naginterfaces.library.quad.md_​sgq_​multi_​vec

naginterfaces.library.quad.md_sgq_multi_vec(ni, f, maxdlv, comm, data=None, spiked_sorder='C')

md_sgq_multi_vec approximates a vector of definite integrals $F$ over the unit hypercube $\Omega ={[0,1]}^d$, given the vector of integrands $f\left(x\right)$.

$$F={\int }_{\Omega }f\left(x\right)dx={\int }_0^1{\int }_0^1\dots {\int }_0^{1}f(x_1,x_2,\dots ,x_d)d_{x_1}{d}_{x_2}\dots d_{x_d}\text{.}$$

The function uses a sparse grid discretisation, allowing for computationally feasible estimations of integrals of high dimension ($d\sim O\left(100\right)$).

Note: this function uses optional algorithmic parameters, see also: opt_set(), opt_get().

For full information please refer to the NAG Library document for d01es

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d01/d01esf.html

Parameters

niint

$n_i$, the number of integrands.

fcallable (fm, iflag) = f(ni, ndim, xtr, icolzp, irowix, xs, qs, iflag, data=None)

$f$ must return the value of the integrands $f_j$ at a set of $n_x$ $d$-dimensional points $x_i$, implicitly supplied as columns of a matrix $X(d,n_x)$.

If $X$ was supplied explicitly you would find that most of the elements attain the same value, $x_{\text{tr}}$; the larger the number of dimensions, the greater the proportion of elements of $X$ would be equal to $x_{\text{tr}}$.

So, $X$ is effectively a sparse matrix, except that the ‘zero’ elements are replaced by elements that are all equal to the value $x_{\text{tr}}$.

For this reason $X$ is supplied, as though it were a sparse matrix, in compressed column storage (CCS) format (see the F11 Introduction).

Individual entries $x_{\text{k},i}$ of $X$, for $\text{k}=1,2,\dots ,d$, are either trivially valued, in which case $x_{k,i}=x_{\text{tr}}$, or are non-trivially valued.

For point $i$, the non-trivial row indices and corresponding abscissae values are supplied in elements $c\left(\text{i}\right)=icolzp[\text{i}-1],\dots ,icolzp\left[\text{i}\right]-1$, for $\text{i}=1,2,\dots ,n_x$, of the arrays $irowix$ and $xs$, respectively.

Hence the $i$th column of the matrix $X$ is retrievable as

$$\begin{array}{c}\begin{array}{c}X\left(irowix\right[c\left(i\right)-1],i)=xs[c\left(i\right)-1]\text{,}\\ \\ X(k\notin irowix[c\left(i\right)-1],i)=x_{\text{tr}}\text{.}\end{array}\end{array}$$

An equivalent integer valued matrix $\text{Q}$ is also implicitly provided.

This contains the unique indices $q_{k,i}$ of the underlying one-dimensional quadrature rule corresponding to the individual abscissae $x_{k,i}$.

For trivial abscissae, the implicit index $q_{k,i}=1$. $\text{Q}$ is supplied in the same CCS format as $X$, with the non-trivial values supplied in $qs$.

Parameters

niint

$n_i$, the number of integrands.

ndimint

$\text{d}$, the number of dimensions.

xtrfloat

$x_{\text{tr}}$, the value of the trivial elements of $X$.

icolzpint, ndarray, shape $(\text{nx}+1)$

The set $\left\{icolzp\right[\text{i}-1],\dots ,icolzp[\text{i}]-1\}$ contains the indices of $irowix$ and $xs$ corresponding to the non-trivial elements of column $\text{i}$ of $X$ and hence of the point $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n_x$.

irowixint, ndarray, shape $\left(\text{nntr}\right)$

The row indices corresponding to the non-trivial elements of $X$.

xsfloat, ndarray, shape $\left(\text{nntr}\right)$

$x_{k,i}\ne x_{\text{tr}}$, the non-trivial entries of $X$.

qsint, ndarray, shape $\left(\text{nntr}\right)$

$q_{k,i}\ne 1$, the indices of the underlying quadrature rules corresponding to $x_{k,i}\ne x_{\text{tr}}$.

iflagint

If $iflag=0$, this is the first call to $f$. $n_x=1$, and the entire point $x_1$ will satisfy $x_{\text{k},1}=x_{\text{tr}}$, for $\text{k}=1,2,\dots ,d$. In addition, $\text{nntr}$ contains the total number of abscissae from the underlying one-dimensional quadrature; $xs$ contains the complete set of abscissae and $qs$ contains the corresponding quadrature indices, with $xs\left[0\right]=x_{\text{tr}}$ and $qs\left[0\right]=1$. This will always be called in serial.

