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- W2624229329 abstract "Optimal decision making under uncertainty iscritical for control and optimization of complex systems. However,many techniques for solving problems such as stochastic optimalcontrol and data assimilation encounter the curse of dimensionalitywhen too many state variables are involved. In this thesis, wepropose a framework for computing with high-dimensional functionsthat mitigates this exponential growth in complexity for problemswith separable structure. Our framework tightly integrates twoemerging areas: tensor decompositions and continuous computation.Tensor decompositions are able to effectively compress and operatewith low-rank multidimensional arrays. Continuous computation is aparadigm for computing with functions instead of arrays, and it isbest realized by Chebfun, a MATLAB package for computing withfunctions of up to three dimensions. Continuous computationprovides a natural framework for building numerical algorithms thateffectively, naturally, and automatically adapt to problemstructure. The first part of this thesis describes a compressedcontinuous computation framework centered around a continuousanalogue to the (discrete) tensor-train decomposition called thefunction-train decomposition. Computation with the function-trainrequires continuous matrix factorizations and continuous numericallinear algebra. Continuous analogues are presented for performingcross approximation; rounding; multilinear algebra operations suchas addition, multiplication, integration, and differentiation; andcontinuous, rank-revealing, alternating least squares. Advantagesof the function-train over the tensor-train include the ability toadaptively approximate functions and the ability to compute withfunctions that are parameterized differently. For example, whileelementwise multiplication between tensors of different sizes isundefined, functions in FT format can be readily multipliedtogether. Next, we develop compressed versions of value iteration,policy iteration, and multilevel algorithms for solving dynamicprogramming problems arising in stochastic optimal control. Thesetechniques enable computing global solutions to a broader set ofproblems, for example those with non-affine control inputs, thanpreviously possible. Examples are presented for motion planningwith robotic systems that have up to seven states. Finally, we usethe FT to extend integration-based Gaussian filtering to largerstate spaces than previously considered. Examples are presented fordynamical systems with up to twenty states." @default.
- W2624229329 created "2017-06-15" @default.
- W2624229329 creator A5021570608 @default.
- W2624229329 date "2017-01-01" @default.
- W2624229329 modified "2023-09-23" @default.
- W2624229329 title "Continuous low-rank tensor decompositions, with applications to stochastic optimal control and data assimilation" @default.
- W2624229329 hasPublicationYear "2017" @default.
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