Time-discrete Problem Research Articles

Convergence of a Robust Deep FBSDE Method for Stochastic Control

In this paper, we propose a deep learning based numerical scheme for strongly coupled forward backward stochastic differential equations (FBSDEs), stemming from stochastic control. It is a modification of the deep BSDE method in which the initial value to the backward equation is not a free parameter, and with a new loss function being the weighted sum of the cost of the control problem, and a variance term which coincides with the mean squared error in the terminal condition. We show by a numerical example that a direct extension of the classical deep BSDE method to FBSDEs fails for a simple linear-quadratic control problem, and we motivate why the new method works. Under regularity and boundedness assumptions on the exact controls of time continuous and time discrete control problems, we provide an error analysis for our method. We show empirically that the method converges for three different problems, one being the one that failed for a direct extension of the deep BSDE method.

A numerical assessment of partitioned implicit methods for thermomechanical problems

Implicit partitioned methods for solving thermomechanical problems possess various advantages over monolithic schemes, such as flexibility and simplicity of implementation, at the cost of performance and robustness. This contribution’s primary purpose is to explore suitable partitioned implicit techniques for solving the thermomechanical problem with different sources of coupling. The spatial and time-discrete problem is reformulated as an equivalent root-finding problem using single-field solvers. This work considers the fixed-point method, Aitken relaxation, multisecant quasi-Newton methods, Newton–Krylov method, and polynomial vector extrapolation techniques in cycling mode, which are selected from the relevant literature. These techniques are employed to solve three different problems, including thermoelastic and thermoplastic constitutive behaviors, as well as thermomechanical contact. The methods are compared based on the CPU runtime and the number of residual evaluations, considering as well the memory requirements and the implementation effort. Overall, Broyden’s method can be regarded as the best performing and most advantageous method, with the Aitken relaxation also providing a very efficient and simple alternative. This work brings new insights into the application of partitioned implicit methods to thermomechanical problems and can motivate the selection of these techniques, which are not currently widespread in the community.

Duality results and regularization schemes for Prandtl–Reuss perfect plasticity

We consider the time-discretized problem of the quasi-static evolution problem in perfect plasticity posed in a non-reflexive Banach space. Based on a novel equivalent reformulation in a reflexive Banach space, the primal problem is characterized as a Fenchel dual problem of the classical incremental stress problem. This allows to obtain necessary and sufficient optimality conditions for the time-discrete problems of perfect plasticity. Furthermore, the consistency of a primal-dual stabilization scheme is proven. As a consequence, not only stresses, but also displacements and strains are shown to converge to a solution of the original problem in a suitable topology. The corresponding dual problem has a simpler structure and turns out to be well-suited for numerical purposes. For the resulting subproblems an efficient algorithmic approach in the infinite-dimensional setting based on the semismooth Newton method is proposed.

Approximation Properties and Tight Bounds for Constrained Mixed-Integer Optimal Control

We extend recent work on solving mixed-integer nonlinear optimal control problems (MIOCPs) to the case of integer control functions subject to constraints that involve a pointwise coupling of the state with the integer controls. We extend a theorem due to [S. Sager, H. Bock, and M. Diehl, Math. Program. Ser. A, 133 (2012), pp. 1--23] to the case of MIOCPs with constraints on the integer control and show that the integrality gap vanishes in function space when the coarseness of the rounding grid is driven to zero even after adding constraints of this type. For the time-discretized problem, we extend a sum-up rounding (SUR) scheme due to [S. Sager, C. Reinelt, and H. Bock, Math. Program. Ser. A, 118 (2009), pp. 109--149] to the new problem class. Our scheme permits one to constructively obtain an $\varepsilon$-feasible and $\varepsilon$-optimal binary feasible control. We derive new, tight upper bounds on the integer control approximation error made by SUR. For unconstrained binary controls on equidistant grids, we reduce the approximation error bound from $\mathcal{O}(|\Omega|)$ to $\mathcal{O}(\log |\Omega|)$ asymptotically for $|\Omega| \to \infty$ and a fixed coarseness of the rounding grid, where $|\Omega|$ is the number of binary controls. For constrained binary controls, we show that the approximation problem is more difficult, and we give a proof of an approximation error bound of complexity $\mathcal{O}(|\Omega|)$. A numerical example compares our approach to a state-of-the-art mixed-integer nonlinear programming solver and illustrates the applicability of our results when solving MIOCPs using the direct and simultaneous approach.

Introduction to Solving Quant Finance Problems with Time-Stepped FBSDE and Deep Learning

In this introductory paper, we discuss how quantitative finance problems under some common risk factor dynamics for some common instruments and approaches can be formulated as time-continuous or time-discrete forward-backward stochastic differential equations (FBSDE) final-value or control problems, how these final value problems can be turned into control problems, how time-continuous problems can be turned into time-discrete problems, and how the forward and backward stochastic differential equations (SDE) can be time-stepped. We obtain both forward and backward time-stepped time-discrete stochastic control problems (where forward and backward indicate in which direction the $Y$ SDE is time-stepped) that we will solve with optimization approaches using deep neural networks for the controls and stochastic gradient and other deep learning methods for the actual optimization/learning. We close with examples for the forward and backward methods for an European option pricing problem. Several methods and approaches are new.

