Discretization Of Partial Differential Equations Research Articles

Collocation methods for nonlinear differential equations on low-rank manifolds

We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

Stability and Convergence Analysis of the Discrete Dynamical System for Simulating a Moving Bed

The efficiency of controlling the simulated moving bed (SMB) has long been a critical issue in the chemical engineering industry. Most existing research relies on finite element methods, which often result in lower control efficiency and are unable to achieve online control. To enhance control over the SMB process, this paper employs the Crank–Nicolson method to develop a discrete dynamical model. This approach allows for the investigation of system stability and convergence, fundamentally addressing the sources of error. During the discretization of partial differential equations (PDEs), two main types of errors arise: intrinsic errors from the method itself and truncation errors due to derivative approximations and the discretization process. Research indicates that for the former, the iterative process remains convergent as long as the time and spatial steps are sufficiently small. Regarding truncation errors, studies have demonstrated that they exhibit second-order behavior relative to time and spatial steps. The theoretical validation shows that the iteration works effectively, and simulations confirm that the finite difference method is stable and performs well with varying SMB system parameters and controller processes. This provides a solid theoretical foundation for practical, real-time online control.

Dynamic optimal energy flow of integrated electricity and gas systems in continuous space

The intrinsic flexibility of integrated electricity and gas systems (IEGSs) hinges on the gas flow dynamics dictated by partial differential equations (PDEs). However, conventional numerical approaches for PDEs grapple with inefficiencies and inaccuracies for operational analysis, chiefly due to the discretization of PDE invoked. For the first time, this study pioneers an analytical methodology for IEGS's analysis, effectively eliminating the pitfalls of discretization in solving gas flow dynamics. Based on this, the dynamic optimal energy flow problem in IEGS is reformulated in continuous space, significantly simplifying complex models while bolstering analytical precision. Case studies affirm the remarkable advantages of the proposed analytical approach. Under comparable scenarios, the proposed method improves computational efficiency dozens of times and exhibits concurrently heightened levels of accuracy.

An efficient ADAM-type algorithm with finite elements discretization technique for random elliptic optimal control problems

We consider an optimal control problem governed by an elliptic partial differential equation (PDE) with random coefficient, and introduce an efficient numerical method for the problem under concern. Based on a finite element discretization of the constraint PDE and objective functional, an ADAM-type iteration scheme is proposed to compute the approximated optimal control, which equips the stochastic gradient iteration with modified momentum acceleration and adaptive step size. We provide theoretical bounds on both the discretization and iteration error of the proposed numerical method. Our method is tested on two examples, and the numerical results show the efficiency and performance of the method.

Negative order sobolev cubatures: preconditioners of partial differential equation learning tasks circumventing numerical stiffness

We present a variational approach aimed at enhancing the training of physics-informed neural networks (PINNs) and more general surrogate models for learning partial differential equations (PDE). In particular, we extend our formerly introduced notion of Sobolev cubatures to negative orders, enabling the approximation of negative order Sobolev norms. We mathematically prove the effect of negative order Sobolev cubatures in improving the condition number of discrete PDE learning problems, providing balancing scalars that mitigate numerical stiffness issues caused by loss imbalances. Additionally, we consider polynomial surrogate models (PSMs), which maintain the flexibility of PINN formulations while preserving the convexity structure of the PDE operators. The combination of negative order Sobolev cubatures and PSMs delivers well-conditioned discrete optimization problems, solvable via an exponentially fast convergent gradient descent for λ-convex losses. Our theoretical contributions are supported by numerical experiments, addressing linear and non-linear, forward and inverse PDE problems. These experiments show that the Sobolev cubature-based PSMs emerge as the superior state-of-the-art PINN technique.

Discrete and continuous mathematical models of sharp-fronted collective cell migration and invasion.

Mathematical models describing the spatial spreading and invasion of populations of biological cells are often developed in a continuum modelling framework using reaction-diffusion equations. While continuum models based on linear diffusion are routinely employed and known to capture key experimental observations, linear diffusion fails to predict well-defined sharp fronts that are often observed experimentally. This observation has motivated the use of nonlinear degenerate diffusion; however, these nonlinear models and the associated parameters lack a clear biological motivation and interpretation. Here, we take a different approach by developing a stochastic discrete lattice-based model incorporating biologically inspired mechanisms and then deriving the reaction-diffusion continuum limit. Inspired by experimental observations, agents in the simulation deposit extracellular material, which we call a substrate, locally onto the lattice, and the motility of agents is taken to be proportional to the substrate density. Discrete simulations that mimic a two-dimensional circular barrier assay illustrate how the discrete model supports both smooth and sharp-fronted density profiles depending on the rate of substrate deposition. Coarse-graining the discrete model leads to a novel partial differential equation (PDE) model whose solution accurately approximates averaged data from the discrete model. The new discrete model and PDE approximation provide a simple, biologically motivated framework for modelling the spreading, growth and invasion of cell populations with well-defined sharp fronts. Open-source Julia code to replicate all results in this work is available on GitHub.