In subsequent calls to $f$, $iflag=1$.

Subsequent calls may be made from within an OpenMP parallel region.

See Parallelism and Performance for details.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

fmfloat, array-like, shape $(ni,\text{nx})$

$fm[\text{p}-1,\text{i}-1]=f_{\text{p}}\left(x_{\text{i}}\right)$, for $\text{p}=1,2,\dots ,n_{\text{i}}$, for $\text{i}=1,2,\dots ,n_x$.

iflagint

Set $iflag<0$ if you wish to force an immediate exit from md_sgq_multi_vec with $errno$ = -1.

maxdlvint, array-like, shape $\left(\text{ndim}\right)$

$m$, the array of maximum levels for each dimension. $maxdlv[\text{j}-1]$, for $\text{j}=1,2,\dots ,d$, contains $m_{\text{j}}$, the maximum level of quadrature rule dimension j will support.

The default value, $min(m_q,L)$ will be used if either $maxdlv[j-1]\le 0$ or $maxdlv[j-1]\ge min(m_q,L)$ (for details on the default values for $m_q$ and $L$ and on how to change these values see the options ‘Maximum Level’, ‘Maximum Quadrature Level’ and ‘Quadrature Rule’).

If $maxdlv[j-1]=1$ for all $j$, only one evaluation will be performed, and as such no error estimation will be possible.

Note: setting non-default values for some dimensions makes the assumption that the contribution from the omitted subspaces is $0$. The integral and error estimates will only be based on included subspaces, which if the $0$ contribution assumption is not valid will be erroneous.

Suggested value: $maxdlv[j-1]=0$ for all $j=1,2,\dots ,d$.

commdict, communication object

Communication structure.

This argument must have been initialized by a prior call to opt_set().

dataarbitrary, optional

User-communication data for callback functions.

spiked_sorderstr, optional

If $fm$ in $f$ is spiked (i.e., has unit extent in all but one dimension, or has size $1$), $spiked\_sorder$ selects the storage order to associate with it in the NAG Engine:

spiked_sorder = $\text{'C'}$

row-major storage will be used;

spiked_sorder = $\text{'F'}$

column-major storage will be used.

Returns

dinestfloat, ndarray, shape $\left(ni\right)$

$dinest[\text{p}-1]$ contains the final estimate ${\hat{F}}_{\text{p}}$ of the definite integral $F_p$, for $\text{p}=1,2,\dots ,n_i$.

errestfloat, ndarray, shape $\left(ni\right)$

$errest[\text{p}-1]$ contains the final error estimate $E_p$ of the definite integral $F_p$, for $\text{p}=1,2,\dots ,n_i$.

ivalidint, ndarray, shape $\left(ni\right)$

$ivalid[\text{p}-1]$ indicates the final state of integral $\text{p}$, for $\text{p}=1,2,\dots ,n_i$.

$ivalid[p-1]=0$

The error estimate for integral $p$ was below the requested tolerance.

$ivalid[p-1]=1$

The error estimate for integral $p$ was below the requested tolerance. The final level used was non-isotropic.

$ivalid[p-1]=2$

The error estimate for integral $p$ was above the requested tolerance.

$ivalid[p-1]=3$

The error estimate for integral $p$ was above $max(0.1\left|{\hat{F}}_p\right|,0.01)$.

$ivalid[p-1]<0$

You aborted the evaluation before an error estimate could be made.

Other Parameters

‘Absolute Tolerance’float

Default $=\sqrt{ϵ}$

$r={ϵ}_a$, the absolute tolerance required.

