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  "body": "\n## Definition\n\nLet $d \\ge 1$ and let $X^1 \\subseteq X^2 \\subseteq X^3 \\subseteq \\cdots$ be a nested sequence of finite subsets of an interval $I$.\nFor an integer $q \\ge d$, the *sparse grid* of level $q$ in dimension $d$ is the set\n$$H(q, d) = \\bigcup_{\\substack{\\mathbf{i} \\in \\N^d,\\ i_1, \\dots, i_d \\ge 1 \\\\ i_1 + \\cdots + i_d \\le q}} X^{i_1} \\times X^{i_2} \\times \\cdots \\times X^{i_d} \\subseteq I^d.$$\nLet $U^i$ be a [[quadrature rule|ADSVMCY77A]] on $I$ with nodes $X^i$, let $U^0 = 0$, and let $\\Delta^i = U^i - U^{i-1}$.\nThe Smolyak rule is\n$$A(q, d) = \\sum_{i_1 + \\cdots + i_d \\le q} \\Delta^{i_1} \\otimes \\cdots \\otimes \\Delta^{i_d}.$$\nIt is a rule on $I^d$ whose nodes lie in $H(q, d)$.\n\n## Remarks\n\nThe full tensor grid $X^{q - d + 1} \\times \\cdots \\times X^{q - d + 1}$ is the union of the products with every $i_j \\le q - d + 1$; the sparse grid replaces this box condition on $\\mathbf{i}$ by the simplex condition $i_1 + \\cdots + i_d \\le q$.\nWrite $\\ell = q - d + 1$ for the finest level in each direction.\nWhen $X^i$ has about $2^i$ points and $d$ is fixed, the sparse grid has $O(2^\\ell \\ell^{d-1})$ points, while the full grid $X^\\ell \\times \\cdots \\times X^\\ell$ has about $2^{\\ell d}$ points.\n\nExample: take $I = [-1, 1]$, $X^1 = \\set{0}$ and $X^2 = \\set{-1, 0, 1}$, and $q = d + 1$.\nThe allowed $\\mathbf{i}$ are $(1, \\dots, 1)$ and those with one entry $2$ and the others $1$.\nSo $H(d + 1, d)$ consists of the origin and the $2d$ points $\\pm e_j$, which makes $2d + 1$ points.\n\nNon-example: the full grid $\\set{-1, 0, 1}^d$ is not $H(d + 1, d)$ for $d \\ge 2$, since it contains $(1, 1, 0, \\dots, 0)$, which would need $\\mathbf{i}$ with two entries equal to $2$ and so $i_1 + \\cdots + i_d = d + 2 \u003e q$.\n",
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