The Experts below are selected from a list of 306 Experts worldwide ranked by ideXlab platform
James M Calvin - One of the best experts on this subject based on the ideXlab platform.
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A lower bound on complexity of optimization under the r-fold integrated Wiener Measure
Journal of Complexity, 2020Co-Authors: James M CalvinAbstract:AbstractWe consider the problem of approximating the global minimum of an r-times continuously differentiable function on the unit interval, based on sequentially chosen function and derivative evaluations. Using a probability model based on the r-fold integrated Wiener Measure, we establish a lower bound on the expected number of function evaluations required to approximate the minimum to within ϵ on average
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an adaptive univariate global optimization algorithm and its convergence rate under the Wiener Measure
Informatica (lithuanian Academy of Sciences), 2011Co-Authors: James M CalvinAbstract:We describe an adaptive algorithm for approximating the global minimum of a continuous univariate function. The convergence rate of the error is studied for the case of a random objective function distributed according to the Wiener Measure.
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a lower bound on complexity of optimization under the r fold integrated Wiener Measure
Journal of Complexity, 2011Co-Authors: James M CalvinAbstract:We consider the problem of approximating the global minimum of an r-times continuously differentiable function on the unit interval, based on sequentially chosen function and derivative evaluations. Using a probability model based on the r-fold integrated Wiener Measure, we establish a lower bound on the expected number of function evaluations required to approximate the minimum to within @e on average.
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a one dimensional optimization algorithm and its convergence rate under the Wiener Measure
Journal of Complexity, 2001Co-Authors: James M CalvinAbstract:In this paper we describe an adaptive algorithm for approximating the global minimum of a continuous function on the unit interval, motivated by viewing the function as a sample path of a Wiener process. It operates by choosing the next observation point to maximize the probability that the objective function has a value at that point lower than an adaptively chosen threshold. The error converges to zero for any continuous function. Under the Wiener Measure, the error converges to zero at rate e?n?n, where {?n} (a parameter of the algorithm) is a positive sequence converging to zero at an arbitrarily slow rate.
Bruce K Driver - One of the best experts on this subject based on the ideXlab platform.
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absolute continuity of heat kernel Measure with pinned Wiener Measure on loop groups
Annals of Probability, 2001Co-Authors: Bruce K Driver, Vikram K SrimurthyAbstract:Let t > 0, K be a connected compact Lie group equipped with an Ad K - invariant inner product on the Lie Algebra of K. Associated to this data are two Measures μ 0 t and νt 0 on L(K) - the space of continuous loops based at e e K. The Measure μ 0 t is pinned Wiener Measure with variance t while the Measure ν 0 t is a heat kernel Measure on L(K). The Measure μ 0 t is constructed using a K - valued Brownian motion while the Measure ν 0 t is constructed using a L(K) - valued Brownian motion. In this paper we show that ν 0 t is absolutely continuous with respect to μ 0 t and the Radon-Nikodym derivative dν 0 t /dμ 0 t is bounded.
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equivalence of heat kernel Measure and pinned Wiener Measure on loop groups
Comptes Rendus De L Academie Des Sciences Serie I-mathematique, 2000Co-Authors: Shigeki Aida, Bruce K DriverAbstract:Abstract We show that heat kernel Measure and pinned Wiener Measure on loop groups over simply connected compact Lie groups are equivalent.
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finite dimensional approximations to Wiener Measure and path integral formulas on manifolds
Journal of Functional Analysis, 1999Co-Authors: Lars Andersson, Bruce K DriverAbstract:Abstract Certain natural geometric approximation schemes are developed for Wiener Measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite dimensional manifolds H P ( M ) consisting of piecewise geodesic paths adapted to partitions P of [0, 1]. The finite dimensional manifolds H P ( M ) carry both an H 1 and a L 2 type Riemannian structures, G 1 P and G 0 P , respectively. It is proved that (1/ Z i P ) e −(1/2) E ( σ ) d Vol G i P ( σ )→ ρ i ( σ ) dν ( σ ) as mesh ( P )→0, where E ( σ ) is the energy of the piecewise geodesic path σ ∈ H P ( M ), and for i =0 and 1, Z i P is a “normalization” constant, Vol G i P is the Riemannian volume form relative to G i P , and ν is Wiener Measure on paths on M . Here ρ 1 ( σ )≡1 and ρ 0 ( σ )=exp(− 1 6 ∫ 1 0 Scal( σ ( s )) ds ) where Scal is the scalar curvature of M . These results are also shown to imply the well known integration by parts formula for the Wiener Measure.
