Advances in Computational Mathematics最新文献

In this article, we propose an efficient hybrid method to calculate the highly oscillatory Bessel integral (int _{0}^{1} frac{f(x)}{x-tau } J_{m} (omega x^{gamma } )textrm{d}x) with the Cauchy type singular point, where ( 0< tau < 1, m ge 0, 2gamma in N^{+}. ) The hybrid method is established by combining the complex integration method with the Clenshaw– Curtis– Filon– type method. Based on the special transformation of the integrand and the additivity of the integration interval, we convert the integral into three integrals. The explicit formula of the first one is expressed in terms of the Meijer G function. The second is computed by using the complex integration method and the Gauss– Laguerre quadrature rule. For the third, we adopt the Clenshaw– Curtis– Filon– type method to obtain the quadrature formula. In particular, the important recursive relationship of the required modified moments is derived by utilizing the Bessel equation and the properties of Chebyshev polynomials. Importantly, the strict error analysis is performed by a large amount of theoretical analysis. Our proposed methods only require a few nodes and interpolation multiplicities to achieve very high accuracy. Finally, numerical examples are provided to verify the validity of our theoretical analysis and the accuracy of the proposed methods.

Two-dimensional transport codes for the simulation of tokamak plasma are reduced version of full 3D fluid models where plasma turbulence has been smoothed out by averaging. One of the main issues nowadays in such reduced models is the accurate modelling of transverse transport fluxes resulting from the averaging of stresses due to fluctuations. Transverse fluxes are assumed driven by local gradients, and characterised by ad hoc diffusion coefficients (turbulent eddy viscosity), adjusted by hand in order to match numerical solutions with experimental measurements. However, these coefficients vary substantially depending on the machine used, type of experiment and even the location inside the device, reducing drastically the predictive capabilities of these codes for a new configuration. To mitigate this issue, we recently proposed an innovative path for fusion plasma simulations by adding two supplementary transport equations to the mean-flow system for turbulence characteristic variables (here the turbulent kinetic energy k and its dissipation rate (epsilon )) to estimate the turbulent eddy viscosity. The remaining free parameters are more driven by the underlying transport physics and hence vary much less between machines and between locations in the plasma. In this paper, as a proof of concept, we explore, on the basis of digital twin experiments, the efficiency of the assimilation of data to fix these free parameters involved in the transverse turbulent transport models in the set of averaged equations in 2D.

We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues (lambda (n)) of shapes with n edges that are of the form (lambda (n) sim xsum _{k=0}^{infty } frac{C_k(x)}{n^k}) where x is the limiting eigenvalue for (nrightarrow infty ). Expansions of this form have previously only been known for regular polygons with Dirichlet boundary conditions and (quite surprisingly) involve Riemann zeta values and single-valued multiple zeta values, which makes them interesting to study. We provide numerical evidence for closed-form expressions of higher order (C_k(x)) and give more examples of shapes for which such expansions are possible (including regular polygons with Neumann boundary condition, regular star polygons, and star shapes with sinusoidal boundary).

With the advent of massive data sets, much of the computational science and engineering community has moved toward data-intensive approaches in regression and classification. However, these present significant challenges due to increasing size, complexity, and dimensionality of the problems. In particular, covariance matrices in many cases are numerically unstable, and linear algebra shows that often such matrices cannot be inverted accurately on a finite precision computer. A common ad hoc approach to stabilizing a matrix is application of a so-called nugget. However, this can change the model and introduce error to the original solution. It is well known from numerical analysis that ill-conditioned matrices cannot be accurately inverted. In this paper, we develop a multilevel computational method that scales well with the number of observations and dimensions. A multilevel basis is constructed adapted to a kd-tree partitioning of the observations. Numerically unstable covariance matrices with large condition numbers can be transformed into well-conditioned multilevel ones without compromising accuracy. Moreover, it is shown that the multilevel prediction exactly solves the best linear unbiased predictor (BLUP) and generalized least squares (GLS) model, but is numerically stable. The multilevel method is tested on numerically unstable problems of up to 25 dimensions. Numerical results show speedups of up to 42,050 times for solving the BLUP problem, but with the same accuracy as the traditional iterative approach. For very ill-conditioned cases, the speedup is infinite. In addition, decay estimates of the multilevel covariance matrices are derived based on high dimensional interpolation techniques from the field of numerical analysis. This work lies at the intersection of statistics, uncertainty quantification, high performance computing, and computational applied mathematics.

