A new proof of geometric convergence for the adaptive generalized weighted analog sampling (GWAS) method
-
Rong Kong
Abstract
Generalized Weighted Analog Sampling is a variance-reducing method for solving radiative transport problems that makes use of a biased (though asymptotically unbiased) estimator. The introduction of bias provides a mechanism for combining the best features of unbiased estimators while avoiding their limitations. In this paper we present a new proof that adaptive GWAS estimation based on combining the variance-reducing power of importance sampling with the sampling simplicity of correlated sampling yields geometrically convergent estimates of radiative transport solutions. The new proof establishes a stronger and more general theory of geometric convergence for GWAS.
Funding statement: The second author gratefully acknowledges partial support from NIH NIBIB Laser Microbeam and Medical Program (LAMMP, P41EB015890) and from NIH R25GM103818.
A Preliminary estimates
The following formula from probability theory can be found in [12] and is used in many of our proofs.
For any random variables X and Y,
As well, the following set-theoretic result arises in estimating probabilities:
Assume that A and B are two subsets of a probability space
and, for two small positive numbers
Then
Proof of Lemma 1.
We will use the following formulas, for any random variables X and Y:
For
where we have used
As for
which is (5.2).
To calculate the variance of
The first term of the right-hand side is equal to zero because, under the
condition
where it can be easily verified that
or
which is (5.3). To calculate
Proof of Lemma 2.
From (2.11) and (5.1), we obtain
We then have
Using the first inequality of (5.5) and also applying Proposition 2, we obtain
which means
as both sides of the inequality are deterministic. Estimate (5.6) is proved.
Now, we derive an upper bound of
or
Since
or
Taking the maximum value of both sides, we obtain
In (A.3), notice that
where κ is defined by (2.4). Therefore, from (A.3), we have
Now using the second inequality of (5.5) and Proposition 2, we have
or
which means
because both sides of the inequality are deterministic and
The proof of Lemma 2 is completed. ∎
Proof of Corollary 3.
According to Chebyshev’s
inequality, for any W and
which means
Using (5.6), we have
Now we just choose
to complete the proof of Corollary 3. ∎
Proof of Lemma 4.
From the definition (3.7) of ζ and equation (3.9) about
Noticing (4.5) and (4.7) about the definition of
Therefore,
which can be estimated by
Obviously, if
and, therefore,
On the other hand, using (2.4),
Applying (A.7) to (A.6), we obtain
Using condition (5.8) and Proposition 2, we obtain
The proof of Lemma 4 is completed. ∎
Proof of Theorem 6.
Noticing the definition (3.7) of ζ and the equation (3.9) satisfied by
Using (2.11), we obtain
Noticing the notations
Using (A.5) in the proof of Lemma 4, we obtain
Since, for any a and b,
Noticing the definition of
Using (5.10) together with Proposition 2, we obtain (5.11).
The proof is completed. ∎
B Estimation of the bias
Proof of Theorem 1.
Recalling the definition of
Note that
Now, from Corollary 3, for
Therefore, from (B.1),
or
According to (4.9), the definition of
where the subscript w indicates that
As for the random variable Z, we can prove it similarly. Notice
Then using (B.2), we obtain the first inequality of (6.1).
This completes the proof of Theorem 1. ∎
C Estimation of second moments
Proof of Theorem 1.
We first estimate
i.e.,
We have
Now, from Corollary 3, for
Therefore, from (C.1),
or
Because of the independence of the random walks for different i and j, the second sum in the braces of (C.2) vanishes. We then have
or
According to condition (7.3), we can apply Lemma 6, specifically inequality (5.12):
Now combining (C.3) and (C.4), and using Proposition 2, we obtain
which is the second inequality of (7.4) (after redefining the constants
As for V, we note
Similarly, we can obtain the first inequality of (7.4).
In order to derive (7.5), we notice that
and
Combining (C.5), (C.6) and (C.7), we obtain
Now using (7.2) and Proposition 2, we obtain
where
Noticing the second
inequality of (7.1), since
which is the first inequality of (7.5). The second inequality can be obtained similarly.
This completes the proof. ∎
D General geometric convergence
Proof of Theorem 1.
We will prove by induction that ,
for any
We have added two additional inequalities to the list (first two inequalities of (D.1)). The reason for doing so is that, to prove the third inequality, we need the first two inequalities.
For
where we have taken
We need to estimate
and
From (D.2), (D.3) and (D.4), we obtain
Applying the first inequality (D.1) and Proposition 2, we obtain
where
or after we replace
Combining (D.5) with (D.6) and noticing Proposition 2, we obtain
Thus (D.1) is proved for
We now prove (D.1) for the general case. Again, according to Chebyshev’s inequality,
or after replacing
or
According to Theorem 1,
where
or by (2.6),
Noticing (4.2), we can find a
We then have
In order to prove the second inequality of (D.1), we notice
Combining (D.11) with (D.12) and applying Proposition 2 (about the lower bound),
Combining (D.8) with (D.13) and applying Proposition 2, we obtain
Applying (D.10), we obtain
We can then pick a
Combining (D.14) with (D.15), when
which is the second inequality of (D.1).
The third inequality of (D.1) can be easily proved. All we have to do
is to go over the steps from (D.2) through (D.7) with the
superscript 1 replaced by m and 0 replaced by
Thus, when
The proof of Theorem 1 is completed. ∎
E Geometric convergence for expansion in basis functions
Proof of Theorem 1.
Using the expressions of the true
solution
Therefore, estimating
The proof is similar to Theorem 1. We will prove by induction
that , for any
We have added two additional inequalities to the list (first two inequalities of (E.2)). The reason for doing so is that, to prove the third inequality, we need the first two inequalities.
For
where we have taken
We need to estimate
and
From (E.3), (E.4) and (E.5), we obtain
Applying the first inequality (E.2) and Proposition 2, we obtain
where
or after we replace
Combining (E.1) with (E.7) and (E.8) and noticing Proposition 2, we obtain
Thus (E.2) is proved for
We now prove (E.2) for the general case. Again, according to Chebyshev’s inequality,
or after replacing
which leads to
or using Proposition 2,
According to Theorem 1,
where
or by (2.6),
Noticing (4.2), we can find a
We then have
In order to prove the second inequality of (E.2), we notice
Combining (E.13) with (E.14) and applying Proposition 2 (about the lower bound),
Combining (E.10) with (E.15) and applying Proposition 2, we obtain
Applying (E.12), we obtain
We can pick a sufficiently large N to make
Combining (E.16) with (E.17), when
which is the second inequality of (E.2).
The third inequality of (E.2) can be easily proved. All we have to do
is to go over the steps from (E.6) through (E.9) with the
superscript 1 replaced by m and 0 replaced by
Thus, when
The proof of Theorem 1 is completed. ∎
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© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- A new proof of geometric convergence for the adaptive generalized weighted analog sampling (GWAS) method
- Ninomiya–Victoir scheme: Strong convergence, antithetic version and application to multilevel estimators
- Numerical computation for backward doubly SDEs with random terminal time
- Vector Monte Carlo stochastic matrix-based algorithms for large linear systems
Articles in the same Issue
- Frontmatter
- A new proof of geometric convergence for the adaptive generalized weighted analog sampling (GWAS) method
- Ninomiya–Victoir scheme: Strong convergence, antithetic version and application to multilevel estimators
- Numerical computation for backward doubly SDEs with random terminal time
- Vector Monte Carlo stochastic matrix-based algorithms for large linear systems
