Advances in Computational Mathematics最新文献

Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising, e.g., as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for deep neural operator and generalized polynomial chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable Hilbert spaces. Operator in- and outputs from function spaces are assumed to be parametrized by stable, affine representation systems. Admissible representation systems comprise orthonormal bases, Riesz bases, or suitable tight frames of the spaces under consideration. Algebraic expression rate bounds are established for both, deep neural and spectral operator surrogates acting in scales of separable Hilbert spaces containing domain and range of the map to be expressed, with finite Sobolev or Besov regularity. We illustrate the abstract concepts by expression rate bounds for the coefficient-to-solution map for a linear elliptic PDE on the torus.

Regularized pairwise ranking with Gaussian kernels is one of the cutting-edge learning algorithms. Despite a wide range of applications, a rigorous theoretical demonstration still lacks to support the performance of such ranking estimators. This work aims to fill this gap by developing novel oracle inequalities for regularized pairwise ranking. With the help of these oracle inequalities, we derive fast learning rates of Gaussian ranking estimators under a general box-counting dimension assumption on the input domain combined with the noise conditions or the standard smoothness condition. Our theoretical analysis improves the existing estimates and shows that a low intrinsic dimension of input space can help the rates circumvent the curse of dimensionality.

We develop a sparse spectral method for a class of fractional differential equations, posed on (mathbb {R}), in one dimension. These equations may include sqrt-Laplacian, Hilbert, derivative, and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the second kind in conjunction with their Hilbert transforms. The former functions are supported on ([-1,1]) whereas the latter have global support. The global approximation space may contain different affine transformations of the basis, mapping ([-1,1]) to other intervals. Remarkably, not only are the induced linear systems sparse, but the operator decouples across the different affine transformations. Hence, the solve reduces to solving K independent sparse linear systems of size (mathcal {O}(n)times mathcal {O}(n)), with (mathcal {O}(n)) nonzero entries, where K is the number of different intervals and n is the highest polynomial degree contained in the sum space. This results in an (mathcal {O}(n)) complexity solve. Applications to fractional heat and wave equations are considered.

We consider a signal composed of several periods of a periodic function, of which we observe a noisy reparametrization. The phase estimation problem consists of finding that reparametrization and, in particular, the number of observed periods. Existing methods are well suited to the setting where the periodic function is known or, at least, simple. We consider the case when it is unknown, and we propose an estimation method based on the shape of the signal. We use the persistent homology of sublevel sets of the signal to capture the temporal structure of its local extrema. We infer the number of periods in the signal by counting points in the persistence diagram and their multiplicities. Using the estimated number of periods, we construct an estimator of the reparametrization. It is based on counting the number of sufficiently prominent local minima in the signal. This work is motivated by a vehicle positioning problem, on which we evaluated the proposed method.

Consider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable and isotropic elastic solid, which is immersed in a homogeneous compressible air/fluid. By the Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the model is formulated as a boundary value problem of acoustic-elastic interaction. Based on a duality argument technique, an a posteriori error estimate is derived for the finite element method with the truncated DtN boundary operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN boundary operator, where the latter decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed for solving the acoustic-elastic interaction problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

In this paper, optimal estimates on the decaying rates of Jacobi expansion coefficients are obtained by fractional calculus for functions with algebraic and logarithmic singularities. This is inspired by the fact that integer-order derivatives fail to deal with singularity of fractional-type, while fractional calculus can. To this end, we first introduce new fractional Sobolev spaces defined as the range of the (L^p)-space under the Riemann-Liouville fractional integral. The connection between these new spaces and classical fractional-order Sobolev spaces is then elucidated. Under this framework, the optimal decaying rate of Jacobi expansion coefficients is obtained, based on which the projection errors under different norms are given. This work is expected to introduce fractional calculus into traditional fields in approximation theory and to explore the possibility in solving classical problems by this ‘new’ tool.

An adaptive time-stepping scheme is developed for the Zakharov-Rubenchik system to resolve the multiple time scales accurately and to improve the computational efficiency during long-time simulations. The Crank-Nicolson formula and the Fourier pseudo-spectral method are respectively utilized for the temporal and spatial approximations. The proposed numerical method is proved to preserve the mass and energy conservative laws in the discrete levels exactly so that the magnetic field, the density of mass, and the fluid speed are stable on a general class of nonuniform time meshes. With the aid of the priori estimates derived from the discrete invariance and the newly proved discrete Gronwall inequality on variable time grids, sharp convergence analysis of the fully discrete scheme is established rigorously. Error estimate shows that the suggested adaptive time-stepping method can attain the second-order accuracy in time and the spectral accuracy in space. Extensive numerical experiments coupled with an adaptive time-stepping algorithm are presented to show the effectiveness of our numerical method in capturing the multiple time scale evolution for various velocity cases during the interactions of solitons.

Multilevel Stein variational gradient descent is a method for particle-based variational inference that leverages hierarchies of surrogate target distributions with varying costs and fidelity to computationally speed up inference. The contribution of this work is twofold. First, an extension of a previous cost complexity analysis is presented that applies even when the exponential convergence rate of single-level Stein variational gradient descent depends on iteration-varying parameters. Second, multilevel Stein variational gradient descent is applied to a large-scale Bayesian inverse problem of inferring discretized basal sliding coefficient fields of the Arolla glacier ice. The numerical experiments demonstrate that the multilevel version achieves orders of magnitude speedups compared to its single-level version.

In the triangular/tetrahedral spectral finite elements, we apply a bilinear/trilinear transformation to map a reference square/cube to a triangle/tetrahedron, which consequently maps the (varvec{Q_k}) polynomial space on the reference element to a finite element space of rational/algebraic functions on the triangle/tetrahedron. We prove that the resulting finite element space, even under this singular referencing mapping, can retain the property of optimal-order approximation. In addition, we prove that the standard Gauss-Legendre numerical integration would provide sufficient accuracy so that the finite element solutions converge at the optimal order. In particular, the finite element method, with singular mappings and numerical integration, preserves (varvec{P_k}) polynomials. That is, the (varvec{Q_k}) finite element solution is exact if the true solution is a (varvec{P_k}) polynomial. Numerical tests are provided, verifying all theoretic findings.

Many problems in data science can be treated as recovering structural signals from a set of linear measurements, sometimes perturbed by dense noise or sparse corruptions. In this paper, we develop a unified framework of considering a nonsmooth formulation with sparse or low-rank constraint for meeting the challenges of mixed noises—bounded noise and sparse noise. We show that the nonsmooth formulations of the problems can be well solved by the projected subgradient methods at a rapid rate when initialized at any points. Consequently, nonsmooth loss functions ((ell _1)-minimization programs) are naturally robust against sparse noise. Our framework simplifies and generalizes the existing analyses including compressed sensing, matrix sensing, quadratic sensing, and bilinear sensing. Motivated by recent work on the stochastic gradient method, we also give some experimentally and theoretically preliminary results about the projected stochastic subgradient method.