We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator $A$, a time $t>0$, an arbitrary initial vector $u_0$ and an error tolerance $\epsilon>0$, the algorithm computes $\exp(tA)u_0$ with error bounded by $\epsilon$. The algorithm is based on a combination of a regularized functional calculus, suitable contour quadrature rules, and the adaptive computation of resolvents in infinite dimensions. As a particular case, we show that it is possible, even when only allowing pointwise evaluation of coefficients, to compute, with error control, semigroups on the unbounded domain $L^2(\mathbb{R}^d)$ that are generated by partial differential operators with polynomially bounded coefficients of locally bounded total variation. For analytic semigroups (and more general Laplace transform inversion), we provide a quadrature rule whose error decreases like $\exp(-cN/\log(N))$ for $N$ quadrature points, that remains stable as $N\rightarrow\infty$, and which is also suitable for infinite-dimensional operators. Numerical examples are given, including: Schr\"odinger and wave equations on the aperiodic Ammann--Beenker tiling, complex perturbed fractional diffusion equations on $L^2(\mathbb{R})$, and damped Euler--Bernoulli beam equations.
翻译:我们开发了一种算法, 在无限的Hilbert 空格上计算强烈连续的半组, 并有明确的错误控制。 如果有发电机 $A, 时间 > 0 美元, 任意初始矢量 $_ 0 美元 和错误容忍 $\ epsilon> 0 美元, 算法计算 $\ exm( tA) u_ 0 美元, 由 $\ epsilon 受 $\ epsilon 约束的错误构成。 算法基于常规功能计算、 合适的等等等功能的计算组合, 以及无限尺寸的固态计算。 具体的例子是, 即使只允许对系数进行点度评估, 也只能用错误控制来计算, 在无界域的 $L2( mathbrickral) 中半组 美元(rickal-rickal- developlemental $Nral- develople) 。