Hamilton-Jacobi Partial Differential Equations Research Articles

Efficient Computation of State-Constrained Reachability Problems Using Hopf–Lax Formulae

This paper considers two state-constrained reachability problems: computing (1) control-invariant and (2) reach-avoid sets, both under state constraints. Prior research has developed Hamilton-Jacobi (HJ) partial differential equations (PDEs) that characterize the optimal cost functions of these two problems. Unfortunately, solving the HJ PDEs by grid-based methods, such as level-set methods, suffers from exponentially growing computational complexity in the system state dimension. In order to alleviate this computational issue, this paper proposes a Hopf-Lax formula for each reachability problem's HJ PDE. The advantage of the Hopf-Lax formulae is that they have more favorable convexity conditions than the corresponding problems. Thus, direct methods may be used to solve Hopf-Lax formulae and thus efficiently compute the optimal solution of the reachability problems under specified conditions. This paper provides an example for each reachability problem and demonstrates the performance of the proposed Hopf-Lax method.

A Hamilton–Jacobi PDE Associated with Hydrodynamic Fluctuations from a Nonlinear Diffusion Equation

We study a class of Hamilton–Jacobi partial differential equations in the space of probability measures. In the first part of this paper, we prove comparison principles (implying uniqueness) for this class. In the second part, we establish the existence of a solution and give a representation using a family of partial differential equations with control. A large part of our analysis exploits special structures of the Hamiltonian, which might look mysterious at first sight. However, we show that this Hamiltonian structure arises naturally as limit of Hamiltonians of microscopical models. Indeed, in the third part of this paper, we informally derive the Hamiltonian studied before, in a context of fluctuation theory on the hydrodynamic scale. The analysis is carried out for a specific model of stochastic interacting particles in gas kinetics, namely a version of the Carleman model. We use a two-scale averaging method on Hamiltonians defined in the space of probability measures to derive the limiting Hamiltonian.

On Bayesian Posterior Mean Estimators in Imaging Sciences and Hamilton–Jacobi Partial Differential Equations

Variational and Bayesian methods are two widely used set of approaches to solve image denoising problems. In a Bayesian setting, these approaches correspond, respectively, to using maximum a posteriori estimators and posterior mean estimators for reconstructing images. In this paper, we propose novel theoretical connections between Hamilton–Jacobi partial differential equations (HJ PDEs) and a broad class of posterior mean estimators with quadratic data fidelity term and log-concave prior. Where solutions to some first-order HJ PDEs with initial data describe maximum a posteriori estimators, here we show that solutions to some viscous HJ PDEs with initial data describe a broad class of posterior mean estimators. We use these connections to establish representation formulas and various properties of posterior mean estimators. In particular, we use these connections to show that some Bayesian posterior mean estimators can be expressed as proximal mappings of smooth functions and derive representation formulas for these functions.

A Parametric Level Set Method for Topology Optimization Based on Deep Neural Network

Abstract This paper proposes a new parametric level set method for topology optimization based on deep neural network (DNN). In this method, the fully connected DNN is incorporated into the conventional level set methods to construct an effective approach for structural topology optimization. The implicit function of level set is described by fully connected DNNs. A DNN-based level set optimization method is proposed, where the Hamilton–Jacobi partial differential equations (PDEs) are transformed into parametrized ordinary differential equations (ODEs). The zero-level set of implicit function is updated through updating the weights and biases of networks. The parametrized reinitialization is applied periodically to prevent the implicit function from being too steep or too flat in the vicinity of its zero-level set. The proposed method is implemented in the framework of minimum compliance, which is a well-known benchmark for topology optimization. In practice, designers desire to have multiple design options, where they can choose a better conceptual design base on their design experience. One of the major advantages of the DNN-based level set method is capable to generate diverse and competitive designs with different network architectures. Several numerical examples are presented to verify the effectiveness of the proposed DNN-based level set method.

On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton–Jacobi partial differential equations

We propose novel connections between several neural network architectures and viscosity solutions of some Hamilton–Jacobi (HJ) partial differential equations (PDEs) whose Hamiltonian is convex and only depends on the spatial gradient of the solution. To be specific, we prove that under certain assumptions, the two neural network architectures we proposed represent viscosity solutions to two sets of HJ PDEs with zero error. We also implement our proposed neural network architectures using Tensorflow and provide several examples and illustrations. Note that these neural network representations can avoid curve of dimensionality for certain HJ PDEs, since they do not involve neither grids nor discretization. Our results suggest that efficient dedicated hardware implementation for neural networks can be leveraged to evaluate viscosity solutions of certain HJ PDEs.

