- Josef Dick and Friedrich Pillichshammer, Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010. © Cambridge University Press.

is now available here. The final published version can be obtained directly from Cambridge University Press here.

The preprint version differs of course from the final version, for instance, the page numbers are different. However, the numbering of Chapters, Sections, Theorems, Lemmas, Corollaries, Definitions and Examples is the same in both versions. The list of corrections is for the published version. We do not have a separate list for the preprint version (though the corrections for the published version also apply to the preprint version).

Filed under: Quasi-Monte Carlo, Research Tagged: cyclic digital nets, digital net, discrepancy, discrepancy theory, duality theory, Duality Theory for Digital Nets, fast component by component, geometric discrepancy, Higher order digital net, higher order digital sequence, higher order polynomial lattice rule, higher order Sobol sequence, hyperplane nets, Niederreiter sequence, numerical integration, polynomial lattice rule, Propagation Rule, quasi-Monte Carlo, randomised quasi-Monte Carlo, Sobol sequence, uniform distribution, Walsh function ]]>

- Josef Dick, Daniel Rudolf and Houying Zhu, Discrepancy bounds for uniformly ergodic Markov chain quasi-Monte Carlo. ArXiv, 19 March 2013. Submitted, 2013.

In a sense, this paper is an extension of the paper

- Su Chen, Josef Dick, Art Owen, Consistency of Markov Chain quasi-Monte Carlo for continuous state spaces. Ann. Stat., 39, 679–701, 2011. For a blog entry and preprint version see here.

In this paper we prove bounds on the discrepancy of Markov chains which are generated by a deterministic driver sequence . We also prove a Koksma-Hlawka inequality. The main assumption which we use is uniform ergodicity of the transition (Markov) kernel. We describe the essential ingredients and results in the following.

**The transition kernel of a Markov chain**

Let be a probability space. We want to generate a Markov chain in the state space whose distribution converges to the target distribution . The idea of the transition kernel is the following. Assume that is the th point of the Markov chain. Let be an arbitrary set. Then the probability that the next point of the Markov chain is in , is given by . For a precise definition of the transition kernel see Definition 2 on page 6 here.

Let be the starting point of our Markov chain. Then we use to sample the next point of our Markov chain. The point is then sampled from the distribution , and so on. Asymptotically we then get that the distribution of converges in some sense to the target distribution . One can also write down the transition kernel for going steps, denoted by , namely it can be defined recursively by

where is the indicator function of the set . Here, just means that we integration with respect to the probability measure .

This method is an important algorithm in simulation problems where it is not possible to generate samples which have distribution . Instead one generates a Markov chain which has target distribution and uses the chain to as the points for the simulation. Such algorithms are called Markov chain Monte Carlo.

There are many methods for generating the Markov chain, see for instance the Metropolis-Hastings algorithm or the slice sampler. However, those algorithms will not play a role in the following. We just assume that there is a function , which we call update function. This function is used to generate the Markov chain by , where is a uniformly distributed random number in .

**Uniformly ergodic Markov chains**

Uniform ergodicity is a property of the transition kernel which tells us how fast the distribution of converges to the target distribution . The assumption is of the following form

where and are constants. This assumption is actually very natural as we explain for a special case in the following.

If the transition kernel satisfies certain assumptions, we have

where and the function is in fact a reproducing kernel with some additional properties. The kernel has a Karhunen-Loeve expansion

where are the eigenvalues and are orthonormal eigenfunctions of the integral operator (we assume that this operator is a compact operator). We have

Let now

Then it can be shown using the recursive formula for and the Karhunen-Loeve expansion of that

Thus converges exponentially fast to . (In fact, under certain assumptions, one can choose .) But we have , the stationary distribution.

**The discrepancy of the Markov chain**

We measure the discrepancy of the empirical distribution of the first points of the Markov chain from the target distribution . The empirical distribution for a set is

where is the indicator function of the set . The local discrepancy is

Note that for arbitrary sets one cannot get convergence in general (if say and is just the point set , then the local discrepancy is ). One example is to consider boxes of the form . For instance, one can consider the set of sets . We call the set the set of test sets. The discrepancy is now defined by

**Main result**

The main result of the paper is the proof that for any starting point there exists a finite driver sequence such that the Markov chain generated by this driver sequence is bounded by

for some constant independent of . See Corollary 5 here.

