PDE Problems Research Articles

Convergence rate of free boundary of numerical scheme for American option

Based on the optimal estimate of convergence rate $O(\Delta x)$ of the value function ofan explicit finite difference scheme for the American put option problem in [6], an $O(\sqrt{\Delta x})$ rate of convergence of the free boundary resulting from a general compatible numerical scheme to the true free boundary is proven. A new criterion for the compatibility of a generic numerical scheme to the PDE problem is presented. A numerical example is also included.

Recovery of harmonic functions from partial boundary data respecting internal pointwise values

Abstract We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply connected domain with Lipschitz boundary ∂ ⁡ Ω ${\partial\Omega}$ . Assuming Dirichlet and Neumann data available on Γ ⊂ ∂ ⁡ Ω ${\Gamma\subset\partial\Omega}$ to be real-valued functions in W 1 / 2 , 2 ⁢ ( Γ ) ${W^{1/2,2}(\Gamma)}$ and L 2 ⁢ ( Γ ) ${L^{2}(\Gamma)}$ classes, respectively, we develop a non-iterative method for solving this ill-posed Cauchy problem choosing L 2 ${L^{2}}$ bound of the solution on ∂ ⁡ Ω ∖ Γ ${\partial\Omega\setminus\Gamma}$ as a regularizing parameter. The present complex-analytic approach also naturally allows imposing additional pointwise constraints on the solution which, on practical side, can help incorporating outlying boundary measurements without changing the boundary into a less regular one.

Hopf bifurcation in structural population models

We study a nonlinear PDE problem motivated by the peculiar patterns arising in myxobacteria, namely, counter‐migrating cell density waves. We rigorously prove the existence of Hopf bifurcations for some specific values of the parameters of the system. This shows the existence of periodic solutions for the systems under consideration. Copyright © 2016 John Wiley & Sons, Ltd.

Optimal recovery of integral operators and its applications

In this paper we present the solution to a problem of recovering a rather arbitrary integral operator based on incomplete information with error. We apply the main result to obtain optimal methods of recovery and compute the optimal error for the solutions to certain integral equations as well as boundary and initial value problems for various PDE’s.

Concentration Wave for a Class of Reaction Chromatography System with Pulse Injections

By using fluid dynamics theory with the effects of adsorption and reaction, the chromatography model with a reaction A → B was established as a system of two hyperbolic partial differential equations (PDE’s). In some practical situations, the reaction chromatography model was simplified a semi-coupled system of two linear hyperbolic PDE’s. In which, the reactant concentration wave model was the initial-boundary value problem of a self-closed hyperbolic PDE, while the resultant concentration wave model was the initial-boundary value problem of hyperbolic PDE coupling reactant concentration. The general explicit expressions for the concentration wave of the reactants and resultants were derived by Laplace transform. The δ-pulse and wide pulse injections were taken as the examples to discuss detailedly, and then the stability analysis between the resultant solutions of the two modes of pulse injection was further discussed. It was significant for further analysis of chromatography, optimizing chromatographic separation, determining the physical and chemical characters.

An Optimal Solver for Linear Systems Arising from Stochastic FEM Approximation of Diffusion Equations with Random Coefficients

This paper discusses the design and implementation of efficient solution algorithms for symmetric linear systems associated with stochastic Galerkin approximation of elliptic PDE problems with correlated random data. The novel feature of our preconditioned MINRES solver is the incorporation of error control in the natural “energy” norm in combination with a reliable and efficient a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. The MATLAB codes used in the numerical studies are available online.

Discrete approximations to local times for reflected diffusions

We propose a discrete analogue for the boundary local time of reflected diffusions in bounded Lipschitz domains. This discrete analogue, called the discrete local time, can be effectively simulated in practice and is obtained pathwise from random walks on lattices. We establish weak convergence of the joint law of the discrete local time and the associated random walks as the lattice size decreases to zero. A cornerstone of the proof is the local central limit theorem for reflected diffusions developed in [7]. Applications of the join convergence result to PDE problems are illustrated.

Efficient Adaptive Stochastic Galerkin Methods for Parametric Operator Equations

This paper is concerned with the design and implementation of efficient solution algorithms for elliptic PDE problems with correlated random data. The energy orthogonality that is built into stochastic Galerkin approximations is cleverly exploited to give an innovative energy error estimation strategy that utilizes the tensor product structure of the approximation space. An associated error estimator is constructed and shown theoretically and numerically to be an effective mechanism for driving an adaptive refinement process. The codes used in the numerical studies are available online.

