Nonlinear Parabolic PDEs Research Articles

Error analysis for the approximation of axisymmetric Willmore flow by $C^1$-finite elements

We consider the Willmore flow of axially symmetric surfaces subject to Dirichlet boundary conditions. The corresponding evolution is described by a nonlinear parabolic PDE of fourth order for the radius function. A suitable weak form of the equation, which is based on the first variation of the Willmore energy, leads to a semidiscrete scheme, in which we employ piecewise cubic C^1 -finite elements for the one-dimensional approximation in space.We prove optimal error bounds in Sobolev norms for the solution and its time derivative and present numerical test examples.

An efficient approach for the numerical solution of the Monge–Ampère equation

In this paper, we present a new method to compute the numerical solution of the elliptic Monge–Ampère equation. This method is based on solving a parabolic Monge–Ampère equation for the steady state solution. We study the problem of global existence, uniqueness, and convergence of the solution of the fully nonlinear parabolic PDE to the unique solution of the elliptic Monge–Ampère equation. Some numerical experiments are presented to show the convergence and the regularity of the numerical solution.

Wave front behavior of traveling wave solutions for a PDE having square-root dynamics

AbstractOur main goal is to investigate the asymptotic behavior of traveling wave solutions to a nonlinear parabolic PDE having square-root dynamics in its reaction term. To calculate this result, we apply the method of dominant balance.

Max-Plus Stochastic Control and Risk-Sensitivity

In the Maslov idempotent probability calculus, expectations of random variables are defined so as to be linear with respect to max-plus addition and scalar multiplication. This paper considers control problems in which the objective is to minimize the max-plus expectation of some max-plus additive running cost. Such problems arise naturally as limits of some types of risk sensitive stochastic control problems. The value function is a viscosity solution to a quasivariational inequality (QVI) of dynamic programming. Equivalence of this QVI to a nonlinear parabolic PDE with discontinuous Hamiltonian is used to prove a comparison theorem for viscosity sub- and super-solutions. An example from mathematical finance is given, and an application in nonlinear H-infinity control is sketched.

Uncertain Volatility Model: A Monte-Carlo Approach

The uncertain volatility model has long ago attracted the attention of practitioners as it provides worst-case pricing scenario for the sell-side. The valuation of a financial derivative based on this model requires solving a fully non-linear PDE. One can rely on finite difference schemes only when the number of variables (that is, underlyings and path-dependent variables) is small - in practice no more than three. In all other cases, numerical valuation seems out of reach. In this paper, we outline two accurate, easy-to-implement Monte-Carlo-like methods which hardly depend on dimensionality. The first method requires a parameterization of the optimal covariance matrix and consists in a series of backward low-dimensional optimizations. The second method relies heavily on a recently established connection between second-order backward stochastic differential equations and non-linear second-order parabolic PDEs. Both methods are illustrated by numerical experiments.

On the Mathematical Analysis and Numerical Approximation of a System of Nonlinear Parabolic PDEs

In this paper we consider a boundary value problem for a system of 2 nonlinear parabolic PDEs e.g. arising in the context of flow and transport in porous media. The flow model is based on tho nonlinear Richard’s equation problem and is combined with the transport equation through saturation and Darcy’s velocity (discharge) terms. The convective terms are approximated by means of the method of characteristics initiated by P. Pironneau [Num. Math. 38 (1982), 871–885] and R. Douglas and T. F. Russel [SIAM J. Num. Anal. 19 (1982), 309–332]. The nonlinear terms in Richard’s equation are approximated by means of a relaxation scheme applied by W. Jäger and J. Kačur [RAIRO Model. Math. Anal. Num. 29 (1995), 605–627] and J. Kačur [IMA J. Num. Anal. 19 (1999), 119–154; SIAM J. Num. Anal. 39 (1999), c 290–316]. The convergence of the approximation method is proved.

Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods

We derive a posteriori error estimates, which exhibit optimal global order, for a class of time stepping methods of any order that include Runge–Kutta Collocation (RK-C) methods and the continuous Galerkin (cG) method for linear and nonlinear stiff ODEs and parabolic PDEs. The key ingredients in deriving these bounds are appropriate one-degree higher continuous reconstructions of the approximate solutions and pointwise error representations. The reconstructions are based on rather general orthogonality properties and lead to upper and lower bounds for the error regardless of the time-step; they do not hinge on asymptotics.

