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  "body": "\n## Definition\n\nLet $F$ be a set, let $G$ be a [[normed vector space|03J5RAKB30]], and let $S \\colon F \\to G$ be a map, the *solution operator*.\nLet $\\Lambda$ be a set of maps $F \\to \\R$, the *admissible information functionals*, for example the evaluations $f \\mapsto f(t)$.\nFor $n \\ge 0$, an *algorithm using $n$ pieces of information* is a map $A = \\phi \\circ N$ where:\n\n1. $N(f) = \\big(L_1(f), \\ldots, L_n(f)\\big)$, with $L_1 \\in \\Lambda$ and each $L_k \\in \\Lambda$ chosen as a function of the values $L_1(f), \\ldots, L_{k-1}(f)$;\n2. $\\phi$ is any map from the set of possible values of $N$ to $G$.\n\nThe *worst-case error* of $A$ is $e(A) = \\sup_{f \\in F} \\norm{S(f) - A(f)}$.\nThe *$n$-th minimal error* is $e(n) = \\inf_A e(A)$ over all algorithms $A$ using $n$ pieces of information.\nFor $\\varepsilon \u003e 0$, the *information complexity* of the problem is\n$$n(\\varepsilon) = \\min \\set{n \\ge 0 : e(n) \\le \\varepsilon},$$\nwith $n(\\varepsilon) = \\infty$ if no such $n$ exists.\n*Information-based complexity* is the study of $e(n)$ and $n(\\varepsilon)$, and of the total cost $\\mathrm{comp}(\\varepsilon)$ that also counts the arithmetic operations of $\\phi$.\n\n## Remarks\n\nExample: let $F$ be the set of functions $f \\colon [0,1] \\to \\R$ with $\\abs{f(s) - f(t)} \\le \\abs{s - t}$, let $S(f) = \\int_0^1 f(t) \\, dt \\in \\R$, and let $\\Lambda$ be the function evaluations.\nThe midpoint rule $A(f) = \\frac{1}{n} \\sum_{k=1}^{n} f\\big(\\frac{2k-1}{2n}\\big)$ has $e(A) = \\frac{1}{4n}$, since on each subinterval of length $1/n$ the error is at most $\\int \\abs{t - c} \\, dt = \\frac{1}{4n^2}$, where $c$ is its midpoint.\nNo algorithm does better, so $e(n) = \\frac{1}{4n}$ and $n(\\varepsilon) = \\ceil{1/(4\\varepsilon)}$.\nNon-example: the map $A(f) = \\int_0^1 f$ itself has error $0$, but it uses infinitely many pieces of information.\nIndeed two functions in $F$ that agree at finitely many points can have different integrals.\nBesides the worst-case setting, there are average-case, probabilistic and randomized settings, where the supremum over $F$ is replaced by an expectation or the algorithm may use random choices.\nOften $\\Lambda$ is the class of all continuous [[linear functionals|07YQ1C78G0]] on a normed space containing $F$.\nA problem is called *polynomially tractable* when $n(\\varepsilon)$ for the $d$-variate version grows at most polynomially in $d$ and $1/\\varepsilon$.\n",
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