Several further extensions would be interesting. One is to remove the dependence of the driver sequence on the starting point. Further one would like to have a driver sequence which can be used for a large class of update functions. In practice one wants to have explicit constructions of such driver sequences. Ultimately one would also like to have driver sequences for which one can achieve a rate of convergence beyond the Monte Carlo rate, at least, for some restricted class of problems. Answers to these questions would be very interesting, but they seem to be quite challenging, at least at the moment.

Filed under: Research Tagged: concentration inequality, discrepancy theory, Markov chain Monte Carlo, Markov chain quasi Monte Carlo ]]>

A -net in base 3 is a set of 9 points in the square which have the following property:

Each rectangle

for nonnegative integers , , , contains exactly one point of the point set.

Such point sets can be generated using Sudoku. Consider the following example of a Sudoku: The square can be partitioned into 9 rows, each containing the numbers 1 to 9, it can be partitioned such that each column contains the numbers 1 to 9 and it can also be partitioned into squares such that each one contains the numbers 1 to 9. This is illustrated in the following: each of the shaded regions contains exactly the numbers 1 to 9:

See also here for more information.

By putting a point in each small square where there is a 1 and leaving the other small squares empty we get the following point set:

This can be understood as a point set in which has the -net property in base 3 described above. This follows since the point set inherits the structure of the Sudoku.

**A higher order Sudoku**

The Sudoku presented above has a number of further properties, as illustrated in the following. Note that the shaded region again contains all the numbers from 1 to 9.

Thus, by replacing 1 with a point and leave the other areas empty, the point set above also inherits this structure. These additional conditions are similar to higher order nets.

**Higher order Sudoku puzzles**

In analogy to higher order nets, we call a Sudoku which satisfies all the constraints a higher order Sudoku.

In the following we present a (very difficult) higher order Sudoku puzzle. A printable pdf file can be found here.

Filed under: Diversions Tagged: Higher order digital net, Sudoku ]]>

This page is currently hosted at the Katholieke Universiteit Leuven.

If you would like to contribute, please sign up with your true name at http://roth.cs.kuleuven.be/w-mcqmc/index.php?title=Special:UserLogin&returnto=Main_Page. After logging in, you can update information, contribute new pages and generally help to make the MCQMC Wiki a success.

Filed under: Diversions, Quasi-Monte Carlo Tagged: MCQMC, MCQMC Wiki, Wiki ]]>

http://www.regmed.ru/getfile.asp?CatId=db&Name=IMSobol_1969_book.pdf The book can be cited using:

- @preamble{

“\def\cprime{$’$} ”

}

@book {Sobol1969,

AUTHOR = {{{\cyr{T}}sobol{\cprime}, I. M.}},

TITLE = {Mnogomernye kvadraturnye formuly i funktsii {K}haara},

PUBLISHER = {Izdat. “Nauka”, Moscow},

YEAR = {1969},

PAGES = {288}

}

The english translation of the book title is ‘Multidimensional quadrature formulae and Haar functions’.

* Update 01/12/2010:* Professor Sobol provided a response which is added at the end of this entry.

The book consists of two parts. In the first part, the author deals with Haar functions. Here the author considers functions defined on These functions are an important tool in the second part of the book which analyses numerical integration of functions defined on using Sobol sequences. Sobol’s sequences have become a staple tool for numerical integration of high dimensional functions defined on where can be large. These are widely used in applications.

* Update 01/12/2010* Below is a response from Professor Sobol:

Below is a very brief abstract for my 1969 book, and

I must inform you that the online link http://www.regmed.ru/getfile.asp?CatId=db&Name=IMSobol_1969_book.pdf

will be closed at 29 December 2010.

I.M.Sobol, Multidimensional Quadrature Formulas and Haar Functions, 1969, Moscow, Nauka, 288pp, in Russian.

