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It is known that each of the successive quotient groups of the grope andsolvable filtrations of the knot concordance group has an infinite ranksubgroup. The generating knots of these subgroups are constructed usingiterated doubling operators. In this paper, for each of the successivequotients of the filtrations we give a new infinite rank subgroup whichtrivially intersects the previously known infinite rank subgroups. Instead ofiterated doubling operators, the generating knots of these new subgroups areconstructed using the notion of algebraic $n$-solutions, which was introducedby Cochran and Teichner. Moreover, for each slice knot $K$ whose Alexanderpolynomial has degree greater than 2, we construct the generating knots suchthat they have the same derived quotients and higher-order Alexander invariantsup to a certain order.  In the proof, we use an $L^2$-theoretic obstruction for a knot to being$n.5$-solvable given by Cha, which is based on $L^2$-theoretic techniquesdeveloped by Cha and Orr. We also generalize the notion of algebraic$n$-solutions to the notion of $R$-algebraic $n$-solutions where $R$ is eitherrationals or the field of $p$ elements for a prime $p$.

Amenable signatures, algebraic solutions, and filtrations of the knot concordance group