation simplicity of interior penalty methods for discontinuous Galerkin discretizations. We apply it to several initial- and boundary-value problems, as well as eigenvalue problems arising in electromagnetics. In each case our formulations are shown to be well posed thanks to an appropriate stabilization of the jumps across the interfaces, and the solutions are extremely robust with respect to the stabilization parameter. Finally we describe a construction using tensor-product splines on mapped cartesian patches, and we detail the associated matrix operators. Our numerical experiments confirm the accuracy and stability of this discrete framework, and they allow us to verify that expected structure-preserving properties such as divergence or harmonic constraints are respected to floating-point accuracy.

A dynamic hysteresis model of heat and mass transfer for hygrothermal bio-based materials

The hygrothermal behaviour of porous building materials is significantly determined by the moisture and heat transfer phenomena. However, most of existing models used to predict this phenomenon neglect moisture hysteresis effect and dynamic capillarity effect, limiting their accuracy in assessing the efficiency and sustainability of bio-based building materials. To bridge this gap, the model presented in this paper considers both dynamic and hysteresis effects. The Finite Element Method (FEM) is adopted for the discrete approximation of the partial differential equations governing heat and moisture transfer. The model is validated with respect to experiment performed on bio-based material subjected to thermo-hygrical boundaries conditions. Results show unequivocally that dynamic capillarity effect and moisture hysteresis effect should be considered simultaneously to quantify hygrothermal behaviour of porous building materials. This critical insight underscores the significance of this research, offering a comprehensive model for evaluating the efficiency and sustainability of these materials by addressing the often-neglected aspects of moisture behaviour, i.e., dynamic and hysteresis effects. This model is marked by its precision and thorough consideration of essential factors, making it an indispensable contribution to the field and advances the understanding of the hygrothermal behaviour of bio-based building materials in real conditions. It would help in designing optimized building envelop.

Reduced Lagrange multiplier approach for non-matching coupling of mixed-dimensional domains

Many physical problems involving heterogeneous spatial scales, such as the flow through fractured porous media, the study of fiber-reinforced materials, or the modeling of the small circulation in living tissues — just to mention a few examples — can be described as coupled partial differential equations defined in domains of heterogeneous dimensions that are embedded into each other. This formulation is a consequence of geometric model reduction techniques that transform the original problems defined in complex three-dimensional domains into more tractable ones. The definition and the approximation of coupling operators suitable for this class of problems is still a challenge. We develop a general mathematical framework for the analysis and the approximation of partial differential equations coupled by non-matching constraints across different dimensions, focusing on their enforcement using Lagrange multipliers. In this context, we address in abstract and general terms the well-posedness, stability, and robustness of the problem with respect to the smallest characteristic length of the embedded domain. We also address the numerical approximation of the problem and we discuss the infsup stability of the proposed numerical scheme for some representative configuration of the embedded domain. The main message of this work is twofold: from the standpoint of the theory of mixed-dimensional problems, we provide general and abstract mathematical tools to formulate coupled problems across dimensions. From the practical standpoint of the numerical approximation, we show the interplay between the mesh characteristic size, the dimension of the Lagrange multiplier space, and the size of the inclusion in representative configurations interesting for applications. The latter analysis is complemented with illustrative numerical examples.

Stochastic differential equation approximations of generative adversarial network training and its long-run behavior

AbstractThis paper analyzes the training process of generative adversarial networks (GANs) via stochastic differential equations (SDEs). It first establishes SDE approximations for the training of GANs under stochastic gradient algorithms, with precise error bound analysis. It then describes the long-run behavior of GAN training via the invariant measures of its SDE approximations under proper conditions. This work builds a theoretical foundation for GAN training and provides analytical tools to study its evolution and stability.

Convergence Rates for the Truncated Euler-Maruyama Method for Nonlinear Stochastic Differential Equations

In this paper, our main aim is to investigate the strong convergence rate of the truncated Euler-Maruyama approximations for stochastic differential equations with superlinearly growing drift coefficients. When the diffusion coefficient is polynomially growing or linearly growing, the strong convergence rate of arbitrarily close to one half is established at a single time T or over a time interval [0,T], respectively. In both situations, the common one-sided Lipschitz and polynomial growth conditions for the drift coefficients are not required. Two examples are provided to illustrate the theory.

