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280
Analysis of Bounded Variation Penalty Methods for IllPosed Problems
 INVERSE PROBLEMS
, 1994
"... This paper presents an abstract analysis of bounded variation (BV) methods for illposed operator equations Au = z. Let T (u) def = kAu \Gamma zk 2 + ffJ(u); where the penalty, or "regularization", parameter ff ? 0 and the functional J(u) is the BV norm or seminorm of u, also known as the tota ..."
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Cited by 101 (1 self)
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This paper presents an abstract analysis of bounded variation (BV) methods for illposed operator equations Au = z. Let T (u) def = kAu \Gamma zk 2 + ffJ(u); where the penalty, or "regularization", parameter ff ? 0 and the functional J(u) is the BV norm or seminorm of u, also known as the total variation of u. Under mild restrictions on the operator A and the functional J(u), it is shown that the functional T (u) has a unique minimizer which is stable with respect to certain perturbations in the data z, the operator A, the parameter ff, and the functional J(u). In addition, convergence results are obtained which apply when these perturbations vanish and the regularization parameter is chosen appropriately.
Solving ForwardBackward Stochastic Differential Equations Explicitly – a Four Step Scheme
 Prob. Th. Rel. Fields
, 1994
"... Abstract. The problem of nding adapted solutions to systems of coupled linear forwardbackward stochastic di erential equations (FBSDEs, for short) is investigated. A necessary condition of solvability leads to a reduction of general linear FBSDEs to a special one. By some ideas from controllability ..."
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Cited by 88 (11 self)
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Abstract. The problem of nding adapted solutions to systems of coupled linear forwardbackward stochastic di erential equations (FBSDEs, for short) is investigated. A necessary condition of solvability leads to a reduction of general linear FBSDEs to a special one. By some ideas from controllability in control theory, using some functional analysis, we obtain a necessary and su cient condition for the solvability of the linear FBSDEs with the processes Z (serves as a correction, see x1) being absent in the drift. Then a Riccati type equation for matrixvalued (not necessarily square) functions is derived using the idea of the FourStepScheme (introduced in [11] for general FBSDEs). The solvability of such a Riccati type equation is studied which leads to a representation of adapted solutions to linear FBSDEs. Keywords. Linear forwardbackward stochastic di erential equations, adapted solution, Riccati type equation. AMS Mathematics subject classi cation. 60H10.
Exact Multiplicity of Positive Solutions for a Class of Semilinear Problem, II
 JOURNAL DIFFERENTIAL EQUATIONS
, 1999
"... We consider the positive solutions to the semilinear problem: An + %f ( I,) = 0, in B”. 1, = 0, on aB”. i where B ” is the unit ball in R”.)I 2 I. and I. is a positive parameter. It is well known that iffis a smooth function, then any positive solution to the equation is radially symmetric, and al ..."
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Cited by 59 (33 self)
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We consider the positive solutions to the semilinear problem: An + %f ( I,) = 0, in B”. 1, = 0, on aB”. i where B ” is the unit ball in R”.)I 2 I. and I. is a positive parameter. It is well known that iffis a smooth function, then any positive solution to the equation is radially symmetric, and all solutions can be pdrameterized by their maximum values. We develop a unitied approach to obtain the exact multiplicity of the positive solutions for a wide class ol ’ nonlinear functions./1ll). and the precise shape of the global bifurcation diagrams are rigorously proved. Our technique combines the birurcation analysis. stability analysis, and topological methods. We show that the shape of the bil’urcdtion curve depends on the shape of the graph of (unction Jrr)/fr as well as the growth rate of,/: cj 1999 Academic Press 1.
Error Estimate for Approximate Solutions of a Nonlinear ConvectionDiffusion Problem
, 2002
"... This paper proves the estimate ku" uk L 1 (Q T ) C" 1=5 , where, for all " > 0, u" is the weak solution of (u" ) t + div(q f(u" )) ('(u")+"u") = 0 with initial and boundary conditions, u is the entropy weak solution of u t + div(qf(u)) ('(u)) = 0 with the same initial and boundary conditions, a ..."
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Cited by 36 (13 self)
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This paper proves the estimate ku" uk L 1 (Q T ) C" 1=5 , where, for all " > 0, u" is the weak solution of (u" ) t + div(q f(u" )) ('(u")+"u") = 0 with initial and boundary conditions, u is the entropy weak solution of u t + div(qf(u)) ('(u)) = 0 with the same initial and boundary conditions, and C > 0 does not depend on ". The domain
A Variable Metric Proximal Point Algorithm for Monotone Operators
, 1997
"... The Proximal Point Algorithm (PPA) is a method for solving inclusions of the form 0 2 T (z) where T is a monotone operator on a Hilbert space. The algorithm is one of the most powerful and versatile solution techniques for solving variational inequalities, convex programs, and convexconcave mini ..."