Authors Summary

The first part of the book contains Haar functions, Fourier-Haar series and their applications in quadrature formulas and approximation problems.

The second part contains multiple Fourier-Haar series, Multidimensional quadrature formulas, exact error bounds for certain classes of integrands, -nets and -sequences, infinite-dimensional quadrature formulas.

Remark. In 1987, -sequences got a second name: -sequences in base (here and is the dimension). However, in most papers a third name is used: Sobol sequences. Now in 2010, such sequences having additional uniformity properties remain among the most popular quasi-random sequences.

Filed under: Quasi-Monte Carlo, Research Tagged: Sobol, Sobol sequence, Sobol's book ]]>

**The algorithm**

In the following we describe the algorithm. A Matlab implementation is given in the next section. The algorithm is of the form

where the integrand are weights and are the quadrature points. We now describe how to obtain the quadrature points and the corresponding weights.

To obtain the quadrature points, we use a digital -net over a finite field of prime order Let In the following we consider only the case Choose the number of quadrature points, i.e. choose

The algorithm is illustrated by the following picture.

In step a), we choose a partitioning of the real line How to choose this partition depends on to the problem at hand. For instance, in the numerical example below we chose points where denotes the inverse error function. Then the points partition the real line into intervals. We now put points in the interval and For the interval these are given by where Analoguously for the other interval, which we label by from left to right. We continue putting points in the intervals and using the same method. Continuing doing so we obtain points This is step b) and c) above.

The last step d) now describes how to obtain the point For and we define

*Example*

Consider the picture above. Here we have Assume we are given the point of a digital -net over . Then and and therefore and Thus and Therefore

Now we consider the construction of the weights With the construction of the points, each point falls into an interval of the form Let denote the number of points which are in the interval that is Then the weight is simply given by

With this definition, constant functions in the interval are integrated exactly. However, it is maybe not so obvious how to actually compute the weights. The volume of can be computed since it is clear in which interval a point lies. The number of points in is a bit more tricky to compute. To do so, we need Theorem 15 in the paper. For each interval we know that there are points inside. The number of points in the interval is then given by where

provided that where is the quality parameter of underlying digital net. Using this formulae we can calculate the number of points in provided that the interval contains at least points.

Now we discuss the remaining case of intervals where there are less than points. The analysis of the integration error shows that points which fall into intervals with less than points are not significant. Thus, conceivably, giving them weight would not alter the outcome. More generally, any weight in the interval is possible. Below we choose those weights to be

In the following section we present a Matlab implementation of the algorithm and we use it to estimate the integral

This integral can also be computed analytically

This allows us to compute the integration error. Using a substitution, the above integral can also be written as

We compare our algorithm with the algorithm

where for are the first points of a Sobol sequence.

**Matlab code**

% Digital nets in R^s numerical experiment.

format long;

mmin = 13;

mmax = 23;

X=6;

s = 3;

tSobol = 1;

trs=zeros(1,mmax-mmin+1);

tother=zeros(1,mmax-mmin+1);

error_Rs=zeros(1,mmax-mmin+1);

error_other_method=zeros(1,mmax-mmin+1);

Pointcounter=zeros(1,mmax-mmin+1); % Counts the number of points which fall into intervals for which

for m = mmin:mmax

m

N = pow2(m); % Number of points;

P = sobolset(s); % Get Sobol sequence;

sobolpoints = net(P,N); % Get net from Sobol sequence with N points;

tic

% Elementary intervals on R;

Y = erfinv(1-2.^(-(0:1:m)))*X;

% One dimensional projections;

Z=zeros(1,N); % Points in R

W=zeros(1,N); % Weights of interval length in R

M=zeros(1,N); % Stores the values of m_i

R=1;

for k=0:m-2

for l =0:2^(m-2-k)-1

Z(R+l) = Y(k+1) + l*2^(-(m-2-k))*[Y(k+2)-Y(k+1)];

Z(R+2^(m-2-k)+l) = -Y(k+2)+l*2^(-(m-2-k))*[Y(k+2)-Y(k+1)];

W(R+l) = Y(k+2)-Y(k+1);