‘Index Level’int

Default $=4$

The maximum level at which function values are stored for later use. Larger values use increasingly more memory, and require more time to access specific elements. Lower levels result in more repeated computation. The ‘Maximum Quadrature Level’, $m_q$ is the effective upper limit on ‘Index Level’. If $i\ge m_q$, md_sgq_multi_vec will use $m_q$ as the value of ‘Index Level’.

Constraint: $i\ge 1$.

‘Maximum Level’int

Default $=5$

$i=L$, the maximum number of levels to evaluate.

Constraint: $1<i\le 20$.

Note: the maximum allowable level in any single dimension, $m_q$, is governed by the ‘Quadrature Rule’ selected. If a value greater than $m_q$ is set, only a subset of subspaces from higher levels will be used. Should this subset be empty for a given level, computation will consider the preceding level to be the maximum level and will terminate.

‘Maximum Nx’int

Default $=128$

$i=max{\text{n}}_x$, the maximum number of points to evaluate in a single call to $f$.

Constraint: $1\le i\le 16384$.

‘Maximum Quadrature Level’int

Queriable only

$i=m_q$, the maximum level of the underlying one-dimensional quadrature rule (see ‘Quadrature Rule’).

‘Minimum Level’int

Default $=2$

The minimum number of levels which must be evaluated before an error estimate is used to determine convergence.

Constraint: $i>1$.

Note: if the minimum level is greater than the maximum computable level, the maximum level will be used.

‘Quadrature Rule’str

Default $=\text{'Gauss-Patterson'}$

The underlying one-dimensional quadrature rule to be used in the construction. Open rules do not require evaluations at boundaries.

$\text{‘Quadrature Rule'}=\text{'Gauss-Patterson'}$ or $\text{'GP'}$

The interlacing Gauss–Patterson rules. Level $\ell$ uses $2^{\ell }-1$ abscissae. All levels are open. These rules provide high order accuracy. $m_q=9$.

$\text{‘Quadrature Rule'}=\text{'Clenshaw-Curtis'}$ or $\text{'CC'}$

The interlacing Clenshaw–Curtis rules. Level $\ell$ uses $2^{\ell -1}+1$ abscissae. All levels above level $1$ are closed. $m_q=12$.

‘Relative Tolerance’float

Default $=\sqrt{ϵ}$

$r={ϵ}_a$, the relative tolerance required.

‘Summation Precision’str

Default $=\text{'HIGHER'}$

Determines whether md_sgq_multi_vec uses working precision or higher-than-working precision to accumulate the actions over subspaces.

$\text{‘Summation Precision'}=\text{'HIGHER'}$ or $\text{'H'}$

Higher-than-working precision is used to accumulate the action over a subspace, and for the accumulation of all such actions. This is more expensive computationally, although this is probably negligible in comparison to the cost of evaluating the integrands and the overall runtime. This significantly reduces variation in the result when changing the number of threads.

$\text{‘Summation Precision'}=\text{'WORKING'}$ or $\text{'W'}$

Working precision is used to accumulate the actions over subspaces. This may provide some speedup, particularly if $n_i$ or $n_t$ is large. The results of parallel simulations will however be more prone to variation.

‘Serial Levels’int

Default $=1$

$i=s_l$, the number of levels to be evaluated in serial before initializing parallelization. For relatively trivial integrands, this may need to be set greater than the default to reduce parallel overhead.

Raises

NagValueError

(errno $11$)

On entry, $ni=⟨value⟩$.

Constraint: $ni\ge 1$.

(errno $21$)

On entry, $\text{ndim}=⟨value⟩$.

Constraint: $\text{ndim}\ge 1$.

(errno $1001$)

Either the option arrays $comm$[‘iopts’] and $comm$[‘opts’] have not been initialized for md_sgq_multi_vec, or they have become corrupted.

Warns

NagAlgorithmicWarning

(errno $-1$)

Exit requested from $f$ with $iflag=⟨value⟩$.

(errno $1$)

The requested accuracy was not achieved for at least one integral.

