Principles of Systematic Upscaling

Despite their dizzying speed, modern supercomputers are still incapable of handling many most vital scientific problems. This is primarily due to the scale gap, which exists between the microscopic scale at which physical laws are given and the much larger scale of phenomena we wish to understand. This gap implies, first of all, a huge number of variables (e.g., atoms or gridpoints or pixels), and possibly even a much larger number of interactions (e.g., one force between every pair of atoms). Moreover, computers simulate physical systems by moving few variables at a time; each such move must be extremely small, since a larger move would have to take into account all the motions that should in parallel be performed by all other variables. Such a computer simulation is particularly incapable of moving the system across large-scale energy barriers, which can each be crossed only by a large, coherent motion of very many variables. This type of obstacles makes it impossible, for example, to calculate the properties of nature’s building blocks (elementary particles, atomic nuclei, etc.), or to computerize chemistry and materials science, so as to enable the design of materials, drugs and processes, with enormous potential benefits for medicine, biotechnology, nanotechnology, agriculture, materials science, industrial processing, etc. With current common methods the amount of computer processing often increases so steeply with problem size, that even much faster computers will not do. Past studies have demonstrated that scale-born slownesses can often be overcome by multiscale algorithms. Such algorithms have first been developed in the form of fast multigrid solvers for discretized PDEs [1], [2], [3], [4], [13], [15], [14]. These solvers are based on two processes: (1) classical relaxation schemes, which are generally slow to converge but fast to smooth the error function; (2) approximating the smooth error on a coarser grid (typically having twice the meshsize), by solving there equations which are derived from the PDE and from the fine-grid residuals; the solution of these coarse-grid equations is obtained by using recursively the same two processes. As a result, large scale changes are effectively calculated on correspondingly coarse grids, based on information gathered from finer grids. Such multigrid solvers yield linear complexity , i.e., the solution work is proportional to the number of variables in the system. In many years of research, the range of applicability of these methods has steadily grown, to cover most major types of linear and nonlinear large systems of equations appearing in sciences and engineering. This has been accomplished by extending the concept of “smoothness” in various ways, finally to stand generally for any poorly locally determined solution component, and by correspondingly diversifying the types of coarse representations, to include for instance grid-free solvers (algebraic multigrid [7], [8], [9], [16]), non-deterministic problems ([10], [20], [21], [11], [12]) and multiple coarse-level representations for wave equations [5]. It has been shown (see survey [29]) that the inter-scale interactions can indeed eliminate all kinds of scale-associated difficulties, such as: slow convergence (in minimization processes, PDE solvers, etc.); critical slowing down (in statistical physics); ill-posedness (e.g., of inverse problems); conflicts between small-scale and large-scale representations (e.g., in wave problems, bridging the gap between wave equations and geometrical optics); numerousness of long-range interactions (in many body problems or integral equations); the need to produce many fine-level solutions (e.g., in optimal control, design and data assimilation problems), or a multitude of