Yosida Approximation Research Articles

Yosida approximations of the cumulative distribution function and applications in survival analysis

The Yosida approximation method is a classic regularization technique in maximal monotone operator theory. In the present paper, however, we apply it to the cumulative distribution function (cdf) and study its properties in the context of statistics. In that case the Yosida approximation transforms a given cdf into a new cdf with better continuity properties, namely the new cdf is Lipschitz continuous, and its distance to the original cdf as well as its Lipschitz constant are both controlled by a parameter.When applied to an empirical cdf, which is arguably the most important case in practice, the Yosida approximation yields a continuous piecewise linear cdf in a systematic way, underpinned by a versatile theoretical framework. This provides a new smoothing technique which to our knowledge has not been explored in the literature yet.After establishing several theoretical statistical properties of Yosida approximations we show possible applications to survival analysis. Finally, we pose two open problems in order to stimulate further research along these lines.

On the Cahn–Hilliard equation with kinetic rate dependent dynamic boundary conditions and non-smooth potentials: Well-posedness and asymptotic limits

We analyze a class of Cahn–Hilliard equations with kinetic rate dependent dynamic boundary conditions that describe possible short-range interactions between the binary mixture and the solid boundary. In the presence of surface diffusion on the boundary, the initial boundary value problem can be viewed as a transmission problem consisting of Cahn–Hilliard-type equations both in the bulk and on the boundary. We first establish the existence, uniqueness, and continuous dependence of global weak solutions. In the construction of weak solutions, an explicit convergence rate in terms of the parameter for the Yosida approximation is obtained. Under some additional assumptions, we further prove the existence and uniqueness of global strong solutions. Next, we study the asymptotic limit as the coefficient of the boundary diffusion goes to zero and show that the limit problem with a forward-backward dynamic boundary condition is well posed in a suitable weak formulation. At last, we investigate the asymptotic limits as the kinetic rate tends to zero and infinity, respectively. Our results are valid for a general class of bulk and boundary potentials with double-well structure, including the physically relevant logarithmic potential and the non-smooth double obstacle potential.

Two-phase flows through porous media described by a Cahn–Hilliard–Brinkman model with dynamic boundary conditions

We investigate a new diffuse-interface model that describes creeping two-phase flows (i.e., flows exhibiting a low Reynolds number), especially flows that permeate a porous medium. The system of equations consists of a Brinkman equation for the volume averaged velocity field and a convective Cahn–Hilliard equation with dynamic boundary conditions for the phase field, which describes the location of the two fluids within the domain. The dynamic boundary conditions are incorporated to model the interaction of the fluids with the wall of the container more precisely. In particular, they allow for a dynamic evolution of the contact angle between the interface separating the fluids and the boundary, and for a convection-induced motion of the corresponding contact line. For our model, we first prove the existence of global-in-time weak solutions in the case where regular potentials are used in the Cahn–Hilliard subsystem. In this case, we can further show the uniqueness of the weak solution under suitable additional assumptions. We further prove the existence of weak solutions in the case of singular potentials. Therefore, we regularize such singular potentials by a Moreau–Yosida approximation, such that the results for regular potentials can be applied, and eventually pass to the limit in this approximation scheme.

Yosida distance and existence of invariant manifolds in the infinite-dimensional dynamical systems

We consider the existence of invariant manifolds to evolution equations u ′ ( t ) = A u ( t ) u’(t)=Au(t) , A : D ( A ) ⊂ X → X A:D(A)\subset \mathbb {X}\to \mathbb {X} near its equilibrium A ( 0 ) = 0 A(0)=0 under the assumption that its proto-derivative ∂ A ( x ) \partial A(x) exists and is continuous in x ∈ D ( A ) x\in D(A) in the sense of Yosida distance. Yosida distance between two (unbounded) linear operators U U and V V in a Banach space X \mathbb {X} is defined as d Y ( U , V ) ≔ lim sup μ → + ∞ ‖ U μ − V μ ‖ d_Y(U,V)≔\limsup _{\mu \to +\infty } \| U_\mu -V_\mu \| , where U μ U_\mu and V μ V_\mu are the Yosida approximations of U U and V V , respectively. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of ∂ A \partial A is continuous in the sense of Yosida distance. The Yosida distance approach allows us to generalize the well-known results with possible applications to larger classes of partial differential equations and functional differential equations. The obtained results seem to be new.

Generalized Yosida inclusion problem involving multi-valued operator with XOR operation

Abstract In this article, we study a generalized Yosida variational inclusion problem involving multi-valued operator with XOR operation. It is shown that the generalized Yosida variational inclusion problem involving multi-valued operator with XOR operation is equivalent to a fixed point equation. We have proved that the generalized Yosida approximation operator is Lipschitz continuous. Finally, we prove an existence and convergence result for our problem.