An Algebraic Multilevel Preconditioner with Low-Rank Corrections for Sparse Symmetric Matrices

This paper describes a multilevel preconditioning technique for solving sparse symmetric linear systems of equations. This “Multilevel Schur Low-Rank” (MSLR) preconditioner first builds a tree structure $\mathcal{T}$ based on a hierarchical decomposition of the matrix and then computes an approximate inverse of the original matrix level by level. Unlike classical direct solvers, the construction of the MSLR preconditioner follows a top-down traversal of $\mathcal{T}$ and exploits a low-rank property that is satisfied by the difference between the inverses of the local Schur complements and specific blocks of the original matrix. A few steps of the generalized Lanczos tridiagonalization procedure are applied to capture most of this difference. Numerical results are reported to illustrate the efficiency and robustness of the MSLR preconditioner with both two- and three-dimensional discretized PDE problems and with publicly available test problems.

Similarity, Mass Conservation, and the Numerical Simulation of a Simplified Glacier Equation

A one-dimensional nonlinear scale-invariant PDE problem with moving boundaries (a model glacier equation) is solved numerically using a moving-mesh scheme based on local conservation and compared with a self-similar scaling solution, showing numerical convergence under mesh refinement. It is also shown, analytically and numerically, that the waiting-time exhibited by this problem for general data ends when the local profile close to the boundary becomes self-similar. ∗Department of Mathematics and Statistics, University of Reading, PO Box 220, Reading, RG6 6AX, UK (m.j.baines@reading.ac.uk)

Parametrically defined nonlinear differential equations and their solutions: Applications in fluid dynamics

The study deals with parametrically defined ordinary differential equations, practically unaddressed in the literature. It finds the general solutions for three classes of first- and second-order nonlinear ODEs of this kind. The solutions are further used to construct new exact solutions to the equations of an unsteady axisymmetric boundary layer with pressure gradient on a body of revolution of arbitrary shape. Also the paper suggests a short list of essential problems for nonlinear ODEs and PDEs defined parametrically that need to be addressed in the future.

Radial basis function partition of unity methods for pricing vanilla basket options

Meshfree methods based on radial basis function (RBF) approximation are becoming widely used for solving PDE problems. They are flexible with respect to the problem geometry and highly accurate. A disadvantage of these methods is that the linear system to be solved becomes dense for globally supported RBFs. A remedy is to introduce localisation techniques such as partition of unity. RBF partition of unity methods (RBF–PUM) allow for a significant sparsification of the linear system and lower the computational effort. In this work we apply a global RBF method as well as RBF–PUM to problems in option pricing. We consider one- and two-dimensional vanilla options. In order to price American options we employ a penalty approach. A penalty term, suitable for American multi-asset call options, has been designed. RBF–PUM is shown to be competitive compared with a finite difference method and a global RBF method. It is as accurate as the global RBF method, but significantly faster. The results for RBF–PUM look promising for extension to higher-dimensional problems.

Carleman weight functions for solving ill-posed Cauchy problems for quasilinear PDEs

This is the first publication in which a numerical method with rigorously guaranteed convergence is presented for some ill-posed Cauchy problems for semilinear PDEs of the second order. The key tool is the use of the Carleman weight functions in a Tikhonov-like functionals. The method works for elliptic, parabolic and hyperbolic semilinear equations of the second order.

On potential spaces related to Jacobi expansions

We investigate potential spaces associated with Jacobi expansions. We prove structural and Sobolev-type embedding theorems for these spaces. We also establish their characterizations in terms of suitably defined fractional square functions. Finally, we present sample applications of the Jacobi potential spaces connected with a PDE problem.

A collage-based approach to solving inverse problems for second-order nonlinear hyperbolic PDEs

A goal of many inverse problems is to find unknown parameter values, λ ∈ Λ, so that the given observed data utrue agrees well with the solution data produced using these parameters uλ. Unfortunately finding uλ in terms of the parameters of the problem may be a difficult or even impossible task. Further, the objective function may be a complicated function of the parameters λ ∈ Λ and may require complex minimization techniques. In recent literature, the collage coding approach to solving inverse problems has emerged. This approach avoids the aforementioned difficulties by bounding the approximation error above by a more readily minimizable distance, thus making the approximation error small. The first of these methods was applied to first-order ordinary differential equations and gets its name from the “collage theorem” used in this setting to achieve an upperbound on the approximation error. A number of related ODE problems have been solved using this method and extensions thereof. More recently, collage-based methods for solving linear and nonlinear elliptic partial differential equations have been developed. In this paper we establish a collage-based method for solving inverse problems for nonlinear hyperbolic PDEs. We develop the necessary background material, discuss the complications introduced by the presence of time-dependence, establish sufficient conditions for using the collage-based approach in this setting and present examples of the theory in practice.

