Continuous Coefficients Research Articles

On the complexity of strong approximation of stochastic differential equations with a non-Lipschitz drift coefficient

We survey recent developments in the field of complexity of pathwise approximation in p-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion. First, we briefly review the case of equations with globally Lipschitz continuous coefficients, for which an error rate of at least 1/2 in terms of the number of evaluations of the driving Brownian motion is always guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate that giving up the global Lipschitz continuity of the coefficients may lead to a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an arbitrary slow decay of the smallest possible error that can be achieved on the basis of finitely many evaluations of the driving Brownian motion. Finally, we turn to recent positive results for equations with a drift coefficient that is not globally Lipschitz continuous. Here we focus on scalar equations with a Lipschitz continuous diffusion coefficient and a drift coefficient that satisfies piecewise smoothness assumptions or has fractional Sobolev regularity and we present corresponding complexity results.

Almost minimizers for the thin obstacle problem with variable coefficients

We study almost minimizers for the thin obstacle problem with variable Hölder continuous coefficients and zero thin obstacle, and establish their C^{1,\beta} regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients.

One-dimensional SDEs with LPS-type singular drift coefficients and Hölder continuous diffusion coefficients

In this paper, under the assumptions of LPS-type singular drift coefficients and Hölder continuous diffusion coefficients, the existence, uniqueness, stability, non-confluent property and strong Feller property for the solution to a one-dimensional stochastic differential equations are obtained. Our proof are based on the tools of local time and Zvonkin's transform.

An exponential split-step double balanced $$\vartheta $$ Milstein scheme for SODEs with locally Lipschitz continuous coefficients

An exponential split-step double balanced $$\vartheta $$ Milstein scheme for SODEs with locally Lipschitz continuous coefficients

Sharp lower error bounds for strong approximation of SDEs with piecewise Lipschitz continuous drift coefficient

We study pathwise approximation of strong solutions of scalar stochastic differential equations (SDEs) at a single time in the presence of discontinuities of the drift coefficient. Recently, it has been shown by Müller-Gronbach and Yaroslavtseva (2022) that for all p∈[1,∞) a transformed Milstein-type scheme reaches an Lp-error rate of at least 3/4 when the drift coefficient is a piecewise Lipschitz-continuous function with a piecewise Lipschitz-continuous derivative and the diffusion coefficient is constant. It has been proven by Müller-Gronbach and Yaroslavtseva (2023) that this rate 3/4 is optimal if one additionally assumes that the drift coefficient is bounded, increasing and has a point of discontinuity. While boundedness and monotonicity of the drift coefficient are crucial for the proof of the matching lower bound from Müller-Gronbach and Yaroslavtseva (2023), we show that both conditions can be dropped. For the proof we apply a transformation technique which was so far only used to obtain upper bounds.

Existence and Uniqueness Theorems for the Pfaff Equation with Continuous Coefficients

Existence and Uniqueness Theorems for the Pfaff Equation with Continuous Coefficients

Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

<p style='text-indent:20px;'>We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relative the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeter sets on stratified Lie groups.</p>

On extensibility and qualitative properties of solutions to Riccati's equation

We consider Riccati's equation on the real axis with continuous coefficients and non-negative discriminant of the right-hand side. We study the extensibility of its solutions to unbounded intervals. We obtain asymptotic formulae for its solutions in their dependence on the initial values and the properties of the functions representing roots of the right-hand side of the equation. We obtain results on the asymptotical behaviour of solutions defined near $\pm\infty$. We study the structure of the set of bounded solutions in the case when the roots of the right-hand side of the equation are $C^1$-functions which are different on the whole of their domain and tend monotonically to some limits as $x\to\pm\infty$. We extend, improve, or refine some well-known results. Bibliography: 47 titles.

On a Problem of Calculating the Solvability Set for a Linear System with Uncertainty

We consider a linear-convex control system defined by a set of differential equations with continuous matrix coefficients. The system may have control parameters, as well as uncertainties (interference) the possible values of which are subject to strict pointwise constraints. For this system, over a finite period of time, taking into account the constraints, we study the problem of guaranteed hitting the target set from a given initial position despite the effect of uncertainty. The main stage of solving the problem is the construction of an alternating integral and a solvability set. To construct the latter, the greatest computational complexity is the calculation of the geometric difference between the target set and the set determined by the uncertainty. A two-dimensional example of this problem is considered for which a method is proposed for finding the solvability set without the need to calculate the convex hull of the difference of the support functions of the sets.

