We adopt the integral definition of the fractional Laplace operator and study, on Lipschitz domains, an optimal control problem that involves a fractional elliptic partial differential equation (PDE) as state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. Regularity estimates for optimal variables are also analyzed. We devise two strategies of finite element discretization: a semidiscrete scheme where the control variable is not discretized and a fully discrete scheme where the control variable is discretized with piecewise constant functions. For both solution techniques, we analyze convergence properties of discretizations and derive error estimates.
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We analyze the local accuracy of the virtual element method. More precisely, we prove an error bound similar to the one holding for the finite element method, namely, that the local $H^1$ error in a interior subdomain is bounded by a term behaving like the best approximation allowed by the local smoothness of the solution in a larger interior subdomain plus the global error measured in a negative norm.
In this paper we analyze a pressure-robust method based on divergence-free mixed finite element methods with continuous interior penalty stabilization. The main goal is to prove an $O(h^{k+1/2})$ error estimate for the $L^2$ norm of the velocity in the convection dominated regime. This bound is pressure robust (the error bound of the velocity does not depend on the pressure) and also convection robust (the constants in the error bounds are independent of the Reynolds number).
This paper approaches the unsupervised learning problem by gradient descent in the space of probability density functions. A main result shows that along the gradient flow induced by a distribution-dependent ordinary differential equation (ODE), the unknown data distribution emerges as the long-time limit. That is, one can uncover the data distribution by simulating the distribution-dependent ODE. Intriguingly, the simulation of the ODE is shown equivalent to the training of generative adversarial networks (GANs). This equivalence provides a new "cooperative" view of GANs and, more importantly, sheds new light on the divergence of GANs. In particular, it reveals that the GAN algorithm implicitly minimizes the mean squared error (MSE) between two sets of samples, and this MSE fitting alone can cause GANs to diverge. To construct a solution to the distribution-dependent ODE, we first show that the associated nonlinear Fokker-Planck equation has a unique weak solution, by the Crandall-Liggett theorem for differential equations in Banach spaces. Based on this solution to the Fokker-Planck equation, we construct a unique solution to the ODE, using Trevisan's superposition principle. The convergence of the induced gradient flow to the data distribution is obtained by analyzing the Fokker-Planck equation.
Motivated by numerical modeling of ultrasound waves, we investigate robust conforming finite element discretizations of quasilinear and possibly nonlocal equations of Westervelt type. These wave equations involve either a strong dissipation or damping of fractional-derivative type and we unify them into one class by introducing a memory kernel that satisfies non-restrictive regularity and positivity assumptions. As the involved damping parameter is relatively small and can become negligible in certain (inviscid) media, it is important to develop methods that remain stable as the said parameter vanishes. To this end, the contributions of this work are twofold. First, we determine sufficient conditions under which conforming finite element discretizations of (non)local Westervelt equations can be made robust with respect to the dissipation parameter. Secondly, we establish the rate of convergence of the semi-discrete solutions in the singular vanishing dissipation limit. The analysis hinges upon devising appropriate energy functionals for the semi-discrete solutions that remain uniformly bounded with respect to the damping parameter.
A class of implicit Milstein type methods is introduced and analyzed in the present article for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. By incorporating a pair of method parameters $\theta, \eta \in [0, 1]$ into both the drift and diffusion parts, the new schemes are indeed a kind of drift-diffusion double implicit methods. Within a general framework, we offer upper mean-square error bounds for the proposed schemes, based on certain error terms only getting involved with the exact solution processes. Such error bounds help us to easily analyze mean-square convergence rates of the schemes, without relying on a priori high-order moment estimates of numerical approximations. Putting further globally polynomial growth condition, we successfully recover the expected mean-square convergence rate of order one for the considered schemes with $\theta \in [\tfrac12, 1], \eta \in [0, 1]$. Also, some of the proposed schemes are applied to solve three SDE models evolving in the positive domain $(0, \infty)$. More specifically, the particular drift-diffusion implicit Milstein method ($ \theta = \eta = 1 $) is utilized to approximate the Heston $\tfrac32$-volatility model and the stochastic Lotka-Volterra competition model. The semi-implicit Milstein method ($\theta =1, \eta = 0$) is used to solve the Ait-Sahalia interest rate model. Thanks to the previously obtained error bounds, we reveal the optimal mean-square convergence rate of the positivity preserving schemes under more relaxed conditions, compared with existing relevant results in the literature. Numerical examples are also reported to confirm the previous findings.
