In this paper, we employed linear algebra and functional analysis to determine necessary and sufficient conditions for oscillation-free and stable solutions to linear and nonlinear parabolic partial differential equations. We applied singular value decomposition and Fourier analysis to various finite difference schemes to extract patterns in the eigenfunctions (sampled by the eigenvectors) and the shape of their eigenspectrum. With these, we determined how the initial and boundary conditions affect the frequency and long term behavior of numerical oscillations, as well as the location of solution regions most sensitive to them.
We investigate the resolution of parabolic PDEs via Extreme Learning Machine (ELMs) Neural Networks, which have a single hidden layer and can be trained at a modest computational cost as compared with Deep Learning Neural Networks. Our approach addresses the time evolution by applying classical ODEs techniques and uses ELM-based collocation for solving the resulting stationary elliptic problems. In this framework, the $θ$-method and Backward Difference Formulae (BDF) techniques are investigated on some linear parabolic PDEs that are challeging problems for the stability and accuracy properties…
How to build an accurate reduced order model (ROM) for multidimensional time dependent partial differential equations (PDEs) is quite open. In this paper, we propose a new ROM for linear parabolic PDEs. We prove that our new method can be orders of magnitude faster than standard solvers, and is also much less memory intensive. Under some assumptions on the problem data, we prove that the convergence rates of the new method is the same with standard solvers. Numerical experiments are presented to confirm our theoretical result.
In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by Führer& Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven. In the present work, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated. The proof of the latter easily extends to a large class of least-squares formulations.
We study a general class of singular degenerate parabolic stochastic partial differential equations (SPDEs) which include, in particular, the stochastic porous medium equations and the stochastic fast diffusion equation. We propose a fully discrete numerical approximation of the considered SPDEs based on the very weak formulation. By exploiting the monotonicity properties of the proposed formulation we prove the convergence of the numerical approximation towards the unique solution. Furthermore, we construct an implementable finite element scheme for the spatial discretization of the very weak…
In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasilinear parabolic partial differential equations (PDEs), which are related to the FBSDEs by the Pardoux-Peng theory. The algorithms rely on a learning process by minimizing the pathwise difference between two discrete stochastic processes, defined by the time discretization of the FBSDEs and the DNN representation of the PDE solutions, respectively. The proposed algorithms are shown to generate DNN solutions for a…
This paper is concerned with the regularity of solutions to parabolic evolution equations. Special attention is paid to the smoothness in the specific anisotropic scale $\ B^{r\mathbf{a}}_{τ,τ}, \ \frac{1}τ=\frac{r}{d}+\frac{1}{p}\ $ of Besov spaces where $\mathbf{a}$ measures the anisotropy. The regularity in these spaces determines the approximation order that can be achieved by fully space-time adaptive approximation schemes. In particular, we show that for the heat equation our results significantly improve previous results by Aimar and Gomez [3].
This paper considers optimal control of fractional parabolic PDEs with both state and control constraints. The key challenge is how to handle the state constraints. Similarly, to the elliptic case, in this paper, we establish several new mathematical tools in the parabolic setting that are of wider interest. For example, existence of solution to the fractional parabolic equation with measure data on the right-hand-side. We employ the Moreau-Yosida regularization to handle the state constraints. We establish convergence, with rate, of the regularized optimal control problem to the original one.…
Exp-ParaDiag: Time-Parallel Exponential Integrators for Parabolic PDEs
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育与研究学院特里凡得琅数学学院）
In this paper, we develop an asymptotic expansion-regularization (AER) method for inverse source problems in two-dimensional nonlinear and nonstationary singularly perturbed partial differential equations (PDEs). The key idea of this approach is the use of the asymptotic-expansion theory, which allows us to determine the conditions for the existence and uniqueness of a solution to a given PDE with a sharp transition layer. As a by-product, we derive a simpler link equation between the source function and first-order asymptotic approximation of the measurable quantities, and based on that equat…
Explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit integrators applied to stiff diffusion problems. In this paper, we provide a fully discrete strong convergence analysis of a family of explicit stabilized methods coupled with finite element methods for a class of parabolic semilinear deterministic and stochastic partial differential equations. Numerical experiments including the semilinear stochastic heat equation with space-time white noise confirm the theoretical findings.
Convergence analysis of the Magnus-Rosenbrock type method for the finite element discretization of semilinear non-autonomous parabolic PDEs with nonsmooth initial data
Western Norway University of Applied Sciences（西挪威应用科学大学）；University of Cape Town（开普敦大学）；Technische Universität Chemnitz（开姆尼茨工业大学）
We propose an approximation scheme for a class of semilinear parabolic equations that are convex and coercive in their gradients. Such equations arise often in pricing and portfolio management in incomplete markets and, more broadly, are directly connected to the representation of solutions to backward stochastic differential equations. The proposed scheme is based on splitting the equation in two parts, the first corresponding to a linear parabolic equation and the second to a Hamilton-Jacobi equation. The solutions of these two equations are approximated using, respectively, the Feynman-Kac …
The goal of this work is to develop deep learning numerical methods for solving option XVA pricing problems given by non-linear PDE models. A novel strategy for the treatment of the boundary conditions is proposed, which allows to get rid of the heuristic choice of the weights for the different addends that appear in the loss function related to the training process. It is based on defining the losses associated to the boundaries by means of the PDEs that arise from substituting the related conditions into the model equation itself. Further, automatic differentiation is employed to obtain accu…
For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains $D$, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete $hp$-time-stepping approach. We combine this se…
We investigate three directions to further improve the highly efficient Space-Time Multigrid algorithm with block-Jacobi smoother introduced in [GanNeu16]. First, we derive an analytical expression for the optimal smoothing parameter in the case of a full space-time coarsening strategy; second, we propose a new and efficient direct coarsening strategy which simplifies the code by preventing changes of coarsening regimes; and third, we also optimize the entire two cycle to investigate if further efficiency gains are possible. Especially, we show that our new coarsening strategy leads to a signi…
We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general structure of infinite-dimensional port-Hamiltonian systems (pHs) for which the Partitioned Finite Element Method (PFEM) straightforwardly applies. The proposed strategy is applied to abstract multidimensional linear hyperbolic and parabolic systems of PDEs. Then we show that instructional model problems based on the wave equation, Mindlin equation and heat equatio…
ETH Zurich（苏黎世联邦理工学院）；University of Münster（明斯特大学）；Nanyang Technological University（南洋理工大学）
In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high-dimensional PDEs. We test the method on different examples from physics, stochastic control and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.
