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结合时间方向稀疏小波展开与空间方向自适应低秩近似的抛物型偏微分方程自适应方法。理论证明了该方法的收敛性，并建立了与现有椭圆问题自适应低秩方法相似的复杂度界，验证了其在高维空间域抛物型问题中的适用性。该构造还为此类问题提供了可计算的严格后验误差界。数值实验验证了理论结果。