Towards A Backward Perturbation Analysis For Data Least Squares Problems

An approximate solution for linear boundary-value problems with slowly varying coefficients

A new approach with piecewise-constant arguments to approximate and numerical solutions of oscillatory problems

The Whitham equation is a non-local model for nonlinear dispersive water waves. Since this equation is both nonlinear and non-local, exact or analytical solutions are rare except for in a few special cases. As such, an analytical method which results in minimal error is highly desirable for general forms of the Whitham equation. We obtain approximate analytical solutions to the non-local Whitham equation for general initial data by way of the optimal homotopy analysis method, through the use of a partial differential auxiliary linear operator. A method to control the residual error of these approximate solutions, through the use of the embedded convergence control parameter, is discussed in the context of optimal homotopy analysis. We obtain residual error minimizing solutions, using relatively few terms in the solution series, in the case of several different kernels and associated initial data. Interestingly, we find that for a specific class of initial data, there exists an exact solution given by the first term in the homotopy expansion. A specific example of initial data which satisfies the condition producing an exact solution is included. These results demonstrate the applicability of optimal homotopy analysis to equations which are simultaneously nonlinear and non-local.

We consider the backward error associated with a given approximate solution of a linear least squares problem. The backward error can be very expensive to compute, as it involves the minimal singular value of a certain matrix that depends on the problem data and the approximate solution. An estimate based on a regularized projection of the residual vector has been proposed in the literature and analyzed by several authors. Although numerical experiments in the literature suggest that it is a reliable estimate of the backward error for any given approximate least squares solution, to date no satisfactory explanation for this behavior had been found. We derive new bounds which confirm this experimental observation.

Real-Time Point Process Filter for Multidimensional Decoding Problems Using Mixture Models

Optimal designs for weighted approximation and integration of stochastic processes on [0,∞)

In this paper, we introduce a new kind of properly approximate efficient solution of vector optimization problems. Some properties for this new classof approximate solutions are established. Also necessary and sufficient conditions via nonlinear scalarizations areobtained for properly approximate solutions. And under theassumption of cone subconvexlike functions, we derive linearscalarizations for properly approximate efficient solutions.

This paper provides some new results on robust approximate optimal solutions for convex optimization problems with data uncertainty. By using robust optimization approach (worst-case approach), we first establish necessary and sufficient optimality conditions for robust approximate optimal solutions of this uncertain convex optimization problem. Then, we introduce a Wolfe-type robust approximate dual problem and investigate robust approximate duality relations between them. Moreover, we obtain some robust approximate saddle point theorems for this uncertain convex optimization problem. We also show that our results encompass as special cases some optimization problems considered in the recent literature.

Purpose – The purpose of this paper is to consider a steady two-dimensional magneto-hydrodynamic (MHD) Falkner-Skan boundary layer flow of an incompressible viscous electrically fluid over a permeable wall in the presence of a magnetic field. Design/methodology/approach – The governing equations of MHD Falkner-Skan flow are transformed into an initial values problem of an ordinary differential equation using the Lie symmetry method which are then solved by He's variational iteration method with He's polynomials. Findings – The approximate solution is compared with the known solution using the diagonal Pad’e approximants and the geometrical behavior for the values of various parameters. The results reveal the reliability and validity of the present work, and this combinational method can be applied to other nonlinear boundary layer flow problems. Originality/value – In this paper, an approximate analytical solution of the MHD Falkner-Skan flow problem is obtained by combining the Lie symmetry method with the variational iteration method and He's polynomials.

An inverse dynamics method is introduced for constructing exact special-case solutions for hybrid coordinate ordinary/partial systems of differential equations (hybrid ODE/PDE systems). The solution is constructed such that it lies near a given approximate numerical solution, and therefore the special-case solutions can be generated in a versatile and physically meaningful fashion and can serve as a benchmark problem to validate approximate solution methods. The exact solution is constructed such that it is a differentiable, continuous-function neighbor of the given approximate numerical solution. This continuous solution is then substituted into the governing system of ODEs/PDEs and a full complement of distributed and boundary forces is determined algebraically to exactly satisfy the differential equations. This process has been automated by computer symbol manipulation. Since the exact special-case algebraic solutions can be evaluated anywhere in space and time, this approach is ideally suited to providing a true exact motion and the corresponding forces for studying the convergence errors in a family of approximate solutions. This methodology makes it possible for one to rigorously determine exact solution errors for a significant class of ODE/PDE systems for which the initial-value problem is not, in general, exactly solvable. We explore the utility of this method in validating numerical solution methods.