Linear quadratic optimal control problems of infinite‐dimensional mean‐field type with jumps

AbstractIn this paper, we formulate and investigate a framework for the theory of the linear quadratic optimal control problem (LQ problem) for infinite‐dimensional mean‐field stochastic evolution systems with jumps. We ensure the well‐posedness of the investigated problems by establishing the existence, uniqueness, and a priori estimates for mild solutions to general infinite‐dimensional mean‐field forward stochastic evolution equations (MF‐SEE) and mean‐field backward stochastic evolution equations (MF‐BSEE) with jumps. Leveraging the Yosida approximation theory, we establish a dual theory between MF‐SEE and MF‐BSEE with jumps, overcoming the challenge posed by the inapplicability of Itô's formula in the context of mild solutions. Our main results regarding the existence and uniqueness of the optimal control, along with corresponding dual characterizations and state feedback representations, are obtained through convex analysis techniques, our established dual theory, and decoupling methods.

On a Non-Convex Lagrange Optimal Control Problem

Abstract In this paper, we are concerned with an iterative differential inclusion governed by the time-dependent maximal monotone operator with perturbation. The approach to solve our problem is based on the Yosida approximation technique. The theoretical result is applied to prove an existence result for a Lagrange optimal control problem without assumptions concerning convexity.

On the rate of convergence of Yosida approximation for the nonlocal Cahn–Hilliard equation

Abstract It is well-known that one can construct solutions to the nonlocal Cahn–Hilliard equation with singular potentials via Yosida approximation with parameter $\lambda \to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $\sqrt{\lambda }$. The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert–Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter $\lambda $ could be linked to the discretization parameters, yielding appropriate error estimates.

L2-convergence of Yosida approximation for semi-linear backward stochastic differential equation with jumps in infinite dimension

PurposeThe main motivation of this paper is to present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L2-convergence rate is established.Design/methodology/approachThe authors establish a result concerning the L2-convergence rate of the solution of backward stochastic differential equation with jumps with respect to the Yosida approximation.Findings The authors carry out a convergence rate of Yosida approximation to the semi-linear backward stochastic differential equation in infinite dimension.Originality/valueIn this paper, the authors present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L2-convergence rate is established.

Extension of monotone operators and Lipschitz maps invariant for a group of isometries

Abstract We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in $L^{p}$ -spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.

Double-parameter regularization for solving the backward diffusion problem with parallel-in-time algorithm

We propose a double-parameter regularization scheme for dealing with the backward diffusion process. Considering the smoothing effect of Yosida approximation for PDE, we propose to regularize this ill-posed problem by modifying original governed system in terms of a pseudoparabolic equation together with a quasi-boundary condition simultaneously, which consequently contains two regularizing parameters. Theoretically, we establish the optimal error estimates between the regularizing solution and the exact one in terms of suitable choice strategy for the regularizing parameters, under a-priori regularity assumptions on the exact solution. The a-posteriori choice strategy for the regularizing parameters based on the discrepancy principle is also studied. To weaken the heavy computational cost for solving the discrete nonsymmetric linear regularizing system by finite difference scheme, especially in higher spatial dimensional cases, the block divide-and-conquer method together with the properties of the Schur complement is applied to decompose the linear system into two half-size linear systems, one of which can be solved by the diagonalization technique, and consequently an efficient parallel-in-time algorithm originally developed for direct problem is applicable. Our proposed method is of much lower complexity than the standard solver for the corresponding linear system. Finally, some numerical examples are presented to verify the efficiency of our proposed method.

Approximate solvability method for nonlocal impulsive evolution equation

Abstract In this article, without assuming the compactness of semigroup, we deal with the existence and uniqueness of a mild solution for semilinear impulsive evolution equation with nonlocal condition in a reflexive Banach space by applying the approximate solvability method and Yosida approximations of the infinitesimal generator of C 0-semigroup.

Generalized BSDEs driven by RCLL martingales with stochastic monotone coefficients

A solution is given to generalized backward stochastic differential equations driven by a real-valued RCLL martingale on an arbitrary filtered probability space. The existence and uniqueness of a solution are proved via the Yosida approximation method when the generators are only stochastic monotone with respect to the y-variable and stochastic Lipschitz with respect to the z-variable, with different linear growth conditions.