A parallel preconditioning Schur complement approach for large scale industrial problems

The purpose of this paper is to introduce a new strategy to improve the convergence and efficiency of the class of domain decomposition known as Schur complement techniques related to interface variables for the simulation of mechanical, electrical and thermal problems in presence of cross points. More precisely, we are interested not only in domain decomposition with two equal parts having the same physical properties but rather in more general splitting components. It is known that in the first case, the optimal convergence with good pre-conditioner is obtained in two iterations and the problem is still challenging in the second case. The primary goal then is to fill part of the gap in such problem domain decomposition techniques and to contribute to solve extremely difficult industrial problems of large scale by using parallel sparse direct solver of the multi-core environment of the whole system and handling each part of the system independently of the change of the mesh or the shifting of the mathematical method of resolution and subsequently, we treat the interface as boundary conditions. The numerical experiments of our algorithm are performed on few models arising from discretization of partial differential equations using Finite Element Method (FEM).

Solving inverse problems in physics by optimizing a discrete loss: Fast and accurate learning without neural networks.

In recent years, advances in computing hardware and computational methods have prompted a wealth of activities for solving inverse problems in physics. These problems are often described by systems of partial differential equations (PDEs). The advent of machine learning has reinvigorated the interest in solving inverse problems using neural networks (NNs). In these efforts, the solution of the PDEs is expressed as NNs trained through the minimization of a loss function involving the PDE. Here, we show how to accelerate this approach by five orders of magnitude by deploying, instead of NNs, conventional PDE approximations. The framework of optimizing a discrete loss (ODIL) minimizes a cost function for discrete approximations of the PDEs using gradient-based and Newton's methods. The framework relies on grid-based discretizations of PDEs and inherits their accuracy, convergence, and conservation properties. The implementation of the method is facilitated by adopting machine-learning tools for automatic differentiation. We also propose a multigrid technique to accelerate the convergence of gradient-based optimizers. We present applications to PDE-constrained optimization, optical flow, system identification, and data assimilation. We compare ODIL with the popular method of physics-informed neural networks and show that it outperforms it by several orders of magnitude in computational speed while having better accuracy and convergence rates. We evaluate ODIL on inverse problems involving linear and nonlinear PDEs including the Navier-Stokes equations for flow reconstruction problems. ODIL bridges numerical methods and machine learning and presents a powerful tool for solving challenging, inverse problems across scientific domains.

Symmetrized Two-Scale Finite Element Discretizations for Partial Differential Equations with Symmetric Solutions

Abstract In this paper, some symmetrized two-scale finite element methods are proposed for a class of partial differential equations with symmetric solutions. With these methods, the finite element approximation on a fine tensor-product grid is reduced to the finite element approximations on a much coarser grid and a univariant fine grid. It is shown by both theory and numerics including electronic structure calculations that the resulting approximations still maintain an asymptotically optimal accuracy. By symmetrized two-scale finite element methods, the computational cost can be reduced further by a factor of 𝑑 approximately compared with two-scale finite element methods when Ω = ( 0 , 1 ) d \Omega=(0,1)^{d} . Consequently, symmetrized two-scale finite element methods reduce computational cost significantly.

Reduced order model predictive control for parametrized parabolic partial differential equations

Model Predictive Control (MPC) is a well-established approach to solve infinite horizon optimal control problems. Since optimization over an infinite time horizon is generally infeasible, MPC determines a suboptimal feedback control by repeatedly solving finite time optimal control problems. Although MPC has been successfully used in many applications, applying MPC to large-scale systems – arising, e.g., through discretization of partial differential equations – requires the solution of high-dimensional optimization problems and thus poses immense computational effort. We consider systems governed by parametrized parabolic partial differential equations and employ the reduced basis method as a low-dimensional surrogate model for the finite time optimal control problem. The reduced order optimal control serves as feedback control for the original large-scale system. We analyze the proposed RB-MPC approach by first developing a posteriori error bounds for the errors in the optimal control and associated cost functional. These bounds can be evaluated efficiently in an offline-online computational procedure and allow us to guarantee asymptotic stability of the closed-loop system using the RB-MPC approach in several practical scenarios. We also propose an adaptive strategy to choose the prediction horizon of the finite time optimal control problem. Numerical results are presented to illustrate the theoretical properties of our approach.