A dynamic programming approach for L0 optimal control design

The present work investigates the optimal control problems with L0-control cost. The value function is characterized as the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. The sparsity properties of optimal controllers induced by L0-penalty is analyzed under different cases of control constraints. The existence of optimal controllers is discussed for the time-discretized problem. The value function and the optimal control are computed by solving the corresponding HJB equation. Numerical examples are presented under different types of control constraints and different penalization parameters with special attention to the sparsity. Comparisons between L0-controller and other types of controllers are also illustrated.

The variational formulation of the fully parabolic Keller–Segel system with degenerate diffusion

We prove the time-global existence of solutions of the degenerate Keller–Segel system in higher dimensions, under the assumption that the mass of the first component is below a certain critical value. What we deal with is the full parabolic–parabolic system rather than the simplified parabolic–elliptic system. Our approach is to formulate the problem as a gradient flow on the Wasserstein space. We first consider a time-discretized problem, in which the values of the solution are determined iteratively by solving a certain minimizing problem at each time step. Here we use a new minimizing scheme at each time level, which gives the time-discretized solutions favorable regularity properties. As a consequence, it becomes relatively easy to prove that the time-discretized solutions converge to a weak solution of the original system as the time step size tends to zero.

Uniform polynomial observability of time-discrete conservative linear systems

In this paper we study time semi-discrete approximations of a class of polynomially observable infinite dimensional systems. By using a method based on the resolvent estimate, we derive uniform polynomial observability inequalities within a class of solutions of the time-discrete problem in which the high frequency components have been filtered. We also present an application of our result to stabilization problems.

Optimal Control of Quasistatic Plasticity with Linear Kinematic Hardening III: Optimality Conditions

In this paper we consider an optimal control problem governed by a rate-independent variational inequality arising in quasistatic plasticity with linear kinematic hardening. Since the solution operator of a variational inequality is not differentiable, the Karush-Kuhn-Tucker system is not a necessary optimality condition. We show a system of weakly stationary type by passing to the limit with the optimality system of a regularized and time-discretized problem.

Optimal Control of Quasistatic Plasticity with Linear Kinematic Hardening II: Regularization and Differentiability

We consider an optimal control problem governed by an evolution variational inequality arising in quasistatic plasticity with linear kinematic hardening. A regularization of the time-discrete problem is derived. The regularized forward problem can be interpreted as system of coupled quasilinear PDEs whose principal parts depend on the gradient of the state. We show the Fréchet differentiability of the solution map of this quasilinear system. As a consequence, we obtain a first order necessary optimality system. Moreover, we address certain convergence properties of the regularization.

Optimal Control of Quasi-static Plasticity with Linear Kinematic Hardening, Part I: Existence and Discretization in Time

In this paper we consider an optimal control problem governed by a time-dependent variational inequality arising in quasi-static plasticity with linear kinematic hardening. We address certain continuity properties of the forward operator, which imply the existence of an optimal control. Moreover, a discretization in time is derived and we show that every local minimizer of the continuous problem can be approximated by minimizers of modified, time-discrete problems. (A correction is attached.)

Time discretisation of monotone nonlinear evolution problems by the discontinuous Galerkin method

A class of discontinuous Galerkin methods is studied for the time discretisation of the initial-value problem for a nonlinear first-order evolution equation that is governed by a monotone, coercive, and hemicontinuous operator. The numerical solution is shown to converge towards the weak solution of the original problem. Furthermore, well-posedness of the time-discrete problem as well as a priori error estimates for sufficiently smooth exact solutions are studied.

Computational aspects in presence of negative strain-rate sensitivity with application to aluminium alloys exhibiting the Portevin–Le Chatelier effect

Loss of ellipticity of the time-discretized boundary-value problem for elastic–viscoplastic solid materials with negative strain-rate sensitivity (SRS) is studied. The physical mechanism responsible for the negative SRS is assumed to be dynamic strain ageing. For the time-discretized problem the negative SRS translates into softening and may possibly lead to loss of ellipticity when the time step is greater than a critical value. The paper provides a method to compute the critical time step and illustrates the results for an aluminium alloy.

Uniqueness, loss of ellipticity and localization for the time‐discretized, rate‐dependent boundary value problem with softening

AbstractThe discretization in time of the initial boundary value problem for rate‐dependent (elastic‐viscoplastic) solid materials in presence of softening is investigated in this paper. The emphasis is put on uniqueness, loss of ellipticity and localization. It is found that the time‐discretized problem resembles the incremental problem for rate‐independent materials and softening may lead to ill‐posedness (loss of ellipticity) if the time step is greater than a critical value. It is well established that the implication of loss of ellipticity for the numerical simulations after spatial discretization is the pathological mesh dependency of the computed results. We furnish a method to compute the critical time step and demonstrate its use for a simple example problem. Copyright © 2010 John Wiley & Sons, Ltd.

