An adaptive method for parabolic partial differential equations that combines sparse wavelet expansions in time with adaptive low-rank approximations in the spatial variables is constructed and analyzed. The method is shown to converge and satisfy similar complexity bounds as existing adaptive low-rank methods for elliptic problems, establishing its suitability for parabolic problems on high-dimensional spatial domains. The construction also yields computable rigorous a posteriori error bounds for such problems. The results are illustrated by numerical experiments.

中文翻译：

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高维抛物型偏微分方程的时空自适应低秩方法

构建并分析了一种抛物型偏微分方程的自适应方法，该方法将时间上的稀疏小波展开与空间变量中的自适应低秩近似相结合。该方法被证明可以收敛并满足与椭圆问题的现有自适应低秩方法类似的复杂性界限，从而确立了其对高维空间域上的抛物线问题的适用性。该构造还为此类问题产生可计算的严格后验误差界限。结果通过数值实验来说明。