相关论文: Analytical approximation schemes for solving exact…
Leximin is a common approach to multi-objective optimization, frequently employed in fair division applications. In leximin optimization, one first aims to maximize the smallest objective value; subject to this, one maximizes the…
We apply polynomial approximation methods -- known in the numerical PDEs context as spectral methods -- to approximate the vector-valued function that satisfies a linear system of equations where the matrix and the right hand side depend on…
A novel three-term Polak-Ribi\`{e}re-Polyak conjugate gradient method is proposed for solving vector optimization problems. It should be emphasized that this is the first extension of three-term conjugate gradient methods from scalar…
In this paper we establish a best approximation property of fully discrete Galerkin finite element solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty$ norm. The discretization method…
Complementarity problems and variational inequalities arise in a wide variety of areas, including machine learning, planning, game theory, and physical simulation. In all of these areas, to handle large-scale problem instances, we need fast…
We consider homogenization problems for linear elliptic equations in divergence form. The coecients are assumed to be a local perturbation of some periodic background. We prove $W^{1,p}$ and Lipschitz convergence of the two-scale expansion,…
In this paper, we propose a proximal iteratively reweighted algorithm with extrapolation based on block coordinate update aimed at solving a class of optimization problems which is the sum of a smooth possibly nonconvex loss function and a…
We consider polynomial approximation over the interval $[-1,1]$ by regularized weighted discrete least squares methods with $\ell_2-$ or $\ell_1-$regularization, respectively. As the set of nodes we use Gauss quadrature points (which are…
Error estimates are proved for finite element approximations to the solution of second-order hyperbolic partial differential equations with coefficients varying in both space and time. Optimal rates of convergence in the energy norm are…
We show how the exact renormalization group for the effective action with a sharp momentum cutoff, may be organised by expanding one-particle irreducible parts in terms of homogeneous functions of momenta of integer degree (Taylor…
Motivated by approximation Bayesian computation using mean-field variational approximation and the computation of equilibrium in multi-species systems with cross-interaction, this paper investigates the composite geodesically convex…
Spectral approximation by polynomials on the unit ball is studied in the frame of the Sobolev spaces $W^{s}_p(\ball)$, $1<p<\infty$. The main results give sharp estimates on the order of approximation by polynomials in the Sobolev spaces…
An adaptive modified weak Galerkin method (AmWG) for an elliptic problem is studied in this paper, in addition to its convergence and optimality. The modified weak Galerkin bilinear form is simplified without the need of the skeletal…
When an Approximation Theorist looks at well-posed PDE problems or operator equations, and standard solution algorithms like Finite Elements, Rayleigh-Ritz or Trefftz techniques, methods of fundamental or particular solutions and their…
We propose a new method to obtain approximate solutions for the Schr\"{o}dinger equation with an arbitrary potential that possesses bound states. This method, relying on the auxiliary field technique, allows in many cases to find analytical…
Andresen and Spokoiny's (2013) ``critical dimension in semiparametric estimation`` provide a technique for the finite sample analysis of profile M-estimators. This paper uses very similar ideas to derive two convergence results for the…
Approximate analytical solutions of a two-term potential are studied for the relativistic wave equations, namely, for the Klein-Gordon and Dirac equations. The results are obtained by solving of a Riemann-type equation whose solution can be…
It has recently been demonstrated that locality of spatial supports in the parametrization of coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial expansions of the corresponding parameter-dependent…
We approximate given potentials by means of the specially introduced reference potentials. On the one hand their parameters may be easily found from the usual WKB integral for the given potential; on the other hand they allow a simple…
The Lipkin-Meshkov-Glick (LMG) model was devised to test the validity of different approximate formalisms to treat many-particle systems. The model was constructed to be exactly solvable and yet non-trivial, in order to capture some of the…