Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExistence of dynamical low-rank approximations to parabolic problems
125
0
0.0
(
0
)
Added by
Markus Bachmayr
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The existence and uniqueness of weak solutions to dynamical low-rank evolution problems for parabolic partial differential equations in two spatial dimensions is shown, covering also non-diagonal diffusion in the elliptic part. The proof is based on a variational time-stepping scheme on the low-rank manifold. Moreover, this scheme is shown to be closely related to practical methods for computing such low-rank evolutions.
rate research
Read More
Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the infinite Newton-type fixed point equation $w = - {mathcal L}^{-1} {mathcal F}(hat{u}) + {mathcal L}^{-1} {mathcal G}(w)$, where ${mathcal L}$ is a linearized operator, ${mathcal F}(hat{u})$ is a residual, and ${mathcal G}(w)$ is a local Lipschitz term. Therefore, the estimations of $| {mathcal L}^{-1} {mathcal F}(hat{u}) |$ and $| {mathcal L}^{-1}{mathcal G}(w) |$ play major roles in the verification procedures. In this paper, using a similar concept as the `Schur complement for matrix problems, we represent the inverse operator ${mathcal L}^{-1}$ as an infinite-dimensional operator matrix that can be decomposed into two parts, one finite dimensional and one infinite dimensional. This operator matrix yields a new effective realization of the infinite-dimensional Newton method, enabling a more efficient verification procedure compared with existing methods for the solution of elliptic PDEs. We present some numerical examples that confirm the usefulness of the proposed method. Related results obtained from the representation of the operator matrix as ${mathcal L}^{-1}$ are presented in the appendix.
The challenge of mastering computational tasks of enormous size tends to frequently override questioning the quality of the numerical outcome in terms of accuracy. By this we do not mean the accuracy within the discrete setting, which itself may also be far from evident for ill-conditioned problems or when iterative solvers are involved. By accuracy-controlled computation we mean the deviation of the numerical approximation from the exact solution of an underlying continuous problem in a relevant metric, which has been the initiating interest in the first place. Can the accuracy of a numerical result be rigorously certified - a question that is particularly important in the context of uncertainty quantification, when many possible sources of uncertainties interact. This is the guiding question throughout this article, which reviews recent developments of low-rank approximation methods for problems in high spatial dimensions. In particular, we highlight the role of adaptivity when dealing with such strongly nonlinear methods that integrate in a natural way issues of discrete and continuous accuracy.
In this paper, we present a numerical verification method of solutions for nonlinear parabolic initial boundary value problems. Decomposing the problem into a nonlinear part and an initial value part, we apply Nakaos projection method, which is based on the full-discrete finite element method with constructive error estimates, to the nonlinear part and use the theoretical analysis for the heat equation to the initial value part, respectively. We show some verified examples for solutions of nonlinear problems from initial value to the neighborhood of the stationary solutions, which confirm us the actual effectiveness of our method.
This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a regularized least squares formulation in a continuous infinite-dimensional setting that is based on stable variational time-space formulations of the parabolic PDE. This allows us to derive a priori as well as a posteriori error bounds for the recovered states with respect to a certain reference solution. In these bounds the regularization parameter is disentangled from the underlying discretization. An important ingredient for the derivation of a posteriori bounds is the construction of suitable Fortin operators which allow us to control oscillation errors stemming from the discretization of dual norms. Moreover, the variational framework allows us to contrive preconditioners for the discrete problems whose application can be performed in linear time, and for which the condition numbers of the preconditioned systems are uniformly proportional to that of the regularized continuous problem. In particular, we provide suitable stopping criteria for the iterative solvers based on the a posteriori error bounds. The presented numerical experiments quantify the theoretical findings and demonstrate the performance of the numerical scheme in relation with the underlying discretization and regularization.
In this paper, we develop a computational multiscale to solve the parabolic wave approximation with heterogeneous and variable media. Parabolic wave approximation is a technique to approximate the full wave equation. One benefit of the method is that: one wave propagation direction can be taken as an evolution direction, and we then can discretize it using a classical scheme like Backward Euler. Consequently, we obtain a set of quasi-gas-dynamic (QGD) models with different heterogeneous permeability fields. Then, we employ constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) to perform spatial discretization for the problem. The resulting system can be solved by combining the central difference in time evolution. Due to the variable media, we apply the technique of proper orthogonal decomposition (POD) to further the dimension of the problem and solve the corresponding model problem in the POD space instead of in the whole multiscale space spanned by all possible multiscale basis functions. We prove the stability of the full discretization scheme and give the convergence analysis of the proposed approximation scheme. Numerical results verify the effectiveness of the proposed method.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