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Cited by 25 (3 self)
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The Proximal Point Algorithm (PPA) is a method for solving inclusions of the form 0 2 T (z) where T is a monotone operator on a Hilbert space. The algorithm is one of the most powerful and versatile solution techniques for solving variational inequalities, convex programs, and convexconcave minimax problems. It possesses a robust convergence theory for very general problem classes and is the basis for a wide variety of decomposition methods called splitting methods. Yet, the classical PPA typically exhibits slow convergence in many applications. For this reason, acceleration methods for the PPA algorithm are of great practical importance. In this paper we propose a variable metric implementation of the proximal point algorithm. In essence, the method is a Newtonlike scheme applied to the MoreauYosida resolvent of the operator T . In this article, we establish the global and linear convergence of the proposed method. In addition, we characterize the superlinear convergence of ...
Large time behavior of nonlocal aggregation models with nonlinear diffusion
, 2006
"... The aim of this paper is to establish rigorous results on the large time behavior of nonlocal models for aggregation, including the possible presence of nonlinear diffusion terms modeling local repulsions. We show that, as expected from the practical motivation as well as from numerical simulations, ..."
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Cited by 24 (2 self)
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The aim of this paper is to establish rigorous results on the large time behavior of nonlocal models for aggregation, including the possible presence of nonlinear diffusion terms modeling local repulsions. We show that, as expected from the practical motivation as well as from numerical simulations, one obtains concentrated densities (Dirac δ distributions) as stationary solutions and large time limits in the absence of diffusion. In addition, we provide a comparison for aggregation kernels with infinite respectively finite support. In the first case, there is a unique stationary solution corresponding to concentration at the center of mass, and all solutions of the evolution problem converge to the stationary solution for large time. The speed of convergence in this case is just determined by the behavior of the aggregation kernels at zero, yielding either algebraic or exponential decay or even finite time extinction. For kernels with finite support, we show that an infinite number of stationary solutions exist, and solutions of the evolution problem converge only in a setvalued sense to the set of stationary solutions, which we characterize in detail. Moreover, we also consider the behavior in the presence of nonlinear diffusion terms, the most interesting case being the one of small diffusion coefficients. Via the implicit function theorem we give a quite general proof of a rather natural assertion for such models, namely that there exist stationary solutions that have the form of a local peak around the center of mass. Our approach even yields the order of the size of the support in terms of the diffusion coefficients. All these results are obtained via a metric gradient flow formulation using the Wasserstein metric for probability measures, and are carried out in the case of a single spatial dimension for convenience.
Dynamical systems and discrete methods for solving nonlinear illposed problems
 Appl.Math.Reviews
, 2000
"... 2. Continuous methods for well posed problems 3. Discretization theorems for wellposed problems ..."
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Cited by 21 (17 self)
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2. Continuous methods for well posed problems 3. Discretization theorems for wellposed problems
Convergence analysis of a colocated finite volume scheme for the incompressible NavierStokes equations on general 2D or 3D meshes
 SIAM Journal on Numerical Analysis
"... Abstract. We study a colocated cell centered finite volume method for the approximation of the incompressible NavierStokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressures, all of them colocated at the center of the cells of a u ..."
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Cited by 21 (10 self)
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Abstract. We study a colocated cell centered finite volume method for the approximation of the incompressible NavierStokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressures, all of them colocated at the center of the cells of a unique mesh; hence the need for a stabilization technique, which we choose of the BrezziPitkäranta type. The scheme features two essential properties: the discrete gradient is the transposed of the divergence terms and the discrete trilinear form associated to nonlinear advective terms vanishes on discrete divergence free velocity fields. As a consequence, the scheme is proved to be unconditionally stable and convergent for the Stokes problem, the steady and the transient NavierStokes equations. In this latter case, for a given sequence of approximate solutions computed on meshes the size of which tends to zero, we prove, up to a subsequence, the L 2convergence of the components of the velocity, and, in the steady case, the weak L 2convergence of the pressure. The proof relies on the study of space and time translates of approximate solutions, which allows the application of Kolmogorov’s theorem. The limit of this subsequence is then shown to be a weak solution of the NavierStokes equations. Numerical examples are performed to obtain numerical convergence rates in both the linear and the nonlinear case.