W(R+2^(m-2-k)+l) = Y(k+2)-Y(k+1);

M(R+l) = m-2-k;

M(R+2^(m-2-k)+l) = m-2-k;

end;

R = R + 2^(m-1-k);

end;

Z(N-1)= Y(m);

Z(N) = -Y(m+1);

W(N-1)= Y(m+1)-Y(m);

W(N) = Y(m+1)-Y(m);

M(N-1)= 0;

M(N)= 0;

%Points in R^s;

sobolindex=sobolpoints*N;

Rspoints=zeros(N,s);

for k=1:N

for l=1:s

Rspoints(k,l)=Z(sobolindex(k,l)+1);

end;

end;

% Volume;

Volume=ones(1,N);

for k=1:N

for l=1:s

Volume(k)=Volume(k)*W(sobolindex(k,l)+1);

end;

end;

% m_i values

Mi=zeros(N,s);

for k=1:N

for l=1:s

Mi(k,l)=M(sobolindex(k,l)+1);

end;

end;

Msum = sum(Mi,2);

Mpoint = m*(1-s)+Msum;

for k=1:N

if Mpoint(k) < tSobol

Mpoint(k)=tSobol; % Gives a small weight to points which lie outside hyperbolic cross

else

Pointcounter(m-mmin+1) = Pointcounter(m-mmin+1)+1;

end;

end;

weight=transpose(Volume).*pow2(-Mpoint);

% Example function f(x,y,z) = exp(2 \sqrt(Pi) * (x+y+z));

correct_result = exp(3);

QRs = 0;

for k=1:N

QRs = QRs + weight(k)*exp(2*sqrt(pi)*[Rspoints(k,1)+Rspoints(k,2)+Rspoints(k,3)]) * exp(-pi*[Rspoints(k,1)^2 + Rspoints(k,2)^2 + Rspoints(k,3)^2]);

end;

error_Rs(m-mmin+1) = QRs – correct_result

trs(m-mmin+1) =toc

% Other method

tic

Q=0;

for k=1:N

Q=Q+ exp(2*sqrt(pi)*[erfinv(2*sobolpoints(k,1)-1) + erfinv(2*sobolpoints(k,2)-1)+ erfinv(2*sobolpoints(k,3)-1)]/sqrt(pi));

end;

Result = Q/N;

error_other_method(m-mmin+1) = Result-correct_result

tother(m-mmin+1) =toc

end

error_Rs

trs

error_other_method

tother

Â© Josef Dick

This algorithm and program is freely available, under the condition that it cannot be patented by any other party. Please cite the paper ‘J. Dick, Quasi-Monte Carlo numerical integration on : digital nets and worst-case error. Submitted, 2010.’ if you make use of it. It is well understood that this algorithm is useful in applications and it is not permitted to obtain a patent for applications of this algorithm or program or modifications thereof.

Filed under: Quasi-Monte Carlo, Research Tagged: bounded variation, digital net, numerical integration over R^s, quasi-Monte Carlo, Walsh model ]]>

Digital higher order nets and sequences have been introduced in

- [1] J. Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal., 46, 1519–1553, 2008. doi: 10.1137/060666639 For a Matlab program generating higher order digital nets, numerical examples and plots of higher order digital nets see here.

whereas higher order nets have been introduced in

- [2] J. Dick and J. Baldeaux, Equidistribution properties of generalized nets and sequences. In: Proceedings of the MCQMC’08 conference, Montreal, Canada, P. L’Ecuyer and A. Owen (eds.), pp. 305–323, 2009. doi: 10.1007/978-3-642-04107-5_19 An earlier version can be found here.

There are several parameters occurring in the definition of higher order nets, namely and for higher order sequences we have the parameters

(Digital) higher order nets and sequences are point sets and sequences such that

where is the number of quadrature points and is the smoothness of the integrand

**Definitions**

For convenience we repeat the definitions of those (digital) higher order nets and sequences here. A discussion of the parameters occurring in these definitions follows in the subsequent section.