Optimal Control under Stochastic Target Constraints

We study a class of Markovian optimal stochastic control problems in which the controlled process $Z^{\nu}$ is constrained to satisfy an almost sure constraint $Z^{\nu}(T)\in G\subset\mathbb{R}^{d+1}$ $\mathbb{P}$-a.s. at some final time $T>0$. When the set is of the form $G:=\{(x,y)\in\mathbb{R}^d\times\mathbb{R}:g(x,y)\geq0\}$, with g nondecreasing in y, we provide a Hamilton–Jacobi–Bellman characterization of the associated value function. It gives rise to a state constraint problem, where the constraint can be expressed in terms of an auxiliary value function w which characterizes the set $D:=\{(t,Z^{\nu}(t))\in[0,T]\times\mathbb{R}^{d+1}:Z^{\nu}(T)\in G$ a.s. for some $\nu\}$. Contrary to standard state constraint problems, the domain D is not given a priori and we do not need to impose conditions on its boundary. It is naturally incorporated in the auxiliary value function w, which itself is a viscosity solution of a nonlinear parabolic PDE. Applying ideas recently developed in Bouchard, Elie, and Touzi [SIAM J. Control Optim., 48 (2009), pp. 3123–3150], our general result also allows us to consider optimal control problems with moment constraints of the form $\mathbb{E}[g(Z^{\nu}(T))]\geq0$ or $\mathbb{P}[g(Z^{\nu}(T))\geq0]\geq p$.

Control of 1D parabolic PDEs with Volterra nonlinearities, Part II: Analysis

For a class of stabilizing boundary controllers for nonlinear 1D parabolic PDEs introduced in a companion paper, we derive bounds for the gain kernels of our nonlinear Volterra controllers, prove the convergence of the series in the feedback laws, and establish the stability properties of the closed-loop system. We show that the state transformation is at least locally invertible and include an explicit construction for computing the inverse of the transformation. Using the inverse, we show L 2 and H 1 exponential stability and explicitly construct the exponentially decaying closed-loop solutions. We then illustrate the theoretical results on an analytically tractable example.

Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolic PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains T n of increasing dimensions n + 1 and with a domain shape in the form of a “hyper-pyramid”, 0 ≤ ξ n ≤ ξ n − 1 ⋯ ≤ ξ 1 ≤ x ≤ 1 . We illustrate our design method with several examples. One of the examples is analytical, while in the remaining two examples the controller is numerically approximated. For all the examples we include simulations, showing blow up in open loop, and stabilization for large initial conditions in closed loop. In a companion paper we give a theoretical study of the properties of the transformation, showing global convergence of the transformation and of the control law nonlinear Volterra operators, and explicitly constructing the inverse of the feedback linearizing Volterra transformation; this, in turn, allows us to prove L 2 and H 1 local exponential stability (with an estimate of the region of attraction where possible) and explicitly construct the exponentially decaying closed loop solutions.

Multiscale problems and homogenization for second-order Hamilton–Jacobi equations

We prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic PDEs. As a special case we cover the iterated homogenization for such equations with oscillating initial data. Explicit examples, among others, are the two-scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n-scale homogenization of uniformly parabolic fully nonlinear PDEs.

BOUNDARY CONTROL LAWS FOR PARABOLIC PDES WITH VOLTERRA NONLINEARITIES—PART I: DESIGN

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolyc PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains Tn of increasing dimensions n +1 and with a domain shape in the form of a “hyper-pyramid,” 0 ≤ ζn ≤ ζn–1... ≤ ζ1 ≤ x ≤ 1. This is the first part of a three-part paper; in the second part we illustrate our design method with several examples and in the third part we provide a theoretical study of the properties of the transformation.

NONLINEAR CONTROL OF PDES: ARE FEEDBACK LINEARIZATION AND GEOMETRIC METHODS APPLICABLE?

Feedback linearization is a natural first step in developing nonlinear control design methods for nonlinear PDEs. Feedback linearization procedures for nonlinear ODEs recursively absorb all the plant nonlinearities into a feedback transformation. The resulting transformation often involves nonlinearities of much higher growth than the plant nonlinearities. For example, systems with n states and only quadratic nonlinearities lead to feedback linearizing controllers of polynomial power up to n +1. Intuitively, one would worry that an infinite-step feedback linearization procedure may result in polynomial powers that go to infinity. We present a framework developed over the last eight years in which a boundary control design for nonlinear parabolic PDEs has a well defined limit. This approach opens the door for future possible developments of differential geometric tests of linearizability and controllability for nonlinear PDEs and for generalizations to various other classes of PDEs, including nonlinear Navier-Stokes models of fluid turbulence. As a starting point in this direction, we review the current status of the Lyapunov stabilization techniques for the linearized Navier-Stokes and magnetohydrodynamic PDEs.