The tangential condition was introduced in [Hanke et al., 95] as a sufficient condition for convergence of the Landweber iteration for solving ill-posed problems. In this paper we present a series of time dependent benchmark inverse problems for which we can verify this condition.
Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encounted in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs. As the number of data grows, PINNs generate a se…
We show how a posteriori goal oriented error estimation can be used to efficiently solve the subproblems occurring in a Model Predictive Control (MPC) algorithm. In MPC, only an initial part of a computed solution is implemented as a feedback, which motivates grid refinement particularly tailored to this context. To this end, we present a truncated cost functional as objective for goal oriented adaptivity and prove under stabilizability assumptions that error indicators decay exponentially outside the support of this quantity. This leads to very efficient time and space discretizations for MPC…
Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability an…
To solve linear PDEs on metric graphs with standard coupling conditions (continuity and Kirchhoff's law), we develop and compare a spectral, a second-order finite difference, and a discontinuous Galerkin method. The spectral method yields eigenvalues and eigenvectors of arbitary order with machine precision and converges exponentially. These eigenvectors provide a Fourier-like basis on which to expand the solution; however, more complex coupling conditions require additional research. The discontinuous Galerkin method provides approximations of arbitrary polynomial order; however computing hig…
The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations
College of Math and Statistics, Nanjing University of Information Science and Technology（南京信息工程大学数学与统计学院）；Department of Mathematics, the University of Kansas（堪萨斯大学数学系）
We generalize previous work by Mardal, Nilssen, and Staff (2007, SIAM J. Sci. Comp. v. 29, pp. 361-375) and Rana, Howle, Long, Meek, and Milestone (2021, SIAM J. Sci. Comp. v. 43, p. 475-495) on order-optimal preconditioners for parabolic PDEs to a larger class of differential equations and methods. The problems considered are those of the forms $u_{t}=-\mathcal{K}u+g$ and $u_{tt}=-\mathcal{K}u+g$, where the operator $\mathcal{K}$ is defined by $\mathcal{K}u:=-\nabla\cdot\left(α\nabla u\right)+βu$ and the functions $α$ and $β$ are restricted so that $α>0$, and $β\ge0$. The methods considered a…
We show that the Strang splitting method applied to a diffusion-reaction equation with inhomogeneous general oblique boundary conditions is of order two when the diffusion equation is solved with the Crank-Nicolson method, while order reduction occurs in general if using other Runge-Kutta schemes or even the exact flow itself for the diffusion part. We prove these results when the source term only depends on the space variable, an assumption which makes the splitting scheme equivalent to the Crank-Nicolson method itself applied to the whole problem. Numerical experiments suggest that the secon…
We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most $r-1$ in $t$, for $r\ge1$ and with maximum step size~$k$. It is well known that the spatial $L_2$-norm of the DG error is of optimal order $k^r$ globally in time, and is, for $r\ge2$, superconvergent of order $k^{2r-1}$ at the nodes. We show that on the $n$th subinterval $(t_{n-1},t_n)$, the dominant term in the DG error is proportional to the local right Radau polynomial of degree $r$. This error profile implies that the DG error is of order…
HEATNETs: Explainable Random Feature Neural Networks for High-Dimensional Parabolic PDEs
Modelling and Engineering Risk and Complexity, Scuola Superiore Meridionale（南方高等学院建模与工程风险与复杂性研究中心）；Hopkins Extreme Materials Institute and Dept. of Chemical and Biomolecular Engineering, Johns Hopkins University（约翰斯·霍普金斯大学霍普金斯极端材料研究所和化学与生物分子工程系）；Department of Statistics and Stochastic Modelling and Applications Laboratory, Athens University of Economics and Business（雅典经济与商业大学统计学与随机建模与应用实验室）
Freie Universität Berlin（柏林自由大学）；BTU Cottbus-Senftenberg（科特布斯-森夫滕贝格勃兰登堡工业大学）；dida Datenschmiede GmbH（dida数据锻造有限公司）；Technische Universität Berlin（柏林工业大学）；Universität Potsdam（波茨坦大学）
High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the curse of dimensionality. In this paper, we argue that tensor trains provide an appealing approximation framework for parabolic PDEs: the combination of reformulations in terms of backward stochastic differential equations and regression-type methods in the tensor format holds the promise of leveraging latent low-rank structures enabling both compression and ef…