Mixed finite element method for electrowetting on dielectric with contact line pinning

We present a mixed finite element method for a model of the flow in a Hele-Shaw cell of 2-D fluid droplets surrounded by air driven by surface tension and actuated by an electric field. The application of interest regards a micro-fluidic device called ElectroWetting on Dielectric (EWOD). Our analysis first focuses on the time discrete (continuous in space) problem and is presented in a mixed variational framework, which incorporates curvature as a natural boundary condition. The model includes a viscous damping term for interface motion, as well as contact line pinning (sticking of the interface) and is captured in our formulation by a variational inequality. The semi-discrete problem uses a semiimplicit time discretization of curvature. We prove the well-posedness of the semi-discrete problem and fully discrete problem when discretized with iso-parametric finite elements. We derive a priori error estimates for the space discretization. We also prove the convergence of an Uzawa algorithm for solving the semi-discrete EWOD system with inequality constraint. We conclude with a discussion about experimental orders of convergence.

Variable time-step ϑ-scheme for nonlinear evolution equations governed by a monotone operator

The single-step ϑ-scheme on a variable time grid is employed for the approximate solution of the initial-value problem for a nonlinear first-order evolution equation. The evolution equation is supposed to be governed by a possibly time-dependent hemicontinuous operator that is (up to a shift) monotone and coercive, and fulfills a certain growth condition. A piecewise constant as well as piecewise linear prolongation of the time-discrete solution is shown to converge towards the exact solution if ϑ≥1/2 (including the Crank-Nicolson scheme). In the appearance of a strongly continuous perturbation of the monotone main part, the method is still convergent if ϑ>1/2 and if the ratio of adjacent step sizes is bounded from above by a power of ϑ/(1−ϑ). Besides convergence also well-posedness of the time-discrete problem as well as a priori error estimates are studied.

Time discrete wave equations: Boundary observability and control

In this paper we study the exact boundary controllability of a trapezoidal time discrete wave equation in a bounded domain. We prove that the projection of the solution in an appropriate filtered space is exactly controllable with uniformly bounded cost with respect to the time-step. In this way, the well-known exact-controllability property of the wave equation can be reproduced as the limit, as the time step $h\rightarrow 0$, of the controllability of projections of the time-discrete one. By duality these results are equivalent to deriving uniform observability estimates (with respect to $h\rightarrow 0$) within a class of solutions of the time-discrete problem in which the high frequency components have been filtered. The later is established by means of a time-discrete version of the classical multiplier technique. The optimality of the order of the filtering parameter is also established, although a careful analysis of the expected velocity of propagation of time-discrete waves indicates that its actual value could be improved.

A Numerical Scheme for the Pore-Scale Simulation of Crystal Dissolution and Precipitation in Porous Media

In this paper we analyze a numerical scheme for a dissolution and precipitation model in porous media. We focus here on the chemistry, which is modeled by a parabolic problem that is coupled through the boundary conditions to an ordinary differential inclusion defined on the boundary. We use a regularization approach for constructing a semi-implicit scheme that is stable and convergent. For dealing with the emerging time discrete nonlinear problems, we propose a simple fixed-point iterative procedure. The paper is concluded by numerical results.

Rate-dependent Thermo-mechanical Modelling of Superelastic Shape-memory Alloys for Seismic Applications

Experimental tests performed on superelastic shape-memory alloys (SMAs) show a significant dependence of the stress—strain relationship on the loading—unloading rate, coupled with a not negligible oscillation of the material temperature. This feature is of particular importance in view of the use of such materials in earthquake engineering, where the loading rate affects the structural response. Motivated by this observation and by the limited number of available works on the modelling of SMAs for seismic applications, the present article addresses a uniaxial constitutive model for representing the system rate-dependent thermo-mechanical behavior of superelastic SMAs. The model is based on a single internal scalar variable, the martensite fraction, for which different rate-independent evolutionary equations in rate form are proposed. Moreover, it takes into account the different elastic properties between austenite and martensite. The whole model is then thermo-mechanically coupled with a thermal balance equation. Hence, it considers mechanical dissipation as well as latent heat and includes temperature as a primary independent variable, which is responsible for the dynamic effects. The article also provides a description for the integration, in time, of the constitutive equation and presents the solution algorithm of the corresponding time-discrete problem. Finally, results from numerical analyses are reported and the ability of the model to simulate experimental data obtained from uniaxial tests performed on superelastic SMA wires and bars at frequency levels of excitation typical of earthquake engineering is assessed.

Some mathematical problems in perfect plasticity

Mathematical problems arising in the study of problems of perfect plasticity are reviewed, and some problems of current interest are discussed. In the context of the dual formulation, for which the problem has been thoroughly explored, the strategy of viscoplastic regularization and its application are reconsidered, with a view to exploring the use of this strategy in the study of the primal problem. It is argued that this approach is not without substantial difficulties, and so alternatives to the study of the primal problem are sought. Details are given of an approach by the authors in which direct methods of the calculus of variations are applied to a sequence of time-discrete problems, the solution to the original problem being recovered in the limit as the time step goes to zero.