A Rate of Convergence of Weak Adversarial Neural Networks for the Second Order Parabolic PDEs

In this paper,we give the first rigorous error estimation of the Weak Adversarial Neural Networks(WAN)in solving the second order parabolic PDEs.By decomposing the error into approximation error and statistical error,we first show the weak solution can be approximated by the ReLU2 with arbitrary accuracy,then prove that the statistical error can also be efficiently bounded by the Rademacher complexity of the network functions,which can be further bounded by some integral related with the covering numbers and pseudo-dimension of ReLU2 space.Finally,by combining the two bounds,we prove that the error of the WAN method can be well controlled if the depth and width of the neural network as well as the sample numbers have been properly selected.Our result also reveals some kind of freedom in choosing sample numbers on∂Ωand in the time axis.

Parabolic Partial Differential Equations as Inverse Moments Problem

We considerer parabolic partial differential equations under the conditions on a region . We will see that we can write the equation in partial derivatives as an Fredholm integral equation of first kind and will solve this latter with the techniques of inverse moments problem. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on moments problem. Also we consider the one- dimensional one-phase inverse Stefan problem.

A FAMILY OF HIGH-ORDER ACCURACY EXPLICIT DIFFERENCE SCHEMES WITH BRANCHING STABILITY FOR SOLVING 3-D PARABOLIC PARTIAL DIFFERENTIAL EQUATION

A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and the truncation error is 0(<Delta>t(2) + Deltax(4)).

Iterative algorithm for parabolic and hyperbolic PDEs with nonlocal boundary conditions

In this paper,we are concerned with the numerical solutions for the parabolic and hyperbolic partial differential equations with nonlocal boundary conditions.Thus,we presented a new iterative algorithm based on the Restarted Adomian Decomposition Method(RADM)for solving the two equations of different types involving dissimilar boundary and nonlocal conditions.The algorithm presented transforms the given nonlocal initial boundary value problem to a local Dirichlet one and then employs the RADM for the numerical treatment.Numerical comparisons were made between our proposed method and the Adomian Decomposition Method(ADM)to demonstrate the efficiency and performance of the proposed method.

Finite Element Error Estimation for Parabolic Optimal Control Problems with Pointwise Observations

In this paper,we consider parabolic distributed control problems with cost functional of pointwise observation type either in space or in time.First,we show the well-posedness of the optimization problems and derive the first order optimality systems,where the adjoint state can be expressed as the linear combination of solutions to two backward parabolic equations that involve the Dirac delta distribution as source either in space or in time.Second,we use a space-time finite element method to discretize the control problems,where the state variable is approximated by piecewise constant functions in time and continuous piecewise linear polynomials in space,and the control variable is discretized by following the variational discretization concept.We obtain a priori error estimates for the control and state variables with order O(k 12+h)up to a logarithmic factor under the L 2-norm.Finally,we perform several numerical experiments to support our theoretical results.

A Numerical Methodology for Enforcing Maximum Principles and the Non-Negative Constraint for Transient Diffusion Equations

Transient diffusion equations arise in many branches of engineering and applied sciences(e.g.,heat transfer and mass transfer),and are parabolic partial differential equations.It is well-known that these equations satisfy important mathematical properties like maximum principles and the non-negative constraint,which have implications in mathematical modeling.However,existing numerical formulations for these types of equations do not,in general,satisfy maximum principles and the nonnegative constraint.In this paper,we present a methodology for enforcing maximum principles and the non-negative constraint for transient anisotropic diffusion equation.The proposed methodology is based on the method of horizontal lines in which the time is discretized first.This results in solving steady anisotropic diffusion equation with decay equation at every discrete time-level.We also present other plausible temporal discretizations,and illustrate their shortcomings in meeting maximum principles and the non-negative constraint.The proposedmethodology can handle general computational grids with no additional restrictions on the time-step.We illustrate the performance and accuracy of the proposed methodology using representative numerical examples.We also perform a numerical convergence analysis of the proposed methodology.For comparison,we also present the results from the standard singlefield semi-discrete formulation and the results froma popular software package,which all will violate maximum principles and the non-negative constraint.

Stochastic ordering by g-expectations

We derive sufficient conditions for the convex and monotonic g-stochastic ordering of diffusion processes under nonlinear g-expectations and g-evaluations.Our approach relies on comparison results for forward-backward stochastic differential equations and on several extensions of convexity,monotonicity,and continuous dependence properties for the solutions of associated semilinear parabolic partial differential equations.Applications to contingent claim price comparison under different hedging portfolio constraints are provided.