Subsolutions of an Isaacs Equation and Efficient Schemes for Importance Sampling: Examples and Numerics

Dupuis, Paul; Wang, Hui

doi:10.21236/ada458952

“…In the next subsection we will prove that a function defined based on (12) satisfies an HJB equation. We will use this fact to prove the convergence of (9) to (12).…”

Section: Solution To the Limit Control Problemmentioning

confidence: 99%

“…distribution. The idea of using subsolutions to construct IS algorithms is from [11], [12], [15], and [32], and is called the subsolution approach to IS.…”

Section: Introductionmentioning

confidence: 99%

Approximation of bounds on mixed-level orthogonal arrays

Özbudak

¹

,

Sezer

²

2011

Advances in Applied Probability

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Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao-and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

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“…The sup in (11) equals V (0, 0). Generalizing (12), for A i ≤ t < A i+1 , we have (15) where the sup is subject to…”

Section: The Limit Hjb Equationmentioning

confidence: 99%

Approximation of bounds on mixed-level orthogonal arrays

Sezer

¹

,

Özbudak

²

2011

Adv. Appl. Probab.

0

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Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao-and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

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“…For ease of notation we will drop the superscript (ǫ, δ) and write W . We would like to prove the following: there is a constant C 1 that only depends on the parameter system such that for all x ∈ R d + H b (DW (x)) ≥ −C 1 exp(−ǫ/δ), where b defined in (6) is the boundary corresponding to x. Let E be the set of effective gradients q such that there is a boundary b ′ ≤ b with effective gradient q.…”

Section: A Proof Of Lemma 42mentioning

confidence: 99%

“…To construct our optimal IS algorithms we use an optimality result from [16] which was obtained using the optimal control/subsolution approach to IS of [12,3,4,6,5]. This result states that to construct optimal IS algorithms for the simulation of a wide range of buffer overflow events of any stable Jackson network it is sufficient to build appropriate smooth subsolutions to a Hamilton Jacobi Bellman (HJB) equation and its boundary conditions (these are given in (7) in the context we study in the current paper).…”

Section: Introductionmentioning

confidence: 99%

Asymptotically optimal importance sampling for Jackson networks with a tree topology

Sezer

¹

2009

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Importance sampling (IS) is a variance reduction method for simulating rare events. A recent paper by Dupuis, Wang and Sezer (Ann. App. Probab. 17(4):1306-1346, 2007 exploits connections between IS and subsolutions to a limit HJB equation and its boundary conditions to show how to design and analyze simple and efficient IS algorithms for various overflow events for tandem Jackson networks. The present paper uses the same subsolution approach to build asymptotically optimal IS schemes for stable open Jackson networks with a tree topology. Customers arrive at the single root of the tree. The rare overflow event we consider is the following: given that initially the network is empty, the system experiences a buffer overflow before returning to the empty state. Two types of buffer structures are considered: 1) A single system-wide buffer of size n shared by all nodes, 2) each node i has its own buffer of size β i n, β i ∈ (0, 1).

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Subsolutions of an Isaacs Equation and Efficient Schemes for Importance Sampling: Examples and Numerics

Cited by 6 publications

References 11 publications

Approximation of bounds on mixed-level orthogonal arrays

Approximation of bounds on mixed-level orthogonal arrays

Approximation of bounds on mixed-level orthogonal arrays

Asymptotically optimal importance sampling for Jackson networks with a tree topology

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