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In this paper we describe methods to approximate functions and differential operators on adaptive sparse (dyadic) grids. We distinguish between several representations of a function on the sparse grid and we describe how finite difference (FD) operators can be applied to these representations. For general variable coefficient equations on sparse grids, genuine finite element (FE) discretizations are not feasible and FD operators allow an easier operator evaluation than the adapted FE operators. However, the structure of the FD operators is complex. With the aim to construct an efficient multigrid procedure, we analyze the structure of the discrete Laplacian in its hierarchical representation and show the relation between the full and the sparse grid case. The rather complex relations, that are expressed by scaling matrices for each separate coordinate direction, make us doubt about the possibility of constructing efficient preconditioners that show spectral equivalence. Hence, we question the possibility of constructing a natural multigrid algorithm with optimal O(N) efficiency. We conjecture that for the efficient solution of a general class of adaptive grid problems it is better to accept an additional condition for the dyadic grids (condition L) and to apply adaptive hp-discretization.
Metagenomic studies use high-throughput sequence data to investigate microbial communities in situ. However, considerable challenges remain in the analysis of these data, particularly with regard to speed and reliable analysis of microbial species as opposed to higher level taxa such as phyla. We here present Genometa, a computationally undemanding graphical user interface program that enables identification of bacterial species and gene content from datasets generated by inexpensive high-throughput short read sequencing technologies. Our approach was first verified on two simulated metagenomic short read datasets, detecting 100% and 94% of the bacterial species included with few false positives or false negatives. Subsequent comparative benchmarking analysis against three popular metagenomic algorithms on an Illumina human gut dataset revealed Genometa to attribute the most reads to bacteria at species level (i.e. including all strains of that species) and demonstrate similar or better accuracy than the other programs. Lastly, speed was demonstrated to be many times that of BLAST due to the use of modern short read aligners. Our method is highly accurate if bacteria in the sample are represented by genomes in the reference sequence but cannot find species absent from the reference. This method is one of the most user-friendly and resource efficient approaches and is thus feasible for rapidly analysing millions of short reads on a personal computer.
The question type STACK in the learning management system Moodle is well suited for mathematical questions in higher education courses. Since theoretical computer science is rather close to mathematics, we developed STACK questions for this topic in order to train our students in using abstract notation. Because STACK and the underlying computer algebra system Maxima are not made for dealing with strings, we needed some special functions for our purposes.
At University of Applied Sciences and Arts Hannover, LON-CAPA is used as a learning management system beside Moodle. LON-CAPA has a strong focus on e-assessment in mathematics and sciences. We used LON-CAPA in Hannover mainly in mathematics courses.
Since theoretical computer science needs a lot of mathematics, this course is also well-suited for e-assessment in LON-CAPA. Beside this, we already used JFLAP as an interactive tool to deal with automata, machines and grammars in theoretical computer science. In LON-CAPA, there exists a possibility of using external graders to grade problems.
We decided to write a grading engine (with JFLAP inside) to grade automata, machines and grammars handed in by students and to couple this with LON-CAPA. This report describes the types of questions that are now possible with this grader and how they can be authored in LON-CAPA.
We have used LON-CAPA for computer-aided assessment of problems in mathematics and computer science for several years and are now in the process to migrate to Moodle. We developed the tool LC2Mdl that allows half-automatic migration of several types of LON-CAPA problems to Moodle STACK XML files. Both systems allow randomization and use Maxima as the computer algebra system in the background. In LON-CAPA most parts are written in Perl which is not supported by Moodle in any way. So, fully automatic migration is not possible. Usually, you will have to do additional work by hand. But LC2Mdl saves a lot of time in this migration process.
Grammars, automata, and machines in theoretical computer science can be viewed as specialized types of programs. Automata and grammars can be graded automatically using sets of words that either belong or don't belong to the corresponding language. For machines, traditional unit testing with predefined input and output can be performed. Additionally, we can verify whether automata, grammars, or machines are of the requested type. Students use JFLAP for constructing and testing automata and machines. Our grading system, GraFLAP, builds upon JFLAP and includes additional testing capabilities. We provide tasks in ProFormA format and utilize Grappa to connect the grader to Moodle.