Overcoming the curse of dimensionality for some Hamilton–Jacobi partial differential equations via neural network architectures

We propose new and original mathematical connections between Hamilton–Jacobi (HJ) partial differential equations (PDEs) with initial data and neural network architectures. Specifically, we prove that some classes of neural networks correspond to representation formulas of HJ PDE solutions whose Hamiltonians and initial data are obtained from the parameters of the neural networks. These results do not rely on universal approximation properties of neural networks; rather, our results show that some classes of neural network architectures naturally encode the physics contained in some HJ PDEs. Our results naturally yield efficient neural network-based methods for evaluating solutions of some HJ PDEs in high dimension without using grids or numerical approximations. We also present some numerical results for solving some inverse problems involving HJ PDEs using our proposed architectures.

Conversion of Certain Stochastic Control Problems into Deterministic Control Problems

A class of nonlinear, stochastic staticization control problems (including minimization problems with smooth, convex, coercive payoffs) driven by diffusion dynamics with constant diffusion coefficient is considered. A fundamental solution form is obtained where the same solution can be used for a limited variety of terminal costs without re-solution of the problem. One may convert this fundamental solution form from a stochastic control problem form to a deterministic control problem form. This yields an equivalence between certain second-order (in space) Hamilton-Jacobi partial differential equations (HJ PDEs) and associated first-order HJ PDEs. This reformulation has substantial numerical implications.

Minimax Solution of Functional Hamilton-Jacobi Equations for Neutral Type Systems

We consider the Cauchy problem for a functional Hamilton-Jacobi equation with coinvariant derivatives corresponding to dynamical systems of the neutral type. A definition of minimax (generalized) solution of this problem is given and its existence, uniqueness, and also continuous dependence on the parameters are proved. The dependence of the minimax solution on information images is established, which, in particular, permits showing the consistency of the introduced definition with the definition of minimax solution for Hamilton-Jacobi partial differential equations.

Approximation of Minimax Solutions of Hamilton—Jacobi Functional Equations for Time-Delay Systems

A minimax solution of the Cauchy problem for a functional Hamilton-Jacobi equation with coinvariant derivatives and a condition at the right end is considered. Hamilton-Jacobi equations of this type arise in dynamical optimization problems for time-delay systems. Their approximation is associated with additional issues of a valid transition from an infinite-dimensional functional argument of the desired solution to a finite-dimensional argument. Earlier, the schemes based on the piecewise linear approximation of the functional argument and the correctness properties of minimax solutions were studied. In this paper, a scheme for the approximation of Hamilton-Jacobi functional equations with coinvariant derivatives by usual Hamilton-Jacobi partial differential equations is proposed and justified. The scheme is based on the approximation of the characteristic functional-differential inclusions used in the definition of the desired minimax solution by ordinary differential inclusions.

Algorithm for overcoming the curse of dimensionality for state-dependent Hamilton-Jacobi equations

In this paper, we develop algorithms to overcome the curse of dimensionality in non-convex state-dependent Hamilton-Jacobi partial differential equations (HJ PDEs) arising from optimal control and differential game problems. The subproblems are independent and they can be implemented in an embarrassingly parallel fashion. This is ideal for perfect scaling in parallel computing. The algorithm is proposed to overcome the curse of dimensionality [1,2] when solving HJ PDE. The major contribution of the paper is to change either the solving of a PDE problem or an optimization problem over a space of curves to an optimization problem of a single vector, which goes beyond the work of [40]. We extend the method in [7,9,15], and conjecture a (Lax-type) minimization principle to solve state-dependent HJ PDE when the Hamiltonian is convex, as well as a (Hopf-type) maximization principle to solve state-dependent HJ PDE when the Hamiltonian is non-convex, as a generalization of the well-known Hopf formula in [18,25,50]. We showed the validity of the formula under restricted assumption for the sake of completeness, and would like to bring our readers to [62] which validates our conjectures in a more general setting. We conjectured the weakest assumption of our formula to hold is a pseudoconvexity assumption similar to one stated in [50]. Our method is expected to have application in control theory, differential game problems and elsewhere.

Quantization of fractional singular Lagrangian systems using WKB approximation

In this paper, the fractional singular Lagrangian system is studied. The Hamilton–Jacobi treatment is developed to be applicable for fractional singular Lagrangian system. The equations of motion are obtained for the fractional singular systems and the Hamilton–Jacobi partial differential equations are obtained and solved to determine the action integral. Then the wave function for fractional singular system is obtained. Besides, to demonstrate this theory, the fractional Christ-Lee model is discussed and quantized using the WKB approximation.