Defintion[Digital -net]

Let let be a real number and let be an integer. Let be the finite field of order , where is a prime power. Let where is the th row vector of for and If for all withthe vectors

are linearly independent over then the digital net with generating matrices is called a digital -net over

For and one obtains a digital -net over

Definition[Digital -sequence]

Let and and let be a real number. Let be the finite field of order , where is a prime power. Let where is the th row vector of the matrix for and Further let denote the left upper submatrix of for If for all the matrices generate a digital -net over then the digital sequence with generating matrices is a digital -sequence over

For one obtains a digital -sequence over

Before we can define the analogue geometrical concepts, we need the notion of a higher order elementary interval (which is actually not an interval anymore, but a union of intervals). We need some notation. Let let let let and let where the components and do not appear in and if By a higher order elementary interval we mean a subset of of the form

where is an integer and where for we have in case and if

An example of a higher order (or generalized) elementary interval can be found in [2].

Definition[-net]

Let be natural numbers, let be a real number, and let be an integer. Let be an integer and let The point set is called a -net in base if for all integers where withwhere for we set the empty sum the higher order elementary interval contains exactly point of for each

For and one obtains a -net in base

Definition[-sequence]

Let and be integers and be a real number. Let be a sequence of numbers in Then is a -sequence in base if for all and all we have that is a -net in base

For one obtains a -sequence in base

**Meaning of the parameters**

Some parameters have the same meaning as for classical -nets and -sequences, and are therefore not discussed further. These are

- the base or the finite field over which the net or sequence is constructed,
- the dimension,
- is equal to where is the number of points and is the logarithm in base

Now to the remaining parameters of (digital) higher order nets and (digital) higher order sequences. There are a few guiding principles which we follow in defining (digital) higher order nets and sequences:

- each parameter in the definition of higher order nets should have an analogue in the definition for digital higher order nets and vice versa,
- each parameter in the definition of higher order sequences should have an analogue in the definition for digital higher order sequences and vice versa,
- (digital) higher order sequences should be a direct extension of (digital) higher order nets,
- (digital) higher order nets should be a finite version of (digital) higher order sequences, and
- (digital) -nets should be a special case of (digital) higher order nets and analogously for sequences.

A consequence of these guiding principles is that all definitions are related to each other and hence one should not consider a definition just on its own. In other words, each definition should only be considered in conjunction with all other definitions.

One could of course ignore one or more of the above principles and arrive at a different definition of some object (which might appear more natural). However, it makes the objects more difficult to relate to each other. For instance, a digital -net over is a -net in base and analogously for (digital) higher order sequences. This is a natural result and holds since in the definitions the parameters of each object have direct analogues in the related object.

In the error bound above we said that denotes the smoothness of the integrand We make one assumption which makes the definition a bit more complicated, namely, we assume that is not a known fixed number, that is, we simply do not know the smoothness of the integrand. Since we assume that we do not know we cannot talk about an order net in this context. So we think of the remaining parameters as a function of and, for a given point set or sequence, we would like to know the value of the remaining parameters for each value of In this sense, it would also be possible to write for instance digital -net instead of digital -net (and analogously for the other three cases).

Using the guiding principles we can explain one more parameter immediately. Namely, is the number of rows of the generating matrices By principle (2) it follows that we also need to have the parameter in the definition of higher order nets (though it does not have such an immediate interpretation as for digital higher order nets).

The parameter is introduced because of principle (4). Namely, when one considers the first points of a digital -sequence, then the left upper submatrices of the generating matrices of the digital higher order sequence generate a digital -net. Hence, by principle (4) Notice that we cannot choose in this case, since is not a fixed number (i.e. the smoothness is not known).

Next we discuss the parameter If one considers only (digital) higher order nets, then one could argue that the parameter is redundant. But to understand one needs to consider (digital) higher order sequences rather than (digital) higher order nets. The parameter then appears in the definition of (digital) higher order nets because of principle (3), we want (digital) higher order sequences to be a direct extension of (digital) higher order nets. If one ignores principle (3) then one could define (digital) higher order nets without using

For sequences, it has been shown that in the definition of (digital) higher order sequences cannot grow faster than as The value of must then satisfy that this sum diverges as fast or faster than as

**Simplification 1: Assume that is known**

This is not really a simplification of the notation, since it does not have the same level of generality anymore as the above concepts. Nonetheless, this case is valuable since it is much simpler to understand.

