Advances in Computational Mathematics最新文献

We study an iterative nonlinear solver for the Oldroyd-B system describing incompressible viscoelastic fluid flow. We establish a range of attributes of the fixed-point-based solver, together with the conditions under which it becomes contractive, and examine the smoothness properties of its corresponding fixed-point function. Under these properties, we demonstrate that the solver meets the necessary conditions for the recent Anderson acceleration (AA) framework, thereby showing that AA enhances the solver’s linear convergence rate. Results from three benchmark tests illustrate how AA improves the solver’s ability to converge as the Weissenberg number is increased.

This article builds on the recently proposed RB-ML-ROM approach for parameterized parabolic PDEs and proposes a novel hierarchical trust region algorithm for solving parabolic PDE constrained optimization problems. Instead of using a traditional offline/online splitting approach for model order reduction, we adopt an active learning or enrichment strategy to construct a multi-fidelity hierarchy of reduced order models on-the-fly during the outer optimization loop. The multi-fidelity surrogate model consists of a full order model, a reduced order model and a machine learning model. The proposed hierarchical framework adaptively updates its hierarchy when querying parameters, utilizing a rigorous a posteriori error estimator in an error-aware trust region framework. Numerical experiments are given to demonstrate the efficiency of the proposed approach.

This work presents structure-preserving Lift & Learn, a scientific machine learning method that employs lifting variable transformations to learn structure-preserving reduced-order models for nonlinear partial differential equations (PDEs) with conservation laws. We propose a hybrid learning approach based on a recently developed energy-quadratization strategy that uses knowledge of the nonlinearity at the PDE level to derive an equivalent quadratic lifted system with quadratic system energy. The lifted dynamics obtained via energy quadratization are linear in the old variables, making model learning very effective in the lifted setting. Based on the lifted quadratic PDE model form, the proposed method derives quadratic reduced terms analytically and then uses those derived terms to formulate a constrained optimization problem to learn the remaining linear reduced operators in a structure-preserving way. The proposed hybrid learning approach yields computationally efficient quadratic reduced-order models that respect the underlying physics of the high-dimensional problem. We demonstrate the generalizability of quadratic models learned via the proposed structure-preserving Lift & Learn method through three numerical examples: the one-dimensional wave equation with exponential nonlinearity, the two-dimensional sine-Gordon equation, and the two-dimensional Klein-Gordon-Zakharov equations. The numerical results show that the proposed learning approach is competitive with the state-of-the-art structure-preserving data-driven model reduction method in terms of both accuracy and computational efficiency.

This paper constructs unique compactly supported functions in Sobolev spaces that have minimal norm, maximal support, and maximal central value, under certain renormalizations. They may serve as optimized basis functions in interpolation or approximation, or as shape functions in meshless methods for PDE solving. Their norm is useful for proving upper bounds for convergence rates of interpolation in Sobolev spaces (H_2^m(mathbb {R}^d)), and this paper gives the correct rate (m-d/2) that arises as convergence like (h^{m-d/2}) for interpolation at meshwidth (hrightarrow 0) or a blow-up like (r^{-(m-d/2)}) for norms of compactly supported functions with support radius (rrightarrow 0). In Hilbert spaces with infinitely smooth reproducing kernels, like Gaussians or inverse multiquadrics, there are no compactly supported functions at all, but in spaces with limited smoothness, compactly supported functions exist and can be optimized in the above way. The construction is described in Hilbert space via projections, and analytically via trace operators. Numerical examples are provided.