The Hamilton–Jacobi partial differential equation and the three representations of traffic flow

This paper applies the theory of Hamilton–Jacobi partial differential equations to the case of first-order traffic flow models. The traffic flow surface is analyzed with respect to the three 2-dimensional coordinate systems arising in the space of vehicle number, time and distance. In each case, the solution to the initial and boundary value problems are presented. Explicit solution methods and examples are shown for the triangular flow-density diagram case. This unveils new models and shows how a number of existing models are cast as special cases.

An implicit geometrical approach to level sets update for 3D non planar X-FEM crack propagation

In the X-FEM framework the crack geometry is described independently of the structural mesh, usually by means of two level set functions, thanks to the versatility and wide applicability of the level set methodology. The main difficulty in the adoption of the level set approach is the formulation of an efficient and robust update scheme allowing to calculate the new level set values associated to the new crack configuration after a propagation step. In this paper an implicit geometrical methodology is formulated in order to overcome the stability and performance issues affecting the resolution of the Hamilton–Jacobi partial differential equations governing the level set update. The calculation of the new level sets values is based on geometrical considerations only, which enormously reduces the computational time and increases the robustness of the update calculations. In contrast to other geometrical approaches, this methodology can be applied both to 2D and 3D problems and the crack front and surface are always implicitly described by means of the two level sets. This avoids both the need to describe somehow the crack surface and front and the creation of algorithms to track their evolution, which improves further the performance of the methodology. The numerical tests reported in the paper show that the proposed methodology is 30 times faster in 2D and 3–6 times faster in 3D than classic approaches based on the resolution of the Hamilton–Jacobi equations, and 2–7 times faster than explicit geometrical approaches. The proposed methodology can also manage the separation of a crack front in several fronts during the propagation and the merging of several fronts into one single front.

A meshfree level-set method for topological shape optimization of compliant multiphysics actuators

This paper proposes a topology optimization method for compliant multiphysics actuators of geometrically nonlinear structures using meshfree Galerkin weak-forms and level set methods. The design boundary is implicitly represented as the zero level set of a higher-dimensional level set function, leading to a level set model capable of handling complex shape and topological changes with flexibilities. A family of compactly supported radial basis functions (CSRBFs) is firstly used to interpolate the level set function of Lipschitz continuity, and then augmented to construct the shape function for meshless approximation by satisfying basic requirements, in particular the predetermined consistency and the Kronecker delta function property. A meshless Galerkin method (MGM) with global weak-forms is established to implement the discretization of the state equations. The design of actuators is transformed into an easier size optimization from a more difficult shape and topology optimization. The design boundary evolution is just a question of advancing the discrete level set function in time by updating the design variables of the size optimization. Compared to most conventional level set methods, the proposed meshless level set method is able to implement the free moving boundary discontinuities without remeshing, and unify two different numerical procedures in propagating the discrete level set function (e.g. Eulerian grid) and approximating the state equation (e.g. Lagrangian mesh), respectively. This method can also avoid numerical difficulties in solving a series of complicate Hamilton–Jacobi partial differential equations (PDEs) with explicit time schemes. Two typical numerical examples are used to demonstrate the effectiveness of the proposed method.

An Ordered Upwind Method with Precomputed Stencil and Monotone Node Acceptance for Solving Static Convex Hamilton-Jacobi Equations

We define a ?-causal discretization of static convex Hamilton-Jacobi Partial Differential Equations (HJ PDEs) such that the solution value at a grid node is dependent only on solution values that are smaller by at least ?. We develop a Monotone Acceptance Ordered Upwind Method (MAOUM) that first determines a consistent, ?-causal stencil for each grid node and then solves the discrete equation in a single-pass through the nodes. MAOUM is suited to solving HJ PDEs efficiently on highly-nonuniform grids, since the stencil size adjusts to the level of grid refinement. MAOUM is a Dijkstra-like algorithm that computes the solution in increasing value order by using a heap to sort proposed node values. If ?>0, MAOUM can be converted to a Dial-like algorithm that sorts and accepts values using buckets of width ?. We present three hierarchical criteria for ?-causality of a node value update from a simplex of nodes in the stencil. The asymptotic complexity of MAOUM is found to be $\mathcal {O}((\hat{\Psi}\rho )^{d} N \log N)$ , where d is the dimension, $\hat{\Psi}$ is a measure of anisotropy in the equation, and ? is a measure of the degree of nonuniformity in the grid. This complexity is a constant factor $(\hat{\Psi}\rho)^{d}$ greater than that of the Dijkstra-like Fast Marching Method, but MAOUM solves a much more general class of static HJ PDEs. Although ? factors into the asymptotic complexity, experiments demonstrate that grid nonuniformity does not have a large effect on the computational cost of MAOUM in practice. Our experiments indicate that, due to the stencil initialization overhead, MAOUM performs similarly or slightly worse than the comparable Ordered Upwind Method presented in (Sethian and Vladimirsky, SIAM J. Numer. Anal. 41:323, 2003) for two examples on uniform meshes, but considerably better for an example with rectangular speed profile and significant grid refinement around nonsmooth parts of the solution. We test MAOUM on a diverse set of examples, including seismic wavefront propagation and robotic navigation with wind and obstacles.