(errno $2$)

No accuracy was achieved for at least one integral.

Notes

md_sgq_multi_vec uses a sparse grid to generate a vector of approximations $\hat{F}$ to a vector of integrals $F$ over the unit hypercube $\Omega ={[0,1]}^d$, that is,

$$\hat{F}\approx F={\int }_{{[0,1]}^d}f\left(x\right)dx\text{.}$$

Comparing Quadrature Over Full and Sparse Grids

Before illustrating the sparse grid construction, it is worth comparing integration over a sparse grid to integration over a full grid.

Given a one-dimensional quadrature rule with $N$ abscissae, which accurately evaluates a polynomial of order $P_N$, a full tensor product over $d$ dimensions, a full grid, may be constructed with $N^d$ multidimensional abscissae. Such a product will accurately integrate a polynomial where the maximum power of any dimension is $P_N$. For example if $d=2$ and $P_N=3$, such a rule will accurately integrate any polynomial whose highest order term is $x_1^3{x}_2^3$. Such a polynomial may be said to have a maximum combined order of $P_N^d$, provided no individual dimension contributes a power greater than $P_N$. However, the number of multidimensional abscissae, or points, required increases exponentially with the dimension, rapidly making such a construction unusable.

The sparse grid technique was developed by Smolyak (Smolyak (1963)). In this, multiple one-dimensional quadrature rules of increasing accuracy are combined in such a way as to provide a multidimensional quadrature rule which will accurately evaluate the integral of a polynomial whose maximum order appears as a monomial. Hence a sparse grid construction whose highest level quadrature rule has polynomial order $P_N$ will accurately integrate a polynomial whose maximum combined order is also $P_N$. Again taking $P_N=3$, one may, theoretically, accurately integrate a polynomial such as $x^3+x^{2}y+y^3$, but not a polynomial such as $x^3{y}^3+xy$. Whilst this has a lower maximum combined order than the full tensor product, the number of abscissae required increases significantly slower than the equivalent full grid, making some classes of integrals of dimension $d\sim O\left(100\right)$ tractable. Specifically, if a one-dimensional quadrature rule of level $\ell$ has $N\sim O\left(2^{\ell }\right)$ abscissae, the corresponding full grid will have $O\left({\left(2^{\ell }\right)}^d\right)$ multidimensional abscissae, whereas the sparse grid will have $O\left(2^{\ell }{d}^{\ell -1}\right)$. Figure [label omitted] demonstrates this using a Gauss–Patterson rule with $15$ points in $3$ dimensions. The full grid requires $3375$ points, whereas the sparse grid only requires $111$.

[figure omitted]

Sparse Grid Quadrature

We now include a brief description of the sparse grid construction, sufficient for the understanding of the use of this routine. For a more detailed analysis, see Gerstner and Griebel (1998).

Consider a one-dimensional $n_{\ell }$-point quadrature rule of level $\ell$, $Q_{\ell }$. The action of this rule on a integrand $f$ is to approximate its definite one-dimensional integral ${}_{1}F$ as,

$${}_{1}F={\int }_0^{1}f\left(x\right)dx\approx Q_{\ell }\left(f\right)=\sum _1^{n_{\ell }}{w}_{\ell ,i}\times f\left(x_{\ell ,i}\right)\text{,}$$

using weights $w_{\ell ,i}$ and abscissae $x_{\ell ,i}$, for $\text{i}=1,2,\dots ,n_{\ell }$.

Now construct a set of one-dimensional quadrature rules, $\{Q_{\ell }|\ell =1,\dots ,L\}$, such that the accuracy of the quadrature rule increases with the level number. In this routine we exclusively use quadrature rules which are completely nested, implying that if an abscissae $x_{\ell ,k}$ is in level $\ell$, it is also in level $\ell +1$. The quantity $L$ denotes some maximum level appropriate to the rules that have been selected.