Numerical methods for the optimal feedback control of high-dimensional dynamical systems typically suffer from the curse of dimensionality. In the current presentation, we devise a mesh-free data-based approximation method for the value function of optimal control problems, which partially mitigates the dimensionality problem. The method is based on a greedy Hermite kernel interpolation scheme and incorporates context knowledge by its structure. Especially, the value function surrogate is elegantly enforced to be 0 in the target state, non-negative and constructed as a correction of a linearized model. The algorithm allows formulation in a matrix-free way which ensures efficient offline and online evaluation of the surrogate, circumventing the large-matrix problem for multivariate Hermite interpolation. Additionally, an incremental Cholesky factorization is utilized in the offline generation of the surrogate. For finite time horizons, both convergence of the surrogate to the value function and for the surrogate vs. the optimal controlled dynamical system are proven. Experiments support the effectiveness of the scheme, using among others a new academic model with an explicitly given value function. It may also be useful for the community to validate other optimal control approaches.

The energy dissipation law and maximum bound principle are significant characteristics of the Allen-Chan equation. To preserve discrete counterpart of these properties, the linear part of the target system is usually discretized implicitly, resulting in a large linear or nonlinear system of equations. The fast Fourier transform is commonly used to solve the resulting linear or nonlinear systems with computational costs of (varvec{mathcal {O}(M^d text {log} M)}) at each time step, where (varvec{M}) is the number of spatial grid points in each direction, and (varvec{d}) is the dimension of the problem. Combining the Saul’yev methods and the stabilization techniques, we propose and analyze novel first- and second-order numerical schemes for the Allen-Cahn equation in this paper. In contrast to the traditional methods, the proposed methods can be solved by components, requiring only (varvec{mathcal {O}(M^d)}) computational costs per time step. Additionally, they preserve the maximum bound principle and original energy dissipation law at the discrete level. We also propose rigorous analysis of their consistency and convergence. Numerical experiments are conducted to confirm the theoretical analysis and demonstrate the efficiency of the proposed methods.

This paper is a continuation of Melenk et al., “Stability analysis for electromagnetic waveguides. Part 1: acoustic and homogeneous electromagnetic waveguides” (2023) [5], extending the stability results for homogeneous electromagnetic (EM) waveguides to the non-homogeneous case. The analysis is done using perturbation techniques for self-adjoint operators eigenproblems. We show that the non-homogeneous EM waveguide problem is well-posed with the stability constant scaling linearly with waveguide length L. The results provide a basis for proving convergence of a Discontinuous Petrov-Galerkin (DPG) discretization based on a full envelope ansatz, and the ultraweak variational formulation for the resulting modified system of Maxwell equations, see Part 1.

We consider the problem of state estimation from a few linear measurements, where the state to recover is an element of the manifold (mathcal {M}) of solutions of a parameter-dependent equation. The state is estimated using prior knowledge on (mathcal {M}) coming from model order reduction. Variational approaches based on linear approximation of (mathcal {M}), such as PBDW, yield a recovery error limited by the Kolmogorov width of (mathcal {M}). To overcome this issue, piecewise-affine approximations of (mathcal {M}) have also been considered, that consist in using a library of linear spaces among which one is selected by minimizing some distance to (mathcal {M}). In this paper, we propose a state estimation method relying on dictionary-based model reduction, where space is selected from a library generated by a dictionary of snapshots, using a distance to the manifold. The selection is performed among a set of candidate spaces obtained from a set of (ell _1)-regularized least-squares problems. Then, in the framework of parameter-dependent operator equations (or PDEs) with affine parametrizations, we provide an efficient offline-online decomposition based on randomized linear algebra, that ensures efficient and stable computations while preserving theoretical guarantees.

POD-DL-ROMs have been recently proposed as an extremely versatile strategy to build accurate and reliable reduced order models (ROMs) for nonlinear parametrized partial differential equations, combining (i) a preliminary dimensionality reduction obtained through proper orthogonal decomposition (POD) for the sake of efficiency, (ii) an autoencoder architecture that further reduces the dimensionality of the POD space to a handful of latent coordinates, and (iii) a dense neural network to learn the map that describes the dynamics of the latent coordinates as a function of the input parameters and the time variable. Within this work, we aim at justifying the outstanding approximation capabilities of POD-DL-ROMs by means of a thorough error analysis, showing how the sampling required to generate training data, the dimension of the POD space, and the complexity of the underlying neural networks, impact on the solutions us to formulate practical criteria to control the relative error in the approximation of the solution field of interest, and derive general error estimates. Furthermore, we show that, from a theoretical point of view, POD-DL-ROMs outperform several deep learning-based techniques in terms of model complexity. Finally, we validate our findings by means of suitable numerical experiments, ranging from parameter-dependent operators analytically defined to several parametrized PDEs.

We explore the potential for using a nonsmooth loss function based on the max-norm in the training of an artificial neural network without hidden layers. We hypothesise that this may lead to superior classification results in some special cases where the training data are either very small or the class size is disproportional. Our numerical experiments performed on a simple artificial neural network with no hidden layer appear to confirm our hypothesis.