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finite dimensional approximations to Wiener Measure and path integral formulas on manifolds
arXiv: Differential Geometry, 1998Co-Authors: Lars Andersson, Bruce K DriverAbstract:Certain natural geometric approximation schemes are developed for Wiener Measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite dimensional manifolds consisting of piecewise geodesic paths adapted to partitions $P$ of $[0,1]$. The finite dimensional manifolds of piecewise geodesics carry both an $H^{1}$ and a $L^{2}$ type Riemannian structures $G^i_P$. It is proved that as the mesh of the partition tends to $0$, $$ 1/Z_P^i e^{- 1/2 E(\sigma)} Vol_{G^i_P}(\sigma) \to \rho_i(\sigma)\nu(\sigma) $$ where $E(\sigma )$ is the energy of the piecewise geodesic path $\sigma$, and for $i=0$ and $1$, $Z_P^i$ is a ``normalization'' constant, $Vol_{G^i_P}$ is the Riemannian volume form relative $G^i_P$, and $\nu$ is Wiener Measure on paths on $M$. Here $\rho_1 = 1$ and $$ \rho_0 (\sigma) = \exp( -1/6 \int_0^1 Scal(\sigma(s))ds ) $$ where $Scal$ is the scalar curvature of $M$. These results are also shown to imply the well know integration by parts formula for the Wiener Measure.
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A Cameron-Martin type quasi-invariance theorem for pinned Brownian motion on a compact Riemannian manifold
Transactions of the American Mathematical Society, 1994Co-Authors: Bruce K DriverAbstract:The results in Driver [13] for quasi-invariance of Wiener Measure on the path space of a compact Riemannian manifold ( M) are extended to the case of pinned Wiener Measure. To be more explicit, let h: [0, 1] -+ ToM be a C1 function where M is a compact Riemannian manifold, o E M is a base point, and ToM is the tangent space to M at o E M. Let W(M) be the space of continuous paths from [0,1] into M, v be Wiener Measure on W(M) concentrated on paths starting at o E M, and Hs (w) denote the stochastic-parallel translation operator along a path to E W(M) up to "time" s. (Note: Hs(w) is only well defined up to v-equivalence.) For c E W(M) let Xh(w) denote the vector field along a) given by Xsh(c) =Hs(cw)h(s) for each s E [0, 1]. One should interpret Xh as a vector field on W(M). The vector field Xh induces a flow Sh(t, -): W(M) -_ W(M) which leaves Wiener Measure (v) quasi-invariant, see Driver [13]. It is shown in this paper that the same result is valid if h(l) = 0 and the Wiener Measure (v) is replaced by a pinned Wiener Measure (ive). (The Measure ve is proportional to the Measure v conditioned on the set of paths which start at 0 E M and end at a fixed end point e E M.) Also as in [13], one gets an integration by parts formula for the vector-fields Xh defined above.
Grzegorz W Wasilkowski - One of the best experts on this subject based on the ideXlab platform.
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probabilistic and average linear widths inl norm with respect tor fold Wiener Measure
Journal of Approximation Theory, 1996Co-Authors: Vitaly Maiorov, Grzegorz W WasilkowskiAbstract:We show that forr-fold Wiener Measure, the probabilistic and average linear widths in theL∞-norm are proportional toformula]andformula], respectively.