Finding analytical approximations for discrete, stochastic, individual-based models of ecology

Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how ’bottom up’, individual-based models can be approximated by ’top down’ models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterize these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.

Efficient and accurate nonlinear model reduction via first-order empirical interpolation

We present a model reduction approach that extends the original empirical interpolation method to enable accurate and efficient reduced basis approximation of parametrized nonlinear partial differential equations (PDEs). In the presence of nonlinearity, the Galerkin reduced basis approximation remains computationally expensive due to the high complexity of evaluating the nonlinear terms, which depends on the dimension of the truth approximation. The empirical interpolation method (EIM) was proposed as a nonlinear model reduction technique to render the complexity of evaluating the nonlinear terms independent of the dimension of the truth approximation. We introduce a first-order empirical interpolation method (FOEIM) that makes use of the partial derivative information to construct an inexpensive and stable interpolation of the nonlinear terms. We propose two different FOEIM algorithms to generate interpolation points and basis functions. We apply the FOEIM to nonlinear elliptic PDEs and compare it to the Galerkin reduced basis approximation and the EIM. Numerical results are presented to demonstrate the performance of the three reduced basis approaches.

A deep learning feed-forward neural network framework for the solutions to singularly perturbed delay differential equations

In this paper, we explore a deep learning feedforward artificial neural network (ANN) framework as a numerical tool for approximating the solutions to singularly perturbed delay differential equations (SPDDE). Our approach trains the network with fewer uniform data points along with linear interpolation as a dependent variable. More importantly, the exact solution is not used for training the network. A mean square or Euclidean norm type total loss function is utilized to assess the deep learning network’s performance. In the training and testing phase, our network adjusts itself to fit the tangent and the curvature by minimizing the total loss function and fine-tuning the network’s hyperparameters during the backpropagation stage. Our focus in this contribution is to investigate an answer to the question- “whether neural networks can learn to solve the differential equations with both delay and boundary layer behavior”? Traditional numerical methods are known to perform poorly in approximating the solutions to SPDDE due to boundary layer behavior, which is the sharp change in the gradient of the solution inside a region of very small width. Our proposed network architecture is used to investigate the above question for three varieties of SPDDE. The numerical results demonstrate that the fine-tuned adaptive deep learning architecture can effectively approximate the solution to SPDDE for varieties of delay and perturbation parameters. It is expected that the current method can be extendable to approximate 2D convection-diffusion partial differential equations with a boundary layer.

Investigation of the sedimentation characterization of magnetorheological fluids

Magnetorheological fluids (MRFs) are smart materials that respond to an external magnetic field by changing their rheological properties. A main drawback of MR fluids is sedimentation which may render the fluids unusable over time. In this study, we introduce a system for the investigation of the sedimentation of magnetorheological fluids in the gravitational field. First of all, differential equation approximations describing the initial stage and the post-saturated stage of the sedimentation are proposed to correlate the experimental results. The custom-developed apparatus is explained, which incorporates a measuring sensor with a lock-in amplifier, a vertically positioned test tube, and a motorized z-axis translating stage. The measurement principle is based on the change of mutual induction between the primary and secondary coils of a sensor, according to the volume fraction-dependent susceptibility. The sensor's coils are driven by a lock-in amplifier, which is able to sense the minor settling distances. The motorized translating axis provides the opportunity for the analysis of the concentration profile of the sample column along its entire length and the device is able to work in fully automated mode. The performance of the device is validated by measurements with Lord MR fluids, for which the sedimentation constant and ratio metrics are provided.

Wong-Zakai approximations for stochastic differential equations with path-dependent coefficients

In this paper, we shall prove an L p ( p > 2 ) L^{p}(p>2) strong convergence rate of the non-adapted Wong-Zakai approximation for stochastic differential equations with path-dependent coefficients under d ∞ d_{\infty } -Lipschitz conditions.