Parsimonious physics-informed random projection neural networks for initial value problems of ODEs and index-1 DAEs.

We present a numerical method based on random projections with Gaussian kernels and physics-informed neural networks for the numerical solution of initial value problems (IVPs) of nonlinear stiff ordinary differential equations (ODEs) and index-1 differential algebraic equations (DAEs), which may also arise from spatial discretization of partial differential equations (PDEs). The internal weights are fixed to ones while the unknown weights between the hidden and output layer are computed with Newton's iterations using the Moore-Penrose pseudo-inverse for low to medium scale and sparse QR decomposition with L 2 regularization for medium- to large-scale systems. Building on previous works on random projections, we also prove its approximation accuracy. To deal with stiffness and sharp gradients, we propose an adaptive step-size scheme and address a continuation method for providing good initial guesses for Newton iterations. The "optimal" bounds of the uniform distribution from which the values of the shape parameters of the Gaussian kernels are sampled and the number of basis functions are "parsimoniously" chosen based on bias-variance trade-off decomposition. To assess the performance of the scheme in terms of both numerical approximation accuracy and computational cost, we used eight benchmark problems (three index-1 DAEs problems, and five stiff ODEs problems including the Hindmarsh-Rose neuronal model of chaotic dynamics and the Allen-Cahn phase-field PDE). The efficiency of the scheme was compared against two stiff ODEs/DAEs solvers, namely, ode15s and ode23t solvers of the MATLAB ODE suite as well as against deep learning as implemented in the DeepXDE library for scientific machine learning and physics-informed learning for the solution of the Lotka-Volterra ODEs included in the demos of the library. A software/toolbox in Matlab (that we call RanDiffNet) with demos is also provided.

Space and chaos‐expansion Galerkin proper orthogonal decomposition low‐order discretization of partial differential equations for uncertainty quantification

AbstractThe quantification of multivariate uncertainties in partial differential equations can easily exceed any computing capacity unless proper measures are taken to reduce the complexity of the model. In this work, we propose a multidimensional Galerkin proper orthogonal decomposition that optimally reduces each dimension of a tensorized product space. We provide the analytical framework and results that define and quantify the low‐dimensional approximation. We illustrate its application for uncertainty modeling with polynomial chaos expansions and show its efficiency in a numerical example.

Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a large-scale and ill-conditioned linear system must be solved, that, in case of Maxwell equations, is particularly challenging, because of the presence of both H(div) and H(curl) discretization spaces. We propose here a parallel preconditioner, that exploits the Schur complement block factorization of the linear system matrix and consists of a Jacobi preconditioner for the H(div) block and an auxiliary space preconditioner for the H(curl) block. Several parallel numerical tests have been performed to study the robustness of the solver with respect to mesh refinement, shape of polyhedral elements, time step size and the VEM stabilization parameter.

Isogeometric Analysis Approximation of Linear Elliptic Equations with L1 Data

Isogeometric Analysis (IgA) is a recent technique for the discretization of Partial Differential Equations (PDEs). The main feature of the method is the ability to maintain the same exact description of the computational domain geometry throughout the analysis process, including refinement. In the present paper, we consider, in dimension d >= 2 the Isogeometric Analysis approximation of second order elliptic equations in divergence form with right-hand side in L1 . We assume that the family of meshes is shape regular and satisfies the discrete maximum principle. When the right-hand side belongs to L1(\Omega), we prove that the unique solution of the discrete problem converges to the unique renormalized solution in W01,q(\Omega), 1 <= q < d/(d-1) . We also prove some error estimates and include numerical tests for data with low smoothness.

A mathematical model that integrates cardiac electrophysiology, mechanics, and fluid dynamics: Application to the human left heart.

We propose a mathematical and numerical model for the simulation of the heart function that couples cardiac electrophysiology, active and passive mechanics and hemodynamics, and includes reduced models for cardiac valves and the circulatory system. Our model accounts for the major feedback effects among the different processes that characterize the heart function, including electro-mechanical and mechano-electrical feedback as well as force-strain and force-velocity relationships. Moreover, it provides a three-dimensional representation of both the cardiac muscle and the hemodynamics, coupled in a fluid-structure interaction (FSI) model. By leveraging the multiphysics nature of the problem, we discretize it in time with a segregated electrophysiology-force generation-FSI approach, allowing for efficiency and flexibility in the numerical solution. We employ a monolithic approach for the numerical discretization of the FSI problem. We use finite elements for the spatial discretization of partial differential equations. We carry out a numerical simulation on a realistic human left heart model, obtaining results that are qualitatively and quantitatively in agreement with physiological ranges and medical images.

