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Figure 1: Examples of three different shape-from-operator problems considered in the paper. Left: shape analogy synthesis as shape-from-difference operator problem (shape X is synthesized such that the intrinsic difference operator between C, X is as close as possible to the difference between A, B). Center: style transfer as shape-from-Laplacian problem. The Laplacian of the leftmost shape (fat man) captures the style, while the initial thin man shape (middle) captures the pose. Right: we deform the human shape (bottom leftmost) such that its Laplacian is diagonalized by the first 10 eigenfunctions of the alien Laplacian (top row). The result (bottom rightmost) is an 'intrinsic hybrid' between the two shapes. AbstractWe formulate the problem of shape-from-operator (SfO), recovering an embedding of a mesh from intrinsic operators defined through the discrete metric (edge lengths). Particularly interesting instances of our SfO problem include: shape-from-Laplacian, allowing to transfer style between shapes; shape-from-difference operator, used to synthesize shape analogies; and shape-from-eigenvectors, allowing to generate 'intrinsic averages' of shape collections. Numerically, we approach the SfO problem by splitting it into two optimization sub-problems: metric-from-operator (reconstruction of the discrete metric from the intrinsic operator) and embedding-from-metric (finding a shape embedding that would realize a given metric, a setting of the multidimensional scaling problem). We study numerical properties of our problem, exemplify it on several applications, and discuss its imitations.

Genomic prediction relies on genotypic marker information to predict the agronomic performance of future hybrid breeds based on trial records. Because the effect of markers may vary substantially under the influence of different environmental conditions, marker-by-environment interaction effects have to be taken into account. However, this may lead to a dramatic increase in the computational resources needed for analyzing large-scale trial data. A high-performance computing solution, called Needles, is presented for handling such data sets. Needles is tailored to the particular properties of the underlying algebraic framework by exploiting a sparse matrix formalism where suited and by utilizing distributed computing techniques to enable the use of a dedicated computing cluster. It is demonstrated that large-scale analyses can be performed within reasonable time frames with this framework. Moreover, by analyzing simulated trial data, it is shown that the effects of markers with a high environmental interaction can be predicted more accurately when more records per environment are available in the training data. The availability of such data and their analysis with Needles also may lead to the discovery of highly contributing QTL in specific environmental conditions. Such a framework thus opens the path for plant breeders to select crops based on these QTL, resulting in hybrid lines with optimized agronomic performance in specific environmental conditions.

We present a new approach for preconditioning the interface Schur complement arising in domain decomposition of second-order scalar elliptic problems. The preconditioners are discrete interpolation norms recently introduced in [3]. In particular, we employ discrete representations of norms for the Sobolev space of index 1/2 to approximate the Steklov-Poincaré operators arising from non-overlapping one-level domain decomposition methods. We use the coercivity and continuity of the Schur complement with respect to the preconditioning norm to derive mesh-independent bounds on the convergence of iterative solvers. We also address the case of non-constant coefficients by considering the interpolation of weighted spaces and the corresponding discrete norms.