Stochastic Quasi-Geostrophic Equation with Jump Noise in Lp Spaces

In this paper, we consider a 2D stochastic quasi-geostrophic equation driven by jump noise in a smooth bounded domain. We prove the local existence and uniqueness of mild Lp(D)-solutions for the dissipative quasi-geostrophic equation with a full range of subcritical powers α∈(12,1] by using the semigroup theory and fixed point theorem. Our approach, based on the Yosida approximation argument and Itô formula for the Banach space valued processes, allows for establishing some uniform bounds for the mild solutions and we prove the global existence of mild solutions in L∞(0,T;Lp(D)) space for all p>22α−1, which is consistent with the deterministic case.

On the strong convergence of continuous Newton-like inertial dynamics with Tikhonov regularization for monotone inclusions

In a Hilbert space H, we study the convergence properties of the trajectories of a Newton-like inertial dynamical system with a Tikhonov regularization term governed by a general maximally monotone operator A:H→2H. The maximally monotone operator enters the dynamics via its Yosida approximation with an appropriate adjustment of the Yosida regularization parameter, by adopting an approach introduced by Attouch and Peypouquet (2019) [7] and further developed by Attouch and László (2021) [5]. We obtain fast rates of convergence for the velocity and the Yosida regularization term towards zero, while the generated trajectories converge weakly towards a zero of A or, depending on the system parameters, strongly towards the zero of minimum norm of A. Our analysis reveals that the damping coefficient, the Yosida regularization parameter and the Tikhonov parametrization are strongly correlated.

Co-Variational Inequality Problem Involving Two Generalized Yosida Approximation Operators

We focus our study on a co-variational inequality problem involving two generalized Yosida approximation operators in real uniformly smooth Banach space. We show some characteristics of a generalized Yosida approximation operator, which are used in our main proof. We apply the concept of nonexpansive sunny retraction to obtain a solution to our problem. Convergence analysis is also discussed.

ON THE LOCAL KREISS RESOLVENT CONDITION

We introduce a local version of the Kreiss resolvent condition for Banach space operators and we relate it to the local growth of powers. Also, we introduce a local Yosida approximation and we establish some local results.

On generalized Yosida inclusion problem with application

Herein, a generalized Yosida inclusion problem involving Φ-relaxed co-accretive mapping is investigated. The resolvent and analogous generalized Yosida approximation operator is explicated and its characteristics are outlined. Picard S-type iterative algorithm is presented and convergence is delineated. Further, an equivalence of generalized Yosida inclusion problem to generalized resolvent equation problem is developed. Moreover, the convergence analysis of a contrived iterative scheme is reported to inspect the generalized resolvent equation problem. Finally, a delay differential equation is investigated and theoretical findings are exemplified.

Convergence Analysis for Generalized Yosida Inclusion Problem with Applications

A new generalized Yosida inclusion problem, involving A-relaxed co-accretive mapping, is introduced. The resolvent and associated generalized Yosida approximation operator is construed and a few of its characteristics are discussed. The existence result is quantified in q-uniformly smooth Banach spaces. A four-step iterative scheme is proposed and its convergence analysis is discussed. Our theoretical assertions are illustrated by a numerical example. In addition, we confirm that the developed method is almost stable for contractions. Further, an equivalent generalized resolvent equation problem is established. Finally, by utilizing the Yosida inclusion problem, we investigate a resolvent equation problem and by employing our proposed method, a Volterra–Fredholm integral equation is examined.

A class of Yosida inclusion and graph convergence on Yosida approximation mapping with an application

The proposed work is presented in two folds. The first aim is to deals with the new notion called generalized ?i?j-Hp(., ., ...)-accretive mappings that are the sum of two symmetric accretive mappings. It is an extension of ??-H(., .)-accretive mapping, studied and analyzed by Kazmi [18]. We define the proximalpoint mapping associated with generalized ?i?j-Hp(., ., ...)-accretive mapping and demonstrate aspects on single-valued property and Lipschitz continuity. The graph convergence of generalized ?i?j-Hp(., ., ...)- accretive mapping is discussed. Second aim is to introduce and study the generalized Yosida approximation mapping and Yosida inclusion problem. Next, we obtain the convergence on generalized Yosida approximation mappings by using the graph convergence of generalized ?i?j-Hp(., ., ...)-accretive mappings without using the convergence of its proximal-point mapping. As an application, we consider the Yosida inclusion problem in q-uniformly smooth Banach spaces and propose an iterative scheme connected with generalized Yosida approximation mapping of generalized ?i?j-Hp(., ., ...)-accretive mapping to find a solution of Yosida inclusion problem and discuss its convergence criteria under appropriate assumptions. Some examples are constructed and demonstrate few graphics for the convergence of proximal-point mapping as well as generalized Yosida approximation mapping linked with generalized ?i?j-Hp(., ., ...)-accretive mappings.