This paper considers the Cauchy problem for the nonlinear dynamic string equation of Kirchhoff-type with time-varying coefficients. The objective of this work is to develop a temporal discretization algorithm capable of approximating a solution to this initial-boundary value problem. To this end, a symmetric three-layer semi-discrete scheme is employed with respect to the temporal variable, wherein the value of a nonlinear term is evaluated at the middle node point. This approach enables the numerical solutions per temporal step to be obtained by inverting the linear operators, yielding a system of second-order linear ordinary differential equations. Local convergence of the proposed scheme is established, and it achieves quadratic convergence concerning the step size of the discretization of time on the local temporal interval.
Generalised hyperbolic (GH) processes are a class of stochastic processes that are used to model the dynamics of a wide range of complex systems that exhibit heavy-tailed behavior, including systems in finance, economics, biology, and physics. In this paper, we present novel simulation methods based on subordination with a generalised inverse Gaussian (GIG) process and using a generalised shot-noise representation that involves random thinning of infinite series of decreasing jump sizes. Compared with our previous work on GIG processes, we provide tighter bounds for the construction of rejection sampling ratios, leading to improved acceptance probabilities in simulation. Furthermore, we derive methods for the adaptive determination of the number of points required in the associated random series using concentration inequalities. Residual small jumps are then approximated using an appropriately scaled Brownian motion term with drift. Finally the rejection sampling steps are made significantly more computationally efficient through the use of squeezing functions based on lower and upper bounds on the L\'evy density. Experimental results are presented illustrating the strong performance under various parameter settings and comparing the marginal distribution of the GH paths with exact simulations of GH random variates. The new simulation methodology is made available to researchers through the publication of a Python code repository.
We study the problem of verification and synthesis of robust control barrier functions (CBF) for control-affine polynomial systems with bounded additive uncertainty and convex polynomial constraints on the control. We first formulate robust CBF verification and synthesis as multilevel polynomial optimization problems (POP), where verification optimizes -- in three levels -- the uncertainty, control, and state, while synthesis additionally optimizes the parameter of a chosen parametric CBF candidate. We then show that, by invoking the KKT conditions of the inner optimizations over uncertainty and control, the verification problem can be simplified as a single-level POP and the synthesis problem reduces to a min-max POP. This reduction leads to multilevel semidefinite relaxations. For the verification problem, we apply Lasserre's hierarchy of moment relaxations. For the synthesis problem, we draw connections to existing relaxation techniques for robust min-max POP, which first use sum-of-squares programming to find increasingly tight polynomial lower bounds to the unknown value function of the verification POP, and then call Lasserre's hierarchy again to maximize the lower bounds. Both semidefinite relaxations guarantee asymptotic global convergence to optimality. We provide an in-depth study of our framework on the controlled Van der Pol Oscillator, both with and without additive uncertainty.
Computing accurate splines of degree greater than three is still a challenging task in today's applications. In this type of interpolation, high-order derivatives are needed on the given mesh. As these derivatives are rarely known and are often not easy to approximate accurately, high-degree splines are difficult to obtain using standard approaches. In Beaudoin (1998), Beaudoin and Beauchemin (2003), and Pepin et al. (2019), a new method to compute spline approximations of low or high degree from equidistant interpolation nodes based on the discrete Fourier transform is analyzed. The accuracy of this method greatly depends on the accuracy of the boundary conditions. An algorithm for the computation of the boundary conditions can be found in Beaudoin (1998), and Beaudoin and Beauchemin (2003). However, this algorithm lacks robustness since the approximation of the boundary conditions is strongly dependant on the choice of $\theta$ arbitrary parameters, $\theta$ being the degree of the spline. The goal of this paper is therefore to propose two new robust algorithms, independent of arbitrary parameters, for the computation of the boundary conditions in order to obtain accurate splines of any degree. Numerical results will be presented to show the efficiency of these new approaches.
The kernel-based method has been successfully applied in linear system identification using stable kernel designs. From a Gaussian process perspective, it automatically provides probabilistic error bounds for the identified models from the posterior covariance, which are useful in robust and stochastic control. However, the error bounds require knowledge of the true hyperparameters in the kernel design and are demonstrated to be inaccurate with estimated hyperparameters for lightly damped systems or in the presence of high noise. In this work, we provide reliable quantification of the estimation error when the hyperparameters are unknown. The bounds are obtained by first constructing a high-probability set for the true hyperparameters from the marginal likelihood function and then finding the worst-case posterior covariance within the set. The proposed bound is proven to contain the true model with a high probability and its validity is verified in numerical simulation.
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