Variational methods for discontinuous potentials. II. Global trial functions

A concept of surplus function for the Schrödinger equation is put forward. A novel quantum Monte Carlo approach, namely, the surplus function method, is suggested with use of a novel trial function of significant physical meaning which is based on the proposed surplus function. The trial function is an iteration type, as given in the text. It is theoretically proved that the energy expectation value obtained from the proposed trial function decreases step by step in iterations. In addition, computation formulas and concrete procedures for energy expectation value are presented. Calculations for H2, LiH, Li2, and H2O molecules indicate that the energy expectation values after only 4–5 iterations achieve over 90% of the correlation energy, indicating that the convergence rates are rapid. The trial function used in the present paper requires no parameter optimization and is of the highest accuracy.

A p-extension of the symmetric boundary element method

Variational Methods in Neutron Thermalization

A variational method in which the trial functions are linear combinations of a finite set of simple basic functions is considered as a means of obtaining simple and sufficiently accurate approximations to the flux distributions in homogeneous circular cylinders which range from purely capturing to purely scattering. In particular, analytic formulae and some numerical results are given for basic functions which are even powers of the distance from the cylinder axis and some formulae are given for trial functions composed of square-pulse functions.It is found that even though the trial function with the parameters determined by the variational principle is not itself a particularly good approximation to the exact flux distribution, it may often be easily iterated in the integral equation to give a satisfactory approximation. The trial function may also be used to evaluate integrals of the flux times arbitrary weighting functions with results equivalent to those given by complex variational methods which require the introduction of an auxiliary equation.

The finite volume method in the context of the finite element method

Some variational methods are considered in which the trial functions used are not required to satisfy prescribed boundary conditions. Applications of these methods are made to a two-body system interacting via a hard-core potential.

ON THE ALMOST SURE RATE OF CONVERGENCE OF TEMPORAL-DIFFERENCE LEARNING ALGORITHMS

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (Electron J Probab 25:55, 2020. https://doi.org/10.1214/20-EJP442). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler–Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is Hin (0,1) and the drift is mathcal {C}^alpha , alpha in [0,1] and alpha >1-1/(2H), we show the strong L_p and almost sure rates of convergence to be ((1/2+alpha H)wedge 1) -varepsilon , for any varepsilon >0. Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016. https://doi.org/10.1016/j.spa.2016.02.002). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence 1/2-varepsilon of the Euler–Maruyama scheme for mathcal {C}^alpha drift, for any varepsilon ,alpha >0.

Calculation of energy eigen value of hydrogen negative ion (H − ) in 2p^2 configuration using the method of variation functions has been done. A work on H − can lead to calculations of electric multipole moments of a hydrogen molecule. The trial function is a linear combination of 8 expansion terms each of which is related to the Chandrasekhar’s basis. This work produces a series of 7 energy eigen values which converges to a value of −0.2468 whereas the value of this convergence is expected to be −0.2523. This deviation from the expected value is mainly due to the elimination of interelectronic distance (u) coordinate. The values of the exponent parameters used in this work contribute also to this deviation. This variational method will be applied to the construction of some energy eigen functions of Hv2 .

Chebyshev collocation method for the free vibration analysis of geometrically exact beams with fully intrinsic formulation

The procedure of quantum mechanical treatment of a three-body problem described in the preceding paper byDiehl, Flugge, Schroder, Volkel andWeiguny has been specialized to the ground state in which the eigenfunction only depends upon the three distances between the particles. The solution of the Schrodinger equation has been approximated by variational methods, using the electronic functions ofFinkelstein andHorowitz, and ofGuillemin andZener, but including the nuclear vibration in the trial function.

SummaryRecently, the method of difference potentials has been extended to linear elastic fracture mechanics. The solution was calculated on a grid boundary belonging to the domain of an auxiliary problem, which must be solved multiple times. Singular enrichment functions, such as those used within the extended finite element method, were introduced to improve the approximation near the crack tip leading to near‐optimal convergence rates. Now, the method is further developed by significantly reducing the computation time. This is achieved via the implementation of a system of basis functions introduced along the physical boundary of the problem. The basis functions form an approximation of the trace of the solution at the physical boundary. This method has been proven efficient for the solution of problems on regular (Lipschitz) domains. By introducing the singularity into the finite element space, the approximation of the crack can be realised by regular functions. Near‐optimal convergence rates are then achieved for the enriched formulation. A solution algorithm using the fast Fourier transform is provided with the aim of further increasing the efficiency of the method.

\n \nWe consider mixed and hybrid variational formulations to the linearized\nelasticity system in domains with cracks. Inequality type conditions are\nprescribed at the crack faces which results in unilateral contact problems. The\nvariational formulations are extended to the whole domain including the cracks\nwhich yields, for each problem, a smooth domain formulation. Mixed\nfinite element methods such as PEERS or BDM methods are designed to avoid\nlocking for nearly incompressible materials in plane elasticity. We study and\nimplement discretizations based on such mixed finite element methods for the\nsmooth domain formulations to the unilateral crack problems. We obtain\nconvergence rates and optimal error estimates and we present some numerical\nexperiments in agreement with the theoretical results.\n

Density matrices are positively semi-definite Hermitian matrices with unit trace that describe the states of quantum systems. Many quantum systems of physical interest can be represented as high-dimensional low rank density matrices. A popular problem in {\it quantum state tomography} (QST) is to estimate the unknown low rank density matrix of a quantum system by conducting Pauli measurements. Our main contribution is twofold. First, we establish the minimax lower bounds in Schatten $p$-norms with $1\leq p\leq +\infty$ for low rank density matrices estimation by Pauli measurements. In our previous paper, these minimax lower bounds are proved under the trace regression model with Gaussian noise and the noise is assumed to have common variance. In this paper, we prove these bounds under the Binomial observation model which meets the actual model in QST. Second, we study the Dantzig estimator (DE) for estimating the unknown low rank density matrix under the Binomial observation model by using Pauli measurements. In our previous papers, we studied the least squares estimator and the projection estimator, where we proved the optimal convergence rates for the least squares estimator in Schatten $p$-norms with $1\leq p\leq 2$ and, under a stronger condition, the optimal convergence rates for the projection estimator in Schatten $p$-norms with $1\leq p\leq +\infty$. In this paper, we show that the results of these two distinct estimators can be simultaneously obtained by the Dantzig estimator. Moreover, better convergence rates in Schatten norm distances can be proved for Dantzig estimator under conditions weaker than those needed in previous papers. When the objective function of DE is replaced by the negative von Neumann entropy, we obtain sharp convergence rate in Kullback-Leibler divergence.

The BEM-based Finite Element Method is one of the promising strategies which are applicable for the approximation of boundary value problems on general polygonal and polyhedral meshes. The flexibility with respect to meshes arises from the implicit definition of trial functions in a Trefftz-like manner. These functions are treated locally by means of Boundary Element Methods (BEM). The following presentation deals with the formulation of higher order trial functions and their application in uniform and adaptive mesh refinement strategies. For the adaptive refinement, a residual based error estimate is used on general polygonal meshes for the higher order, conforming trial functions. The first numerical results, in the context of adaptive refinement, show optimal rates of convergence with respect to the number of degrees of freedom.