The goal of this work is investigating the existence and uniqueness of solutions for a class of non-local Riemann-Liouville fractional q-integro-differential equations, in addition to finding the approximate solutions to the proposed equation. The problem incorporates Riemann-Liouville fractional derivatives, Riemann-Liouville fractional q integral, and a q-nonlocal boundary condition. By leveraging the Schauder fixed-point theorem and the Banach contraction principle, we establish rigorous conditions under which solution exists and is unique. These mathematical methods are vital for determining the existence and uniqueness of a solution for a class of non-local Riemann-Liouville fractional q-integro-differential equation. Five illustrative examples validate the theoretical results, demonstrating the applicability of the proposed framework. Furthermore, a neural network-based approach is employed to approximate solutions, achieving high accuracy with minimal error, as evidenced by detailed simulations. The integration of theoretical analysis and computational methods highlights the effectiveness of the approach in addressing complex fractional systems.

The paper studies convex stochastic optimization problems in a reproducing kernel Hilbert space (RKHS). The objective (risk) functional depends on functions from this RKHS and takes the form of a mathematical expectation (integral) of a nonnegative integrand (loss function) over a probability measure. The problem is generally ill-posed, a difficulty that in statistical learning is addressed through Tihonov regularization, with Monte Carlo approximation of integrals, which also makes it possible to solve the problem by finite dimensional (convex) quadratic optimization. The approximate solutions are referred to as kernel learning estimators and are expressed as a linear combination of kernels evaluated at the sample points. They are functional random variables that depend on the full sample. The paper studies a probabilistic convergence of these approximate solutions under a gradual elimination of the regularization parameter with rising number of observations. Its intended contribution is to derive novel nonasymptotic bounds on the minimization error and exponential bounds on the tail distribution of errors and to establish novel sufficient conditions for uniform convergence of kernel estimators to the true (normal) solution with probability one, jointly with a rule for downward adjustment of the regularization factor with increasing sample size. Applications to least squares, median, and quantile regression estimation, as well as to binary classification, are discussed.

In this paper the inductive interference between power lines and parallel buried irrigation pipelines is investigated, using a hybrid method employing finite element formulation and circuit analysis. A specific configuration taken from the literature is examined during an unbalanced operating condition. This is a common case and may go unnoticed for days, which can result in shocks to persons touching pipeline risers. Also, a general case encountered frequently in the Greek distribution network is studied, producing graphs that may be useful to the engineer. It is found that a detailed knowledge of the overall configuration is needed in order to calculate the inductive coupling with minimum error. Otherwise, computed results are usually on the safe side. Moreover, it is realized that under certain circumstances the current flowing through a person touching a riser may be higher than the acceptable limits.

In this paper, the decomposition method is implemented for solving the bidirectional Sawada- Kotera (bSK) equation with two kinds of initial conditions. As a result, the Adomian polynomials have been calculated and the approximate and exact solutions of the bSK equation are obtained by means of Maple, such as solitary wave solutions, doubly-periodic solutions, two-soliton solutions. Moreover, we compare the approximate solution with the exact solution in a table and analyze the absolute error and the relative error. The results reported in this article provide further evidence of the usefulness of the Adomian decomposition method for obtaining solutions of nonlinear problems

We examine a method introduced by Torsten Carleman a century ago to approximate the solution of non-linear differential systems. His idea consists in replacing the non-linear system with an infinite-dimensional linear one. Carleman approximants are obtained when solving a truncated version. We describe the linearization process and present two different approaches to apply the Carleman technique. In the first one, which is the common case, the Carleman approximants are centred at an equilibrium point in phase space. The resulting linear system is always homogeneous, and higher orders of approximation are usually analytically solvable. The closer the point of interest is to the equilibrium point, the more accurate the approximate solution. In a way, this approach may be viewed as an extension of the so-called local, or qualitative, analysis of non-linear differential systems. In the second one, the approximants are built up around the initial condition. In this novel approach, there is a gain in accuracy at the price of solving a non-homogeneous linear system that allows for an analytical solution too. We found a noteworthy pattern in the analyticity properties of the approximants that explains this improved behaviour. The connection with the Koopman operator and related methods is commented on. A number of systems are worked out intended to provide evidence of the practical utility of the Carleman approach. We also derive approximate solutions for the Susceptible-Infected-Removed epidemiological model, which leads to an extended classification of the evolutionary stage of an epidemic process.