So assume now that is a given number. For digital higher order nets we can set the parameter to always if is known. The same can be done for higher order nets and hence the parameter is not needed in this case. Similarly we can just assume that and hence this parameter is also not needed.

Now consider the parameter In the general case the situation is the following: First choose a sequence , then choose a value of and and then find out what are the parameters such that is a -sequence. But now the situation is different. We can now first fix and then choose a sequence which is good for the given value of . The important point is that, using for instance the construction from [1], we know that there exists a sequence for which we can choose such that Sequences for which we cannot choose are not really of interest since they do not yield the optimal rate of convergence (notice that is not possible, see [Theorem 4, 2]. Hence we can restrict ourselves to sequences for which In fact, we can simply choose and With this understanding we do not need to include the parameters (in the general case this is not possible, since currently there is no sequence for which for all ). Since we can always choose for sequences, we can do the same for nets. Thus we can set and for known

Thus, in this case we could simply speak of -nets and -sequences or (digital) order -nets and (digital) order -sequences, as suggested by Art.

With our current knowledge, this information is usually sufficient, since known propagation rules provide enough information about the remaining parameters. That is, for arbitrary , we can obtain useful values for for (digital) order nets or (digital) order sequences for the case when And this information is competitive with respect to the parameters obtained from the current constructions.

However, this somewhat hides the question whether there are constructions which are better than what can be obtained from the current propagation rules for the cases when In other words, it would be interesting to know whether a sequence exists for which for all Currently this is only possible for in a chosen range.

In the notion of extensibility we can describe this in the following way:

- fixed case: is known and we can define (digital) order nets and (digital) order sequences;
- finitely extensible case: unknown and we know higher order nets and sequences which are optimal for all where is an arbitrarily chosen finite number;
- infinitely extensible case: unknown and we know higher order nets and sequences which are optimal for all

Because of the known propagation rules, for some papers which only deal with the first or second case, the notation of (digital) order nets and (digital) order sequences can be sufficient. A solution to the third case is currently not known.

**Simplification 2: Removing and **

We now go back to the general case where is not known and consider all the parameters in this setting.

We now provide some definitions which do not include the parameters and Though has a natural meaning for digital higher order nets (the number of rows of the generating matrices), one could do without specifying it. For sequences on the other hand, one could simply set in the definition and not mention it as a parameter. In this case the definition could be modified to the following (we only consider the digital case, the other case can be modified analogously).

Defintion[Digital -net]

Let let be a real number and let be an integer. Let be the finite field of order , where is a prime power. Let (for some integer ) where is the th row vector of for and If for all withthe vectors

are linearly independent over then the digital net with generating matrices is called a digital -net over

Definition[Digital -sequence]

Let and and let be a real number. Let be the finite field of order , where is a prime power. Let where is the th row vector of the matrix for and Further let denote the left upper submatrix of for If for all the matrices generate a digital -net over then the digital sequence with generating matrices is a digital -sequence over

This is also a possible definition. It does however hide the dependence of the convergence rate on the number For instance consider the following result, first stated with the usual definition and then with the simplified definition:

- the convergence rate of the integration error using a digital -net cannot, in general, be asymptotically smaller than
- the convergence rate of the integration error using a digital -net with generating matrices cannot, in general, be asymptotically smaller than

I believe the first version is somewhat simpler, since is included as a parameter. This is to illustrate the motivation for including as a parameter. Similarly, some propagation rules also benefit from including the parameter To summarize, yes one can do without specifying but in some instances it is easier if it is included in the notation.

The motivation for including is principle 3, but could reasonably be left out. The reason for writing -nets for higher order nets is guiding principle 1. The case for higher order sequences also follows from the guiding principles.

