数值分析
We prove a saturation theorem for linearized shallow ReLU$^k$ neural networks on the unit sphere $\mathbb S^d$. For any antipodally quasi-uniform set of centers, if the target function has smoothness $r>\tfrac{d+2k+1}{2}$, then the best…
We study the problem of multivariate $L_2$-approximation of functions in a weighted Korobov space using a median lattice-based algorithm recently proposed by the authors. In the original work, the algorithm requires knowledge of the…
We calculate the exact value and find the polynomial of the best weighted polynomial approximation of kernels of the form $\frac {A+Bt}{(t^2+\lambda^2)^{s+1}}$, where $A$ and $B$ are fixed complex numbers, $\lambda>0$, $s\in {\mathbb N}$,…
A surrogate-based topology optimisation algorithm for linear elastic structures under parametric loads and boundary conditions is proposed. Instead of learning the parametric solution of the state (and adjoint) problems or the optimisation…
We present an $\textit{a priori}$ error analysis for the kinematic pressure in a fully-discrete finite-differences/-elements discretization of the unsteady $p$-Stokes equations, modelling non-Newtonian fluids. This system is subject to both…
For gradient flows, the existing structure-preserving schemes are difficult to achieve arbitrary high-order accuracy in time while preserving maximum-principle (MBP) and energy dissipating simultaneously. In this paper, we develop a new…
The active element pattern method is widely employed in beam pattern synthesis of array antenna to account for mutual coupling between antenna elements. Calculating the active element patterns for large number of array requires full-wave…
We analyse and implement a quasi-Monte Carlo (QMC) finite element method (FEM) for the forward problem of uncertainty quantification (UQ) for the Helmholtz equation with random coefficients, both in the second-order and zero-order terms of…
In this paper, we study the problem of multivariate $L_2$-approximation of functions belonging to a weighted Korobov space. We propose and analyze a median lattice-based algorithm, inspired by median integration rules, which have attracted…
Reconstructing cardiac electrical activity from body surface electric potential measurements results in the severely ill-posed inverse problem in electrocardiography. Many different regularization approaches have been proposed to improve…
The Random Batch Method (RBM) proposed in [Jin et al. J Comput Phys, 2020] is an efficient algorithm for simulating interacting particle systems (IPS). In this paper, we investigate the Random Batch Method with replacement (RBM-r), which is…
This paper is devoted to the study of a novel mixed Finite Element Method for approximating the solutions of fourth order variational problems subjected to a constraint. The first problem we consider consists in establishing the convergence…
The conventional rounding error analysis provides worst-case bounds with an associated failure probability and ignores the statistical property of the rounding errors. In this paper, we develop a new statistical rounding error analysis for…
We consider the parallel-in-time solution of scalar nonlinear conservation laws in one spatial dimension. The equations are discretized in space with a conservative finite-volume method using weighted essentially non-oscillatory (WENO)…
Bayesian inverse problems arise in various scientific and engineering domains, and solving them can be computationally demanding. This is especially the case for problems governed by partial differential equations, where the repeated…
We present certain techniques to find completely positive maps between matrix algebras that take prescribed values on given data. To this aim we describe a semidefinite programming approach and another convex minimization method supported…
In this work, we construct a primal-dual forward-backward (PDFB) splitting method for computing a class of cross-diffusion systems that can be formulated as gradient flows under transport distances induced by matrix mobilities. By…
The numerical tools to simulate the bidomain model in cardiac electrophysiology are constantly developing due to the great clinical interest and scientific advances in mathematical models and computational power. The bidomain model consists…
In this paper investigations by the same authors on environmental issues concerning the control of the pollution produced by human activities have been extended to include costs related to environmental interventions. The proposed model…
The paper considers the numerical solution of nonlinear integral equations using the Newton-Kantorovich method with the mpmath library. High-precision quadrature of the kernel K(t, s, u) with respect to the variable s for fixed t increases…