Carleman estimates for the regularization of ill-posed Cauchy problems

This work is a survey of results for ill-posed Cauchy problems for PDEs of the author with co-authors starting from 1991. A universal method of the regularization of these problems is presented here. Even though the idea of this method was previously discussed for specific problems, a universal approach of this paper was not discussed, at least in detail. This approach consists in constructing of such Tikhonov functionals which are generated by unbounded linear operators of those PDEs. The approach is quite general one, since it is applicable to all PDE operators for which Carleman estimates are valid. Three main types of operators of the second order are among them: elliptic, parabolic and hyperbolic ones. The key idea is that convergence rates of minimizers are established using Carleman estimates. Generalizations to nonlinear inverse problems, such as problems of reconstructions of obstacles and coefficient inverse problems are also feasible.

Metric tensor recovery for adaptive meshing

Adaptive computation is now recognized as essential for solving complex PDE problems. Conceptually, such a computation requires at each step the definition of a continuous metric field (mesh size and direction) to govern the generation of adapted meshes. In practice, in the adaptive computation, an appropriate a posteriori error estimation is used and an upper-bounding of the error is expressed in terms of discrete metrics associated with the element vertices. In order to obtain a continuous metric field, the discrete field is recovered in the whole domain mesh using an appropriate interpolation method on each element. In this paper, a new method for interpolating discrete metric fields, based on a so-called “natural decomposition” of metrics, is introduced. The proposed method uses known matrix decompositions and is computationally robust and efficient. Classical interpolation methods are recalled and, from numerical examples on simplicial mesh elements, some qualitative comparisons against the new methodology are made to show its relevance.

An augmented singular transform and its partial Newton method for finding new solutions

To find new solutions through the information provided by previously found solution(s), this paper is first to develop a new singular transform to change the local basin-barrier structure of the original problem in an infinite-dimensional space and then to propose an augmented partial Newton method to solve the transformed problem on the solution set. Mathematical justifications of the new formulation and method are established. Details on algorithm implementation are presented. Examples for two very different variational PDE problems are carried out to illustrate the method. Their numerical results are displayed by profile–contour plots and error/convergence data. The formulation and method developed in this paper can also be applied to solve multiple fixed point problems.

A globally convergent numerical method for a coefficient inverse problem for a parabolic equation

In this work a Multidimensional Coefficient Inverse Problem (MCIP) for a parabolic PDE with the data resulting from a single measurement event is considered. This measured data is obtained using a single position of the point source. The most important property of the method presented here is that even though we do not need any advanced knowledge of a small neighborhood of the solution, we still obtain points in this neighborhood. This is the reason why the numerical algorithm for this method is called globally convergent. In the literature a globally convergent numerical method for the MCIP with single measurement data has inclusively been considered in the book (Beilina and Klibanov, 2012) and some other publications of Beilina and Klibanov. In those publications a globally convergent algorithm for MCIP for a hyperbolic PDE was developed. Here we modify their technique to prove the global convergence property of their method for the parabolic case.

Multi-time-step domain decomposition method with non-matching grids for parabolic problems

Evolution of time dependent physical quantities, such as current, heat etc., in composite materials are modeled by initial boundary value problems for parabolic PDEs. These physical quantities follow different evolution patterns in different parts of the computational domain depending on the material properties, size of constituent material subdomains, coupling scheme, etc. Therefore, the stability and accuracy requirements of a numerical integration scheme may necessitate domain dependent time discretization. Parabolic problems are usually solved by discretizing spatially using finite elements and then integrating over time using discrete solvers. We propose an asynchronous multi-domain time integration scheme for parabolic problems. For efficient parallel computing of large problems, we present the dual decomposition method with local Lagrange multipliers to ensure the continuity of the primary unknowns at the interface between subdomains. The proposed method enables us to use domain dependent Rothe method on different parts of a computational domain and thus provide an efficient and robust approach to solving large scale composite material problems.