Viability Solutions to Structured Hamilton–Jacobi Equations under Constraints

Structured Hamilton–Jacobi partial differential equations are Hamilton–Jacobi equations where the time variable is replaced by a vector-valued variable “structuring” the system. It could be the time-age pair (Hamilton–Jacobi–McKendrick equations) or candidates for initial or terminal conditions (Hamilton–Jacobi–Cournot equations) among a manifold of examples. Here, we define the concept of “viability solution” which always exists and can be computed by viability algorithms. This is a constructive approach allowing us to derive from the tools of viability theory (dealing with sets instead of functions) some known as well as new properties of its solutions. Regarding functions as their epigraphs, we bypass the regularity issues to arrive directly at the concept of Barron–Jensen/Frankowska viscosity solutions. Above all we take into account viability constraints and extend classical boundary conditions to other “internal” conditions. Beyond that, we use the Fenchel–Legendre transform to associate a Lagrangian with the Hamiltonian and uncover an underlying variational problem. It is proved that the viability solution to the structured Hamilton–Jacobi equation is the valuation function of this variational principle, the optimal evolutions of which are regulated by a “regulation map” which is constructed from the viability solution and can be computed by viability algorithms. In other words, the viability solution solves all at once a class of first-order partial differential equations of intertemporal optimization problems and the regulation of their optimal evolutions.

Lax–Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton–Jacobi Equation. Part I: Theory

This article proposes a new approach for computing a semi-explicit form of the solution to a class of Hamilton-Jacobi (HJ) partial differential equations (PDEs), using control techniques based on viability theory. We characterize the epigraph of the value function solving the HJ PDE as a capture basin of a target through an auxiliary dynamical system, called ?characteristic system?. The properties of capture basins enable us to define components as building blocks of the solution to the HJ PDE in the Barron/Jensen-Frankowska sense. These components can encode initial conditions, boundary conditions, and internal ?boundary? conditions, which are the topic of this article. A generalized Lax-Hopf formula is derived, and enables us to formulate the necessary and sufficient conditions for a mixed initial and boundary conditions problem with multiple internal boundary conditions to be well posed. We illustrate the capabilities of the method with a data assimilation problem for reconstruction of highway traffic flow using Lagrangian measurements generated from Next Generation Simulation (NGSIM) traffic data.

Games with Exhaustible Resources

We study N-player continuous-time Cournot games in an oligopoly where firms choose production quantities. These are nonzero-sum differential games, whose value functions may be characterized by systems of nonlinear Hamilton-Jacobi partial differential equations. When resources are in finite supply, such as oil, exhaustibility enters as boundary conditions for the PDEs. We analyze the problem when there is an alternative, but expensive, technology (for example solar power for energy production), and give an asymptotic approximation in the limit of small exhaustibility. We illustrate the two-player problem by numerical solutions, and discuss the impact of limited oil reserves on production and oil prices in the duopoly case.

Stochastic homogenization of Hamilon-Jacobi and "viscous"-Hamilton-Jacobi equations with convex nonlinearities -- Revisited

In this note we revisit the homogenization theory of Hamilton-Jacobi and “viscous”- Hamilton-Jacobi partial differential equations with convex nonlinearities in stationary ergodic envi- ronments. We present a new simple proof for the homogenization in probability. The argument uses some a priori bounds (uniform modulus of continuity) on the solution and the convexity and coer- civity (growth) of the nonlinearity. It does not rely, however, on the control interpretation formula of the solution as was the case with all previously known proofs. We also introduce a new formula for the effective Hamiltonian for Hamilton-Jacobi and “viscous” Hamilton-Jacobi equations.

The problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures

We obtain a linear programming characterization for the minimum cost associated with finite dimensional reflected optimal control problems. In order to describe the value functions, we employ an infinite dimensional dual formulation instead of using the characterization via Hamilton–Jacobi partial differential equations. In this paper we consider control problems with both infinite and finite horizons. The reflection is given by the normal cone to a proximal retract set.