We explore the applicability of quantum annealing to the approximation task of curve fitting. To this end, we consider a function that shall approximate a given set of data points and is written as a finite linear combination of standardized functions, e.g., orthogonal polynomials. Consequently, the decision variables subject to optimization are the coefficients of that expansion. Although this task can be accomplished classically, it can also be formulated as a quadratic unconstrained binary optimization problem, which is suited to be solved with quantum annealing. Given that the size of the problem stays below a certain threshold, we find that quantum annealing yields comparable results to the classical solution. Regarding a real-world use case, we discuss the problem to find an optimized speed profile for a vessel using the framework of dynamic programming and outline how the aforementioned approximation task can be put into play. Similar to the curve fitting task, our findings indicate that quantum annealing is currently only feasible if the routing problem is modeled sufficiently small and sparse.

We propose and analyse residual-based a posteriori error estimates for the virtual element discretisation applied to the thin plate vibration problem in both two and three dimensions. Our approach involves a conforming (C^1) discrete formulation suitable for meshes composed of polygons and polyhedra. The reliability and efficiency of the error estimator are established through a dimension-independent proof. Finally, several numerical experiments are reported to demonstrate the optimal performance of the method in 2D and 3D.

A method for numerical approximation of a class of fractional parabolic stochastic evolution equations is introduced and analysed. This class of equations has recently been proposed as a space-time extension of the SPDE-method in spatial statistics. A truncation of the spectral basis function expansion is used to discretise in space, and then a quadrature is used to approximate the temporal evolution of each basis coefficient. Strong error bounds are proved both for the spectral and temporal approximations. The method is tested, and the results are verified by several numerical experiments.

In this paper, we explore a direct parallel-in-time method based on the diagonalization of the time-stepping matrix to solve the Kawarada equation, which arises from the quenching combustion process. Unlike the traditional time-marching finite difference method, where a nonlinear system may need to be solved at each time step once implicit schemes are employed, here, we form a large sparse linear system involving all unknown variables on the time-space domain, so that numerical solution can be handled all at once. Thanks to the non-uniform step sizes, the diagonalization technique can be introduced into a parallel-in-time process. Therefore, the dynamics near quenching time and quenching point can be described more accurately. We use a time window technique to release the deficiency of diagonalization, leading the accuracy and efficiency of the algorithm to be well-balanced. In numerical analysis, we propose a practical PAMS-framework to verify the effectiveness of the aforementioned method from four aspects including Positivity, Asymptoticity, Monotonicity, and Stability. Finally, several numerical experiments are conducted on 1D and 2D Kawarada equations, which demonstrate that the studied parallel-in-time method is reliable and highly efficient for quenching-type reaction diffusion equations. Meanwhile, computational time is reduced satisfactorily compared to the time-marching method.

Surface reconstruction is a challenging problem when no constraint is imposed on data locations. The problem is ill-posed, and most computational algorithms become overly expensive as the number of sample points increases. This article presents a generalization of a popular integral method called minimum curvature (MC) method, which is based on the numerical solution of a modified biharmonic partial differential equation (PDE). Surface reconstruction through the PDE solution for scattered data can be considered as an interior value problem. A new model is suggested to construct an image surface that satisfies data constraints accurately and conveniently. In order to improve the efficiency of the MC method, an effective initialization scheme is suggested. The resulting algorithm is applied for image zooming, synthetic scattered data, and agricultural data acquired by light detection and ranging (LiDAR) technology.

Given the efficient application of Chebyshev polynomial acceleration techniques in standard symmetric and non-symmetric eigenvalue problems as well as generalized symmetric eigenvalue problems, we extend this technique to generalized non-symmetric eigenvalue problems and propose the Chebyshev-Davidson method. By partitioning the spectrum of the corresponding shifted matrix, we construct four Chebyshev polynomial filters at each iteration to accelerate the convergence of desired eigenvectors while suppressing the convergence of undesired eigenvectors. The introduction of multiple Chebyshev polynomial filters does not significantly increase the computational cost. Furthermore, to compute several eigenvalues and corresponding eigenvectors of generalized non-symmetric eigenvalue problems, we propose the block Chebyshev-Davidson method. Numerical experiments are carried out to demonstrate its superior performance and robustness compared to some state-of-the-art iterative methods.