**Simplification 3: Removing **

Now we turn to This parameter is important for higher order sequences and is included in higher order nets because of guiding principle 3. So let us discuss the meaning of for higher order sequences. The important higher order sequences are those for which see [page 15, 2] for more information. Now depends on and is a parameter which can be chosen independent of Currently there is no higher order sequence known for which for all For the constructions currently known we only have for all for some finite and arbitrarily chosen Currently there is no sequence known for which for all Currently the best known sequences have parameters of the form

for some arbitrary but fixed If there would be explicit constructions of higher order sequences for which for all then the parameter would be redundant. As long as this is not the case, the parameter is important when is not known.

For (digital) higher order nets, the parameter is included because of guiding principle 4.

**Conclusion**

To conclude, yes, some simplifications of the notation for (digital) higher order nets and sequences are possible and make sense in certain situations. Especially the assumption that is known simplifies the notation considerably. If is not known then one could do without and to some extend Still I believe that it is useful to adhere to the guiding principles to keep the notation consistent with each other. Comments are welcome.

Filed under: Open problems, Research Tagged: Higher order digital net, higher order digital sequence, higher order net, higher order sequence ]]>

The next ICIAM conference is held in Vancouver from 18th to 22nd of July 2011. Jan Baldeaux, Michael Gnewuch, Anargyros Papageorgiou and myself are going to propose several mini-symposia on Approximation, Tractability, Discrepancy theory and Applications of Monte Carlo and quasi-Monte Carlo methods. Currently we have 15 speakers. If you would like to join one of our mini-symposia please send me an email well before the 18th of October (which is the deadline for proposing mini-symposia).

The tenth international conference on Monte Carlo and quasi-Monte Carlo conference (MCQMC) will be held at UNSW in Sydney from 13th to 17th of February 2012. The official website can be found here. More information will be posted there in due course.

Filed under: Conferences Tagged: ICIAM 2011, MCQMC 2012 ]]>

- @article {MR0104086,

AUTHOR = {Korobov, N. M.},

TITLE = {Approximate evaluation of repeated integrals},

JOURNAL = {Dokl. Akad. Nauk SSSR},

FJOURNAL = {Doklady Akademii Nauk SSSR},

VOLUME = {124},

YEAR = {1959},

PAGES = {1207–1210},

} - Theorem 18 on page 120 in Korobov’s book from 1963:
@book {MR0157483,

AUTHOR = {Korobov, N. M.},

TITLE = {Teoretiko-chislovye metody v priblizhennom analize},

PUBLISHER = {Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow},

YEAR = {1963},

PAGES = {224},

}

In this post I provide a (rough) translation of the results of Korobov (I do not know Russian, so I only understand the formulae). Indeed, as can be seen below, Korobov used the algorithm which is now called component-by-component construction (and which was re-invented much later by Sloan and Reztsov in Sloan, I. H.; Reztsov, A. V. Component-by-component construction of good lattice rules. Math. Comp. 71 (2002), no. 237, 263–273). He also proved that lattice rules constructed component-by-component achieve the optimal rate of convergence of the integration error for functions in the class (i.e., he showed that where is the number of quadrature points).

Unfortunately many authors were not aware of Korobov’s result, hence Korobov has not received due credit for this result. To rectify this situation somewhat (and help make authors aware of Korobov’s result), we provide the details of Korobov’s result here.

**1. Some comments**

The proof of Korobov’s result is in Section 2. We first provide some comments on Korobov’s approach.

For integers let

Further let Then a classical error criterion for lattice rules is

where and Another, related criterion is

The following inequality is useful

Korobov uses a component-by-component approach on a quantity which is related to . This quantity is given by

(I renamed it in honour of Korobov, Korobov himself used the letter which is also used below.) The cbc algorithm is carried out using the quantity rather than Korobov then shows that and are close (see the Corollary below) and that a component-by-component construction of yields the optimal order of convergence.

**2. Korobov’s result**

In the following we prove Korobov’s result. Note, the result is entirely from Korobov’s book, it does NOT contain any contribution from the author of this blog. Further, Korobov’s result is actually slightly more general than the one stated here.

To prove the result, we prove several lemmas.

Lemma 1

Let be an integer. Then>

**Proof.**

Consider the roots of the polynomial which are given by

The polynomial can be factorized

Hence the roots of are given by Therefore we have

By setting in the last expression we obtain

> where the last equality follows from

This proves the lemma.