Now define the action of the tensor product of $d$ rules as,

$$(Q_{{\ell }_1}\otimes \cdots \otimes Q_{{\ell }_d})\left(f\right)=\sum _1^{n_{{\ell }_1}}\cdots \sum _1^{n_{{\ell }_d}}{w}_{{\ell }_1,i_1}\cdots w_{{\ell }_d,i_d}f(x_{{\ell }_1,i_1},\dots ,x_{{\ell }_d,i_d})\text{,}$$

where the individual level indices ${\ell }_j$ are not necessarily ordered or unique. Each tensor product of $d$ rules defines an action of the quadrature rules $Q_{\ell }$, $\ell =({\ell }_1,{\ell }_2,\dots ,{\ell }_d)$ over a subspace, which is given a level $\left|\ell \right|={\sum }_1^d{\ell }_j$. If all rule levels are equal, this is the full tensor product of that level.

The sparse grid construction of level $\ell$ can then be declared as the sum of all actions of the quadrature differences ${\Delta }_k=(Q_k-Q_{k-1})$, over all subspaces having a level at most $\ell -d+1$,

$${}_{d}F\approx Q_{\ell }^d\left(f\right)=\sum _{\text{level at most}\ell -d+1}({\Delta }_{k_1}\otimes \cdots \otimes {\Delta }_{k_d})\left(f\right)\text{.}$$

By definition, all subspaces used for level $\ell -1$ must also be used for level $\ell$, and as such the difference between the result of all actions over subsequent sparse grid constructions may be used as an error estimate.

Let $L$ be the maximum level allowable in a sparse grid construction. The classical sparse grid construction of $\ell =L$ allows each dimension to support a one-dimensional quadrature rule of level at most $L$, with such a quadrature rule being used in every dimension at least once. Such a construction lends equal weight to each dimension of the integration, and is termed here ‘isotropic’.

Define the set $m=(m_j,j=1,2,\dots ,d)$, where $m_j$ is the maximum quadrature rule allowed in the $j$th dimension, and $m_q$ to be the maximum quadrature rule used by any dimension. Let a subspace be identified by its quadrature difference levels, $k=(k_1,k_2,\dots ,k_d)$.

The classical construction may be extended by allowing different dimensions to have different values $m_j$, and by allowing $m_q\le L$. This creates non-isotropic constructions. These are especially useful in higher dimensions, where some dimensions contribute more than others to the result, as they can drastically reduce the number of function evaluations required.

For example, consider the two-dimensional construction with $L=4$. The classical isotropic construction would have the following subspaces.

Subspaces generated by a classical sparse grid with $L=4$.

Level	Subspaces
$1$	$(1,1)$
$2$	$(2,1)$, $(1,2)$
$3$	$(3,1)$, $(2,2)$, $(1,3)$
$4$	$(4,1)$, $(3,2)$, $(2,3)$, $(1,4)$

If the variation in the second dimension is sufficiently accurately described by a quadrature rule of level $2$, the contributions of the subspaces $(1,3)$ and $(1,4)$ are probably negligible. Similarly, if the variation in the first dimension is sufficiently accurately described by a quadrature rule of level $3$, the subspace $(4,1)$ is probably negligible. Furthermore the subspace $(2,3)$ would also probably have negligible impact, whereas the subspaces $(2,2)$ and $(3,2)$ would not. Hence restricting the first dimension to a maximum level of $3$, and the second dimension to a maximum level of $2$ would probably give a sufficiently acceptable estimate, and would generate the following subspaces.

Subspaces generated by a non-isotropic sparse grid with $L=4$, $m_q=3$ and $m=(3,2)$.

Level	Subspaces
$1$	$(1,1)$
$2$	$(2,1)$, $(1,2)$
$3$	$(3,1)$, $(2,2)$
$4$	$(4,1)$, $(3,2)$

Taking this to the extreme, if the variation in the first and second dimensions are sufficiently accurately described by a level $2$ quadrature rule, restricting the maximum level of both dimensions to $2$ would generate the following subspaces.

Subspaces generated by a sparse grid construction with $L=4$, $m_q=2$ and $m=(2,2)$.

Level	Subspaces
$1$	$(1,1)$
$2$	$(2,1)$, $(1,2)$
$3$	$(2,2)$
$4$	None

Hence one subspace is generated at level $3$, and no subspaces are generated at level $4$. The level $3$ subspace $(2,2)$ actually indicates that this is the full grid of level $2$.

