 ![two localized modes of the Laplace operator (left) and four localized modes of the bi-Laplace operator (right). In both cases, the valley network (displayed in red) obtained in Figure \[fig:cplate\_u\] accurately predicts the number and the location of the localization subregions.[]{data-label="fig:cplate_modes"}](cplate_lap_mode00.jpg "fig:"){width="0.0cm"} ![two localized modes of the Laplace operator (left) and four localized modes of the bi-Laplace operator (right). In both cases, the valley network (displayed in red) obtained in Figure \[fig:cplate\_u\] accurately predicts the number and the location of the localization subregions.[]{data-label="fig:cplate_modes"}](cplate_bil_mode00.jpg "fig:"){width="0.0cm"} ![two localized modes of the Laplace operator (left) and four localized modes of the bi-Laplace operator (right). In both cases, the valley network (displayed in red) obtained in Figure \[fig:cplate\_u\] accurately predicts the number and the location of the localization subregions.[]{data-label="fig:cplate_modes"}](cplate_bil_mode00.jpg "fig:"){width="0.0cm"} 