Lazy Evaluation in Infinite-Dimensional Function Spaces with Wavelet Basis
Vectors in numerical computation, i.e., arrays of numbers, often represent continuous functions. We would like to reflect this with types. One apparent obstacle is that spaces of functions are typically infinite-dimensional, while the code must run in finite time and memory. We argue that this can be overcome: even in an infinite-dimensional space, the vectors can in practice be stored in finite memory. However, dual vectors (corresponding essentially to distributions) require infinite data structure. The distinction is usually lost in the finite dimensional case, since dual vectors are often simply represented as vectors (by implicitly choosing a scalar product establishing the correspondence). However, we shall see that an explicit type-level distinction between functions and distributions makes sense and allows directly expressing useful concepts such as the Dirac distribution, which are problematic in the standard finite-resolution picture. The example implementation uses a very simple local basis that corresponds to a Haar Wavelet transform.
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10:50 - 12:10 | |||
10:50 26mTalk | Generating Efficient FFT GPU Code with Lift FHPNC Link to publication DOI Pre-print File Attached | ||
11:16 26mTalk | Lazy Evaluation in Infinite-Dimensional Function Spaces with Wavelet Basis FHPNC Link to publication Pre-print | ||
11:43 26mTalk | Functional Approach to Acceleration of Monte Carlo Simulation for American Option Pricing (extended abstract) FHPNC Wojciech Michal Pawlak University of Copenhagen, Denmark, Martin Elsman University of Copenhagen, Denmark, Cosmin Oancea University of Copenhagen, Denmark Link to publication |