Using md_sgq_multi_vec

md_sgq_multi_vec uses options, supplied in the option arrays $comm$[‘iopts’] and $comm$[‘opts’]. Before calling md_sgq_multi_vec, these option arrays must be initialized using opt_set(). Once initialized, the required options may be set and queried using opt_set() and opt_get() respectively. A complete list of the options available may be found in Other Parameters.

You may control the maximum level required, $L$, using the option ‘Maximum Level’. Furthermore, you may control the first level at which the error comparison will take place using the option ‘Minimum Level’, allowing for the forced evaluation of a predetermined number of levels before the routine attempts to complete. Completion is flagged when the error estimate is sufficiently small:

$$|{\hat{F}}_d^k-{\hat{F}}_d^{k-1}|\le max({ϵ}_a,{ϵ}_r\times {\hat{F}}_d^k)\text{,}$$

where ${ϵ}_a$ and ${ϵ}_r$ are the absolute and relative error tolerances, respectively, and $k\le L$ is the highest level at which computation was performed. The tolerances ${ϵ}_a$ and ${ϵ}_r$ can be controlled by setting the options ‘Absolute Tolerance’ and ‘Relative Tolerance’.

Owing to the interlacing nature of the quadrature rules used herein, abscissae $x$ required in lower level subspaces will also appear in higher-level subspaces. This allows for calculations which will be repeated later to be stored and re-used. However, this is naturally at the expense of memory. It may also be at the expense of computational time, depending on the complexity of the integrands, as the lookup time for a given value is (at worst) $O\left(d\right)$. Furthermore, as the sparse grid level increases, fewer subsequent levels will require values from the current level. You may control the number of levels for which values are stored by setting the option ‘Index Level’.

Two different sets of interlacing quadrature rules are selectable using the option ‘Quadrature Rule’: Gauss–Patterson and Clenshaw–Curtis. Gauss–Patterson rules offer greater polynomial accuracy, whereas Clenshaw–Curtis rules are often effective for oscillatory integrands. Clenshaw–Curtis rules require function values to be evaluated on the boundary of the hypercube, whereas Gauss–Patterson rules do not. Both of these rules use precomputed weights, and as such there is an effective limit on $m_q$; see the description of the option ‘Quadrature Rule’. The value of $m_q$ is returned by the queriable option ‘Maximum Quadrature Level’.

md_sgq_multi_vec also allows for non-isotropic sparse grids to be constructed. This is done by appropriately setting the array $maxdlv$. It should be emphasised that a non-isometric construction should only be used if the integrands behave in a suitable way. For example, they may decay toward zero as the lesser dimensions approach their bounds of $\Omega$. It should also be noted that setting $maxdlv[k-1]=1$ will not reduce the dimension of the integrals, it will simply indicate that only one point in dimension $k$ should be used. It is also advisable to approximate the integrals several times, once with an isometric construction of some level, and then with a non-isometric construction with higher levels in various dimensions. If the difference between the solutions is significantly more than the returned error estimates, the assumptions of dimensional importance are probably incorrect.

The abscissae in each subspace are generally expressible in a sparse manner, because many of the elements of each abscissa will in fact be the centre point of the dimension, which is termed here the ‘trivial’ element. In this function the trivial element is always returned as $0.5$ owing to the restriction to the $[0,1]$ hypercube. As such, the function $f$ returns the abscissae in Compressed Column Storage (CCS) format (see the F11 Introduction). This has particular advantages when using accelerator hardware to evaluate the required functions, as much less data must be forwarded. It also, potentially, allows for calculations to be computed faster, as any sub-calculations dependent upon the trivial value may be potentially re-used.

References

Caflisch, R E, Morokoff, W and Owen, A B, 1997, Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension, Journal of Computational Finance (1), 27–46

Gerstner, T and Griebel, M, 1998, Numerical integration using sparse grids, Numerical Algorithms (18), 209–232

Smolyak, S A, 1963, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR (4), 240–243