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integration and approximation of multivariate functions average case complexity with isotropic Wiener Measure
Journal of Approximation Theory, 1994Co-Authors: Grzegorz W WasilkowskiAbstract:Abstract We study the average case complexity of multivariate integration and L 2 function approximation for the class F = C ([0, 1] d ) of continuous functions of d variables. The class F is endowed with the isotropic Wiener Measure (Brownian motion in Levy′s sense). For the integration problem, the average case complexity of solving the problem to within ϵ is proportional to ϵ -2/(1 + 1/ d ) . This is a negative result since for a large number d of variables, the average case complexity is close to ϵ −2 ; the latter is also achieved by the classical Monte Carlo method in the randomized worst case setting. Furthermore, Θ(ϵ −2 ) is the highest possible average case complexity among all probability Measures with finite expectation of || ƒ 2 L 2 . Thus, for large d , the average case complexity of the integration problem with isotropic Wiener Measure behaves as the worst possible average complexity. For the function approximation problem, the complexity is even higher since it is proportional to ϵ −2 d . These two negative results are in a sharp contrast to (H. Woźniakowski, Bull. Amer. Math. Soc. 24 , No. 1 (1991), 185-194; Bull. Amer. Math. Soc. , to appear), where, for F endowed with the Wiener sheet Measure, small average case complexities have been proven. Indeed, they are of order ϵ −1 (log ϵ −1 ) ( d −1)/2 and ϵ −2 (log ϵ −1 ) 2( d −1) for the integration and function approximation problems, respectively. cccc
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integration and approximation of multivariate functions average case complexity with isotropic Wiener Measure
arXiv: Numerical Analysis, 1993Co-Authors: Grzegorz W WasilkowskiAbstract:We study the average case complexity of multivariate integration and $L_2$ function approximation for the class $F=C([0,1]^d)$ of continuous functions of $d$ variables. The class $F$ is endowed with the isotropic Wiener Measure (Brownian motion in Levy's sense). Furthermore, for both problems, only function values are used as data.
Peter Zhidkov - One of the best experts on this subject based on the ideXlab platform.
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On the equivalence of the centered Gaussian Measure in L2 with the correlation operator (−d2/dx2)−1 and the conditional Wiener Measure
Rendiconti Del Circolo Matematico Di Palermo, 2020Co-Authors: Peter ZhidkovAbstract:Let w and µ be respectively the conditional Wiener Measure in C0([0, 1]) and the centered Gaussian Measure in L2[0, 1] with the correlation operator (−d2/dx2)−1. We prove the equivalence of these two Measures in the following sense: for any Borel set A ⊂ L2[0, 1] the set A ∩ C0([0, 1]) is a Borel subset of C0([0, 1]) and µ(A) = w(A∩C0([0, 1])).
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on the equivalence of the centered gaussian Measure in l2 with the correlation operator d2 dx2 1 and the conditional Wiener Measure
Rendiconti Del Circolo Matematico Di Palermo, 2009Co-Authors: Peter ZhidkovAbstract:Let w and µ be respectively the conditional Wiener Measure in C0([0, 1]) and the centered Gaussian Measure in L2[0, 1] with the correlation operator (−d2/dx2)−1. We prove the equivalence of these two Measures in the following sense: for any Borel set A ⊂ L2[0, 1] the set A ∩ C0([0, 1]) is a Borel subset of C0([0, 1]) and µ(A) = w(A∩C0([0, 1])).
Fabrice Baudoin - One of the best experts on this subject based on the ideXlab platform.
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integration by parts and quasi invariance for the horizontal Wiener Measure on foliated compact manifolds
Journal of Functional Analysis, 2019Co-Authors: Fabrice Baudoin, Qi Feng, Maria GordinaAbstract:Abstract We prove several sub-Riemannian versions of Driver's integration by parts formula which first appeared in [17] . Namely, our results are for the horizontal Wiener Measure on a totally geodesic Riemannian foliation equipped with a sub-Riemannian structure. It is also shown that the horizontal Wiener Measure is quasi-invariant under the action of flows generated by suitable tangent processes.
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integration by parts and quasi invariance for the horizontal Wiener Measure on a foliated compact manifold
arXiv: Probability, 2017Co-Authors: Fabrice Baudoin, Maria Gordina, Qi FengAbstract:We prove several versions of Driver's integration by parts formula for the horizontal Wiener Measure on a totally geodesic Riemannian foliation and prove that the horizontal Wiener Measure has a quasi-invariance property with respect to flows generated by suitable tangent processes.
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Pinning class of the Wiener Measure by a functional: related martingales and invariance properties
Probability Theory and Related Fields, 2003Co-Authors: Fabrice Baudoin, Michèle ThieullenAbstract:For a given functional Y on the path space, we define the pinning class of the Wiener Measure as the class of probabilities which admit the same conditioning given Y as the Wiener Measure. Using stochastic analysis and the theory of initial enlargement of filtration, we study the transformations (not necessarily adapted) which preserve this class. We prove, in this non Markov setting, a stochastic Newton equation and a stochastic Noether theorem. We conclude the paper with some non canonical representations of Brownian motion, closely related to our study.