Lemma 2

Let . Thenwhere and

**Proof.**

We use the Fourier series expansion

Therefore we have

where

We have

Thus we have

The last inequality follows using the triangular inequality, the estimation , and from

Since we obtain

which we can write as

Lemma 3

Let such that and for Then

**Proof. **

For we have

Now assume that the result holds for We have Then

Lemma 4

Let We havewhere

and and for nonnegative real numbers

Lemma 4 implies the following corollary.

Corollary

Let We havewhere

and and for nonnegative real numbers

**Proof of Lemma 4. **

Set

and

where and let

Then we have

Further we have

Thus the conditions of Lemma 3 are satisfied. Using the notation just introduced, we can write the left-hand side of the expression in Lemma 4 as

since

Using Lemma 3 we obtain therefore

Let Then we have

Thus we obtain

Let be the set of functions given by the Fourier series expansion

for which there exists a constant independent of such that

It is known that the integration error for all functions with is bounded by

where

We can now prove the main result.

Theorem(Korobov, 1959)

Let Let be constructed component by component. That is, set and for choose which minimizes(Here, are fixed from the previous step.) Then we have

where

**Proof. **

We show that for We consider first. We have

where we used Lemma 1.

Now let and assume that the result holds for all We have

Thus we have for

We can now use Lemma 4 to obtain that

From this we obtain that

Since we obtain

Since the expression on the left-hand side is we can use the inequality relating to to obtain the result.

Filed under: Research Tagged: component by component construction, Korobov, lattice rule ]]>

**Lattice rule**

Let us first briefly describe what a lattice rule is. Let be natural numbers and let be an integer vector. For nonnegative real numbers we denote by the fractional part of that is, we have Here denotes the largest integer smaller or equal to For vectors we use this approach component-wise.

Lattice rule

Let be natural numbers and let be an integer vector. Let be a function. A lattice rule is a quadrature rule of the form

A lattice rule is useful to approximate the integral Background on lattice rules can be found in Sloan & Joe, Lattice methods for multiple integration, Oxford, 1994 and Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, SIAM, 1992.

**Lattice rules for **

In practice the integral one needs to approximate are usually defined on rather than One can use some transformation to transform an integral to an integral It is often not clear what transformation one should use and the quality of the approximation can depend a lot on a good choice of the transformation.

In order to avoid finding a transformation, we aim to find a `lattice rule type construction` which are constructed in As for the digital net case, the final point set in is not a classical lattice anymore. We want the point set to have a local structure, i.e., in certain subintervals of we basically have a classical shifted lattice rule and at the same time the whole point set should have a global structure. The goal is then to approximate the integral by a quadrature rule of the form

**A first idea**

The first approach to constructing lattice rules in would be analogous to the digital net case, see the post on Quasi-Monte Carlo rules on This requires several steps:

- First we need to solve the one-dimensional problem. Let and let denote its Fourier transform. We need some assumptions on the integrand. The integrand itself needs to go to zero fast enough as Further, the Fourier transform needs to go to zero fast enough as Finally, the derivative of the integrand needs to go to zero as These three assumptions should be used to give us some idea on how to construct the points in the one-dimensional case. Tilings of the phase plane are likely to be useful for this step (some useful ideas on tilings might be here or in the references in this paper).
- The first step should yield points for each coordinate Next we construct points in in the following way. Let be the generating vector for a lattice rule. For each we define the point by
One question is then to ask what structure such a point set has. Are there cases where one obtains shifted lattice rules again in some subcubes of This would be a useful property to have. The weights should then be chosen such that constant functions are integrated exactly in such subcubes.

- Find a criterion for the generating vector of the lattice rule and introduce a construction algorithm for such lattice rules. Alternatively one could just use a generating vector constructed for Korobov lattice rules.
- Prove error bounds for the integration error.

Whether this approach is feasible is currently not known. Any comments and suggestions are welcome.

Filed under: Open problems, Research Tagged: lattice rule, quasi-Monte Carlo ]]>