Differential Evolution Based Numerical Variable Speed Limit Control Method with a Non-Equilibrium Traffic Model

This paper introduces a numerical variable speed limit (VSL) control method on a motorway, modeled by the system of partial differential equations (PDEs) of a non- equilibrium continuum traffic model. The method consists of a macroscopic simulation (i.e., numerical solution of the system of PDEs of the continuum model), introduction of the solution-based cost function and numerical optimization with a differential evolution algorithm (DE). Due to the numerical solution scheme, the method enables application of a wide range of continuum traffic models without prior discretization of PDEs. In this way, the method overcomes the limitations of the basic continuum models and represents a step towards more accurate traffic modelling in control strategies. In this paper, we determine optimal variable speed limits with the DE algorithm on a motorway section modeled by the modified switching curve model, which is a non-equilibrium continuum model consistent with the three-phase traffic flow theory. The effectiveness of the determined variable speed limits is validated using microsimulations of the test section, which show promising reductions of queue lengths and number of stops.

Accelerating Algebraic Multigrid Methods via Artificial Neural Networks

We present a novel deep learning-based algorithm to accelerate—through the use of Artificial Neural Networks (ANNs)—the convergence of Algebraic Multigrid (AMG) methods for the iterative solution of the linear systems of equations stemming from finite element discretizations of Partial Differential Equations (PDE). We show that ANNs can be successfully used to predict the strong connection parameter that enters in the construction of the sequence of increasingly smaller matrix problems standing at the basis of the AMG algorithm, so as to maximize the corresponding convergence factor of the AMG scheme. To demonstrate the practical capabilities of the proposed algorithm, which we call AMG-ANN, we consider the iterative solution of the algebraic system of equations stemming from finite element discretizations of two-dimensional model problems. First, we consider an elliptic equation with a highly heterogeneous diffusion coefficient and then a stationary Stokes problem. We train (off-line) our ANN with a rich dataset and present an in-depth analysis of the effects of tuning the strong threshold parameter on the convergence factor of the resulting AMG iterative scheme.

Secant Acceleration of Sequential Residual Methods for Solving Large-Scale Nonlinear Systems of Equations

Sequential Residual Methods try to solve nonlinear systems of equations $F(x)=0$ by iteratively updating the current approximate solution along a residual-related direction. Therefore, memory requirements are minimal and, consequently, these methods are attractive for solving large-scale nonlinear systems. However, the convergence of these algorithms may be slow in critical cases; therefore, acceleration procedures are welcome. In this paper, we suggest to employ a variation of the Sequential Secant Method in order to accelerate Sequential Residual Methods. The performance of the resulting algorithm is illustrated by applying it to the solution of very large problems coming from the discretization of partial differential equations.

BAMPHI: Matrix-free and transpose-free action of linear combinations of [formula omitted]-functions from exponential integrators

The time integration of stiff systems of Ordinary Differential Equations (ODEs), usually arising from the spatial discretization of Partial Differential Equations (PDEs), constitutes a hot topic in numerical analysis. In particular, exponential-type integrators have attracted much attention for their capacity to effectively handle the stiffness, allowing integration with large time steps. The efficiency of exponential-type integrators strongly relies on the fast computation of the action of the exponential of matrices which often change only slightly during the integration. The authors exploited this characteristic of exponential integrators to develop a backward stable algorithm, bamphi, which is designed to reuse the information gathered through the exponential integration steps, reaching unmatched levels of speed and accuracy on a variety of numerical experiments.

A Bootstrap Multigrid Eigensolver

This paper introduces bootstrap multigrid methods for solving eigenvalue problems arising from the discretization of partial differential equations. Inspired by the full bootstrap algebraic multigrid (BAMG) setup algorithm that includes an AMG eigensolver, it is illustrated how the algorithm can be simplified for the case of a discretized partial differential equation (PDE), thereby developing a bootstrap geometric multigrid (BMG) approach. We illustrate numerically the efficacy of the BMG method for: (1) recovering eigenvalues having large multiplicity, (2) computing interior eigenvalues, and (3) approximating shifted indefinite eigenvalue problems. Numerical experiments are presented to illustrate the basic components and ideas behind the success of the overall bootstrap multigrid approach. For completeness, we present a simplified error analysis of a two-grid bootstrap algorithm for the Laplace-Beltrami eigenvalue problem.