Hybrid PDE solver for data-driven problems and modern branching

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European Journal of Applied Mathematics (EJAM

Improved Operational Matrices of DP-Ball Polynomials for Solving Singular Second Order Linear Dirichlet-type Boundary Value Problems

Solving Dirichlet-type boundary value problems (BVPs) using a novel numerical approach is presented in this study. The operational matrices of DP-Ball Polynomials are used to solve the linear second-order BVPs. The modification of the operational matrix eliminates the BVP\u27s singularity. Consequently, guaranteeing a solution is reached. In this article, three different examples were taken into consideration in order to demonstrate the applicability of the method. Based on the findings, it seems that the methodology may be used effectively to provide accurate solutions

Modified Sumudu Transform Analytical Approximate Methods For Solving Boundary Value Problems

In this study, emphasis is placed on analytical approximate methods. These methods include the combination of the Sumudu transform (ST) with the homotopy perturbation method (HPM), namely the Sumudu transform homotopy perturbation method (STHPM), the combination of the ST with the variational iteration method (VIM), namely the Sumudu transform variational iteration method (STVIM) and finally, the combination of the ST with the homotopy analysis method (HAM), namely the Sumudu transform homotopy analysis method (STHAM). Although these standard methods have been successfully used in solving various types of differential equations, they still suffer from the weakness in choosing the initial guess. In addition, they require an infinite number of iterations which negatively affect the accuracy and convergence of the solutions. The main objective of this thesis is to modify, apply and analyze these methods to handle the difficulties and drawbacks and find the analytical approximate solutions for some cases of linear and nonlinear ordinary differential equations (ODEs). These cases include second-order two-point boundary value problems (BVPs), singular and systems of second-order two-point BVPs. For the proposed methods, the trial function was employed as an initial approximation to provide more accurate approximate solutions for the considered problems. In addition, for the STVIM method, a new algorithm has been proposed to solve various kinds of linear and nonlinear second-order two-point BVPs. In this algorithm, the convolution theorem has been used to find an optimal Lagrange multiplier

Numerical study on hydrodynamics of ships with forward speed based on nonlinear steady wave

In this paper, an improved potential flow model is proposed for the hydrodynamic analysis of ships advancing in waves. A desingularized Rankine panel method, which has been improved with the added effect of nonlinear steady wave-making (NSWM) flow in frequency domain, is employed for 3D diffraction and radiation problems. Non-uniform rational B-splines (NURBS) are used to describe the body and free surfaces. The NSWM potential is computed by linear superposition of the first-order and second-order steady wave-making potentials which are determined by solving the corresponding boundary value problems (BVPs). The so-called mj terms in the body boundary condition of the radiation problem are evaluated with nonlinear steady flow. The free surface boundary conditions in the diffraction and radiation problems are also derived by considering nonlinear steady flow. To verify the improved model and the numerical method adopted in the present study, the nonlinear wave-making problem of a submerged moving sphere is first studied, and the computed results are compared with the analytical results of linear steady flow. Subsequently, the diffraction and radiation problems of a submerged moving sphere and a modified Wigley hull are solved. The numerical results of the wave exciting forces, added masses, and damping coefficients are compared with those obtained by using Neumann–Kelvin (NK) flow and double-body (DB) flow. A comparison of the results indicates that the improved model using the NSWM flow can generally give results in better agreement with the test data and other published results than those by using NK and DB flows, especially for the hydrodynamic coefficients in relatively low frequency ranges

An explicit substructuring method for overlapping domain decomposition based on stochastic calculus

In a recent paper [{\em F. Bernal, J. Mor\'on-Vidal and J.A. Acebr\'on, Comp.$\&$ Math. App. 146:294-308 (2023)}] an hybrid supercomputing algorithm for elliptic equations has been put forward. The idea is that the interfacial nodal solutions solve a linear system, whose coefficients are expectations of functionals of stochastic differential equations confined within patches of about subdomain size. Compared to standard substructuring techniques such as the Schur complement method for the skeleton, the hybrid approach renders an explicit and sparse shrunken matrix -- hence suitable for being substructured again. The ultimate goal is to push strong scalability beyond the state of the art, by leveraging the scope for parallelisation of stochastic calculus. Here, we present a major revamping of that framework, based on the insight of embedding the domain in a cover of overlapping circles (in two dimensions). This allows for efficient Fourier interpolation along the interfaces (now circumferences) and -- crucially -- for the evaluation of most of the interfacial system entries as the solution of small boundary value problems on a circle. This is both extremely efficient (as they can be solved in parallel and by the pseudospectral method) and free of Monte Carlo error. Stochastic numerics are only needed on the relatively few circles intersecting the domain boundary. In sum, the new formulation is significantly faster, simpler and more accurate, while retaining all of the advantageous properties of PDDSparse. Numerical experiments are included for the purpose of illustration

Constrained portfolio choices in the decumulation phase of a pension plan

This paper deals with a constrained investment problem for a defined contribution (DC) pension fund where retirees are allowed to defer the purchase of the annuity at some future time after retirement. This problem has already been treated in the unconstrained case in a number of papers. The aim of this work is to deal with the more realistic case when constraints on the investment strategies and on the state variable are present. Due to the difficulty of the task, we consider the basic model of [Gerrard, Haberman & Vigna, 2004], where interim consumption and annuitization time are fixed. The main goal is to find the optimal portfolio choice to be adopted by the retiree from retirement to annuitization time in a Black and Scholes financial market. We define and study the problem at two different complexity levels. In the first level (problem P1), we only require no short-selling. In the second level (problem P2), we add a constraint on the state variable, by imposing that the final fund cannot be lower than a certain guaranteed safety level. This implies, in particular, no ruin. The mathematical problem is naturally formulated as a stochastic control problem with constraints on the control and the state variable, and is approached by the dynamic programming method. We give a general result of existence and uniqueness of regular solutions for the Hamilton-Jacobi-Bellman equation and, in a special case, we explicitly compute the value function for the problem and give the optimal strategy in feedback form. A numerical application of the special case - when explicit solutions are available - ends the paper and shows the extent of applicability of the model to a DC pension fund in the decumulation phase.pension fund; decumulation phase; constrained portfolio; stochastic optimal control; dynamic programming; Hamilton-Jacobi-Bellman equation