A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem

A new adaptive local mesh refinement method is presented for thin film flow problems containing moving contact lines. Based on adaptation on an optimal interpolation error estimate in the Lp norm (1<p⩽∞) [L. Chen, P. Sun, J. Xu, Multilevel homotopic adaptive finite element methods for convection dominated problems, in: Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering 40 (2004) 459–468], we obtain the optimal anisotropic adaptive meshes in terms of the Hessian matrix of the numerical solution. Such an anisotropic mesh is optimal for anisotropic solutions like the solution of thin film equations on moving contact lines. Thin film flow is described by an important type of nonlinear degenerate fourth order parabolic PDE. In this paper, we address the algorithms and implementation of the new adaptive finite element method for solving such fourth order thin film equations. By means of the resulting algorithm, we are able to capture and resolve the moving contact lines very precisely and efficiently without using any regularization method, even for the extreme degenerate cases, but with fewer grid points and degrees of freedom in contrast to methods on a fixed mesh. As well, we compare the method theoretically and computationally to the positivity-preserving finite difference scheme on a fixed uniform mesh which has proven useful for solving the thin film problem.

Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs

AbstractFor a d‐dimensional diffusion of the form dXt = μ(Xt)dt + σ(Xt)dWt and continuous functions f and g, we study the existence and uniqueness of adapted processes Y, Z, Γ, and A solving the second‐order backward stochastic differential equation (2BSDE) If the associated PDE has a sufficiently regular solution, then it follows directly from Itô's formula that the processes solve the 2BSDE, where 𝓁 is the Dynkin operator of X without the drift term.The main result of the paper shows that if f is Lipschitz in Y as well as decreasing in Γ and the PDE satisfies a comparison principle as in the theory of viscosity solutions, then the existence of a solution (Y, Z,Γ, A) to the 2BSDE implies that the associated PDE has a unique continuous viscosity solution v and the process Y is of the form Yt = v(t, Xt), t ∈ [0, T]. In particular, the 2BSDE has at most one solution. This provides a stochastic representation for solutions of fully nonlinear parabolic PDEs. As a consequence, the numerical treatment of such PDEs can now be approached by Monte Carlo methods. © 2006 Wiley Periodicals, Inc.

A probabilistic representation for the solutions to some non-linear PDEs using pruned branching trees

The solutions to a large class of non-linear parabolic PDEs are given in terms of expectations of suitable functionals of a tree of branching particles. A sufficient, and in some cases necessary, condition is given for the integrability of the stochastic representation, using a comparison scalar PDE. In cases where the representation fails to be integrable, a sequence of pruned trees is constructed, producing approximate stochastic representations that in some cases converge, globally in time, to the solution of the original PDE.

Existence of weak solutions to a system of nonlinear partial differential equations modelling ice streams

This paper deals with the mathematical analysis of a nonlinear system of three differential equations of mixed type. It describes the generation of fast ice streams in ice sheets flowing along soft and deformable beds. The system involves a nonlinear parabolic PDE with a multivalued term in order to deal properly with a free boundary which is naturally associated to the problem of determining the basal water flux in a drainage system. The other two equations in the system are an ODE with a nonlocal (integral) term for the ice thickness, which accounts for mass conservation and a first order PDE describing the ice velocity of the system. We first consider an iterative decoupling procedure to the system equations to obtain the existence and uniqueness of solutions for the uncoupled problems. Then we prove the convergence of the iterative decoupling scheme to a bounded weak solution for the original system.

Evolution equations for nonlinear degenerate parabolic PDE

We define a convex function on H - 1 ( Ω ) and characterize its subdifferential, by which we introduce an evolution equation as a weak formulation of a class of nonlinear, degenerate, singular and noncoercive parabolic PDEs associated with an arbitrary maximal monotone graph and with the Dirichlet boundary condition.

Stability and convergence of a finite volume method for two systems of reaction-diffusion equations in electro-cardiology

The monodomain equations model the propagation of the action potential in the human heart: a very sharp pulse propagating at a high speed, whose computation requires fine unstructured 3D meshes. It is a non-linear parabolic PDE of reaction-diffusion type, coupled to one or several ODE, with multiple time-scales. Numerical difficulties, such as unstructured meshes and stability are addressed here through the use of a finite volume method. Stability conditions are given for two time-stepping methods, and two example sets of ODEs, convergence is proved and error estimates are computed.

Predictive control of transport-reaction processes

This work focuses on the development of computationally efficient predictive control algorithms for nonlinear parabolic and hyperbolic PDEs with state and control constraints arising in the context of transport-reaction processes. We first consider a diffusion-reaction process described by a nonlinear parabolic PDE and address the problem of stabilization of an unstable steady-state subject to input and state constraints. Galerkin’s method is used to derive finite-dimensional systems that capture the dominant dynamics of the parabolic PDE, which are subsequently used for controller design. Various model predictive control (MPC) formulations are constructed on the basis of the finite dimensional approximations and are demonstrated, through simulation, to achieve the control objectives. We then consider a convection-reaction process example described by a set of hyperbolic PDEs and address the problem of stabilization of the desired steady-state subject to input and state constraints, in the presence of disturbances. An easily implementable predictive controller based on a finite dimensional approximation of the PDE obtained by the finite difference method is derived and demonstrated, via simulation, to achieve the control objective.