24 research outputs found
Discretization of the 3D Monge-Ampere operator, between Wide Stencils and Power Diagrams
We introduce a monotone (degenerate elliptic) discretization of the
Monge-Ampere operator, on domains discretized on cartesian grids. The scheme is
consistent provided the solution hessian condition number is uniformly bounded.
Our approach enjoys the simplicity of the Wide Stencil method, but
significantly improves its accuracy using ideas from discretizations of optimal
transport based on power diagrams. We establish the global convergence of a
damped Newton solver for the discrete system of equations. Numerical
experiments, in three dimensions, illustrate the scheme efficiency
Monotone and Consistent discretization of the Monge-Ampere operator
We introduce a novel discretization of the Monge-Ampere operator,
simultaneously consistent and degenerate elliptic, hence accurate and robust in
applications. These properties are achieved by exploiting the arithmetic
structure of the discrete domain, assumed to be a two dimensional cartesian
grid. The construction of our scheme is simple, but its analysis relies on
original tools seldom encountered in numerical analysis, such as the geometry
of two dimensional lattices, and an arithmetic structure called the
Stern-Brocot tree. Numerical experiments illustrate the method's efficiency
Minimal convex extensions and finite difference discretization of the quadratic Monge-Kantorovich problem
We present an adaptation of the MA-LBR scheme to the Monge-Amp{\`e}re
equation with second boundary value condition, provided the target is a convex
set. This yields a fast adaptive method to numerically solve the Optimal
Transport problem between two absolutely continuous measures, the second of
which has convex support. The proposed numerical method actually captures a
specific Brenier solution which is minimal in some sense. We prove the
convergence of the method as the grid stepsize vanishes and we show with
numerical experiments that it is able to reproduce subtle properties of the
Optimal Transport problem
Eigenvalue problems for fully nonlinear elliptic partial differential equations with transport boundary conditions
Fully nonlinear elliptic partial differential equations (PDEs) arise in a number of applications. From mathematical finance to astrophysics, there is a great deal of interest in solving them. Eigenvalue problems for fully nonlinear PDEs with transport boundary conditions are of particular interest as alternative formulations of PDEs that require data to satisfy a solvability condition, which may not be known explicitly or may be polluted by noisy data. Nevertheless, these have not yet been well-explored in the literature. In this dissertation, a convergence framework for numerically solving eigenvalue problems for fully nonlinear PDEs is introduced. In addition, existing two-dimensional methods for nonlinear equations are extended to handle transport boundary conditions and eigenvalue problems. Finally, new techniques are designed to enable appropriate discretization of a large range of fully nonlinear three-dimensional equations
Monotone discretizations of levelset convex geometric PDEs
We introduce a novel algorithm that converges to level-set convex viscosity
solutions of high-dimensional Hamilton-Jacobi equations. The algorithm is
applicable to a broad class of curvature motion PDEs, as well as a recently
developed Hamilton-Jacobi equation for the Tukey depth, which is a statistical
depth measure of data points. A main contribution of our work is a new monotone
scheme for approximating the direction of the gradient, which allows for
monotone discretizations of pure partial derivatives in the direction of, and
orthogonal to, the gradient. We provide a convergence analysis of the algorithm
on both regular Cartesian grids and unstructured point clouds in any dimension
and present numerical experiments that demonstrate the effectiveness of the
algorithm in approximating solutions of the affine flow in two dimensions and
the Tukey depth measure of high-dimensional datasets such as MNIST and
FashionMNIST.Comment: 42 pages including reference
Monotone and Consistent discretization of the Monge-Ampere operator
International audienceWe introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency
Impulse Control in Finance: Numerical Methods and Viscosity Solutions
The goal of this thesis is to provide efficient and provably convergent
numerical methods for solving partial differential equations (PDEs) coming from
impulse control problems motivated by finance. Impulses, which are controlled
jumps in a stochastic process, are used to model realistic features in
financial problems which cannot be captured by ordinary stochastic controls.
The dynamic programming equations associated with impulse control problems
are Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) Other than
in certain special cases, the numerical schemes that come from the
discretization of HJBQVIs take the form of complicated nonlinear matrix
equations also known as Bellman problems. We prove that a policy iteration
algorithm can be used to compute their solutions. In order to do so, we employ
the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a
byproduct of our analysis, we obtain some new results regarding a particular
family of Markov decision processes which can be thought of as impulse control
problems on a discrete state space and the relationship between w.c.d.d.
matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to
directly use the seminal result of Barles and Souganidis (concerning the
convergence of monotone, stable, and consistent numerical schemes to the
viscosity solution) to prove the convergence of our schemes. We address this
issue by extending the work of Barles and Souganidis to nonlocal PDEs in a
manner general enough to apply to HJBQVIs. We apply our schemes to compute the
solutions of various classical problems from finance concerning optimal control
of the exchange rate, optimal consumption with fixed and proportional
transaction costs, and guaranteed minimum withdrawal benefits in variable
annuities
Impulse Control in Finance: Numerical Methods and Viscosity Solutions
The goal of this thesis is to provide efficient and provably convergent
numerical methods for solving partial differential equations (PDEs) coming from
impulse control problems motivated by finance. Impulses, which are controlled
jumps in a stochastic process, are used to model realistic features in
financial problems which cannot be captured by ordinary stochastic controls.
The dynamic programming equations associated with impulse control problems
are Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) Other than
in certain special cases, the numerical schemes that come from the
discretization of HJBQVIs take the form of complicated nonlinear matrix
equations also known as Bellman problems. We prove that a policy iteration
algorithm can be used to compute their solutions. In order to do so, we employ
the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a
byproduct of our analysis, we obtain some new results regarding a particular
family of Markov decision processes which can be thought of as impulse control
problems on a discrete state space and the relationship between w.c.d.d.
matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to
directly use the seminal result of Barles and Souganidis (concerning the
convergence of monotone, stable, and consistent numerical schemes to the
viscosity solution) to prove the convergence of our schemes. We address this
issue by extending the work of Barles and Souganidis to nonlocal PDEs in a
manner general enough to apply to HJBQVIs. We apply our schemes to compute the
solutions of various classical problems from finance concerning optimal control
of the exchange rate, optimal consumption with fixed and proportional
transaction costs, and guaranteed minimum withdrawal benefits in variable
annuities