Stochastic fractional integrodifferential equations with continuous conditions

In this paper, we use a smoothness method to obtain an existence of a strong solution for a general class of stochastic fractional integrodifferential equations. We established also the uniqueness of solution. We consider the case of continuous coefficients, which satisfy some so-called ρ-condition.

Soft actor-critic-based EMS design for dual motor battery electric bus

Thanks to the outstanding adaptability and relatively simple design process, machine learning-based energy management strategies (EMSs) show their superiority in reducing the energy consumption for multi-power powertrains. Given the popularity of distributed-drive in battery electric vehicles (BEVs), this study selects a dual-motor four-speed electric bus to investigate the proposed soft actor-critic (SAC)-based EMS. Two improvements are made to the traditional SAC algorithm to meet the special requirements of EMS in a distributed-drive electric bus in this study. Firstly, the introduced combination of Gumbel-SoftMax and actor-network allows the SAC agents to explore the hybrid action space (discrete operating modes and continuous torque distribution coefficients). Secondly, a heuristic rule-interposing action controller (HRIAC) is involved to reduce illogical exploration of SAC agents in searching for optimal power distributing ratio. Simulation results demonstrate that Gumbel-SoftMax facilitates agents' exploration and narrows the energy consumption gap between the proposed EMS and the global optimal EMS, meanwhile, HRIAC accelerates the convergence of agent training. The comparative results of proposed EMS performance in unknown driving cycles show that the proposed EMS outperforms other deep reinforcement learning (DRL)-based EMSs in adaptability, which is taken as a solid foundation for energy efficiency improvement of dual-motor electric bus in practice.

Our initial experience of monitoring the autoregulation of cerebral blood flow during cardiopulmonary bypass

Background: Cerebral blood flow (CBF) is believed to be relatively constant within an upper and lower blood pressure limit. Different methods are available to monitor CBF autoregulation during surgery. This study aims to critically analyze the application of the cerebral oxygenation index (COx), one of the commonly used techniques, using a reference to data from a series of clinical registrations. Method: CBF was monitored using near-infrared spectroscopy, while cerebral blood pressure was estimated by recordings obtained from either the radial or femoral artery in 10 patients undergoing cardiopulmonary bypass. The association between CBF and blood pressure was calculated as a moving continuous correlation coefficient. A COx index > 0.4 was regarded as a sign of abnormal cerebral autoregulation (CA). Recordings were examined to discuss reliability measures and clinical feasibility of the measurements, followed by interpretation of individual results, identification of possible pitfalls, and suggestions of alternative methods. Results and Conclusion: Monitoring of CA during cardiopulmonary bypass is intriguing and complex. A series of challenges and limitations should be considered before introducing this method into clinical practice.

Perturbation of singular integral operators with piecewise continuous coefficients

In the paper it is shown that the property of singular integral operators with piecewise continuous coefficients to be Noetherian is stable with respect to their perturbation with certain non-compact operators. An example is constructed showing that the corner points of the integration contour significantly affect the Noetherian property of singular operators with translations. These results are obtained using the symbol of the singular operators on contours with corner points, symbol, which is also determined.

Perturbation of singular integral operators with piecewise continuous coefficients

In the paper it is shown that the property of singular integral operators with piecewise continuous coefficients to be Noetherian is stable with respect to their perturbation with certain non-compact operators. An example is constructed showing that the corner points of the integration contour significantly affect the Noetherian property of singular operators with translations. These results are obtained using the symbol of the singular operators on contours with corner points, symbol, which is also determined.

The modified truncated Euler–Maruyama method for stochastic differential equations with concave diffusion coefficients

Influenced by Gyöngy and Rásonyi (2011), many scholars established the strong convergence of several numerical methods for scalar stochastic differential equations (SDEs) with superlinearly growing drift and Hölder continuous diffusion coefficients. However, their methods depend on the Yamada-Watanabe method and therefore fail to work for multi-dimensional SDEs. In this paper, we study the strong Lp−convergence, for all p⩾2, of the modified truncated Euler–Maruyama method for multi-dimensional SDEs with superlinearly growing drift and concave diffusion coefficients satisfying the Osgood condition. We also discuss an example with computer simulations to illustrate our theoretical results.

