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.

翻译：本文构造并分析了一种针对抛物型偏微分方程的自适应方法，该方法在时间上采用稀疏小波展开，在空间变量上则结合了自适应低秩近似。证明了该方法的收敛性，并表明其满足与现有椭圆问题自适应低秩方法相类似的复杂度界，从而确立了其在高维空间域抛物型问题中的适用性。此外，该构造还为这类问题提供了可计算的严格后验误差界。数值实验验证了所得结果。