Elliptic operators with discontinuous coefficients in meshfree GFDM

AbstractIn phase change simulations, material properties such as density, viscosity, or thermal conductivity may exhibit jump discontinuities, possibly of several orders of magnitude. These jump discontinuities represent interfaces between the phases, and they emerge naturally during the simulation; thus, their exact location is generally unknown a priori. Our goal is to simulate phase change processes with a meshfree generalized finite difference method in a monolithic model without distinguishing between the different phases. There, the material properties mentioned above appear as coefficients inside elliptic operators in divergence form and the jumps must be treated adequately by the numerical method. We present a numerical method for discretizing elliptic operators with discontinuous coefficients without the need for a domain decomposition or tracking of interfaces. Our method facilitates the construction of diagonally dominant diffusion operators that lead to M‐matrices for the discrete Poisson's equation, and thus, satisfy the discrete maximum principle. We demonstrate the applicability of the new method for the case of smooth diffusivity and discontinuous diffusivity. We show that the method is first‐order accurate for discontinuous diffusion problems and provides second‐order and fourth‐order convergence for continuous diffusion coefficients.

The Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals: The general case

In this work, we study the Cauchy problem of Poiseuille flow of the full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of two partial differential equations (PDEs): One is a quasi-linear wave equation for the director field representing the crystallization of the nematics, and the other is a parabolic PDE for the velocity field characterizing the liquidity of the material. We extend the work in Chen et al. (2020) [11] for a special case to the general physical setup. The Cauchy problem is shown to have global solutions beyond singularity formation. Among a number of progresses made in this paper, a particular contribution is a systematic treatment of a parabolic PDE with only Hölder continuous diffusion coefficient and rough (maybe unbounded) nonhomogeneous terms.

Characterization of Ulam-Hyers stability of linear differential equations with periodic coefficients

Let X be a Banach space over R or C. We study Ulam–Hyers stability on the time interval R of linear differential equations with continuous coefficients, considering solutions with values in X for higher order equations, and solutions with values in Xn for first-order systems. Roughly speaking, Ulam–Hyers stability of an equation is its property of having an exact solution close to an approximate solution. For a first-order system x′=Ax, we obtain that its Ulam–Hyers stability is equivalent to another important property, namely for each continuous and bounded function g:R→Xn there exists a bounded solution of x′=Ax+g. Using this result, theory of Jordan forms of matrices and Floquet theory, we characterize the Ulam–Hyers stability of a system x′=Ax with periodic coefficients in terms of its characteristic multipliers. Our main result states that the Ulam–Hyers stability of a linear equation with periodic coefficients of order n with unknown x is equivalent to the Ulam–Hyers stability of the corresponding system satisfied by (x,x′,…,x(n−1)).

Pathwise methods for the integration of a stochastic SVIR model

We propose an approach for the precise numerical integration of a stochastic SVIR model defined by a stochastic differential equation (SDE) with non‐globally Lipschitz continuous coefficients and multiplicative noise. This equation, based on a compartmental epidemic model, describes a continuous vaccination strategy with environmental noise effects. By means of an appropriate invertible continuous transformation, we link the solution to the stochastic SVIR model to the solution of an auxiliary random differential equation (RDE) that has an Ornstein–Uhlenbeck process as the only input parameter of the system. In this way, based on this explicit conjugacy between both equations, new pathwise numerical schemes are constructed for the SVIR model. In particular, we propose an exponential method that outperforms other integrators in the literature and is able to approximate, with high stability, meaningful probabilistic features of the continuous system, including its stationary distribution and ergodicity. A simulation study is presented to illustrate the practical performance of the introduced methods, and a comparative analysis with other integrators commonly used for the simulation of epidemiological models is performed.

A strongly monotonic polygonal Euler scheme

In recent years tamed schemes have become an important technique for simulating SDEs and SPDEs whose continuous coefficients display superlinear growth. The taming method involves curbing the growth of the coefficients as a function of stepsize, but so far has not been adapted to preserve the monotonicity of the coefficients. This has arisen as an issue in [4], where the lack of a strongly monotonic tamed scheme forces strong conditions on the setting. In this article we give a novel and explicit method for truncating monotonic functions in separable real Hilbert spaces, and show how this can be used to define a polygonal (tamed) Euler scheme on finite dimensional space, preserving the monotonicity of the drift coefficient, and converging to the true solution at the same rate as the classical Euler scheme for Lipschitz coefficients. Our construction is the first explicit method for truncating monotone functions we are aware of, and the first in infinite dimensions.