abstract space
- Cauchy Series
in metric space, existing an any positive and small \etta, exist a natural number N, $m, n > N$, which satisfies:
it’s called a Cauchy Series. intuitively, as the numbers goes above, the elements get closer.
- Complete Space
any Caunchy Series in these abstract space, its convergence is still in the original space.
e.g. rational number space is not complete space, as is not in rational number space anymore.
intuitively, a complete space is like a shell without any holes on its surface.
- Linear Space
with linear structural set, which can be described by base vector, so any hyper-point in a linear space can be represented by the linear combination of its base, so also called as vector space. in another way, linear space has only add and scalar multi operator.
- Metric Space
to describe length or distance in linear space, adding normal in the linear space, gives normed linear space or metric space.
- Banach Space
a complete metric space
- Inner Product Space
with inner product feature in metric space. for the infinite space, there are two different sub space, either the inner product of the series converged or not.
the convergence sub space is a completed space.
- Hilbert Space
a completed inner product space
- Eurepean Space
a finite Hilbert Space.
functional analysis
the optimzed control theory problem, is to find the functional, under system dynamic constraints, which gives the extremum point(function) in the infinite normed linear space; in general.
in general, the functional is depends on the system dynamic path, which gives some famuous theorem: Banach fixed point.
PDE theory, partial geometry, optimized control theory are all part of functional analysis. from computational mechanics field, there is a chance go to PDE theory; from modern control field, there is a chance go to optimized control theory; from modern physics field, the students may also get familiar with partial geometry. the beauty of math is show up, all these big concepts finally arrive to one source: functional analysis.
to transfer all these big things into numerial world, which gives CAE solver, DQN solvers e.t.c.
the closest point property of Hilbert Space
A subset of A of a vector space is convex: if for all a, b belongs to A, all \lamba such that 0 < \lamba < 1, the point belongs to
A
assuming A is non-empty closed convex set in Hilbert Space H, for any x belongs to H, there is a unique point y of A which is closer to x than any ohter point of A:
orthogonal expansions
if (e_n) is an orthonormal sequence in a Hilbert space H, for any x belongs to H, (x, e_n) is the nth Fourier coefficient of x with respect to (e_n), the Fourier series of x with respect to the sequence (e_n) is the series:
- pointwise converges
where f_nis a sequence of functions in
- complete orthonormal sequence
given an orthonormal sequence of (e_n) and a vector x belongs to H then:
but only when this sequence is complete, the right-hand side converge to left-hand side workks.
an orthonormal sequence (e_n) in Hilbert Space H is complete if the only member of H which is orthogonal to every e_n is the zero vector.
a complete orthonal sequence in H is also called an orthonormal basis of H.
H is separable if it contains a complete orthonormal sequence,
the orthogonal complement (which named as E^T) of a subset E of H is the set:
for any set E in H, E^T is a closed linear subspace of H.
the knowledge above gives the convergence in Hilbert Space, application like FEA, once created the orthonormal basis of solution space, the solution function represented by the linear combination of these basis is guranted to convergence.
Fourier series
it is a complete orthonormal sequence in L^2(-\pi, \pi), and any function represented by a linear combination of Fourier basis is converged.
but here Fourier basis is the arithmetic means of the nth partial sum of the Fourier series of f, not the direct basis itself.
Dual space
FEA solutions for PDE
PDE can be represented in a differential representation:
or an integral representation:
integral formulation has included existing boundary conditions. by multiplying test function \phi on both side of integral representation, and using Green’s first identity will give the weak formulation(variational formulation) of this PDE.
in FEA, assuming the test function \phi and solution T belong to same Hilbert space, the advantage of Hilbert space, is functions inside can do linear combination, like vectors in a vector space. FEA also provides the error estimates, or bounds for the error.
the weak formulation should be obtained by all test functions in Hilbert space. this is weak formulation due to now it doesn’t exactly requries all points in the domain need meet the differential representation of the PDE, but in a integral sense. so even a discontinuity of first derivative of solution T still works by weak formulation.
Hilbert space and weak convergence
convergence in another words means, the existence of solution. for CAE, DQN algorithms, the solution space is in Hilbert space.
Lagrange and its duality
to transfer a general convex optimization problem with lagrange mulitplier:
=>
we always looks for meaning solution, so L should exist maximum extremum. if not, as x grow, the system goes divergence, no meaningful solution. define:
so the constrainted system equation equals to :
to rewrite the original equation as:
then the system’s duality is defined as :
for a system, if the original state equation and its duality equation both exist extremum solution, then :
the benefits of duality problems is
the duality problem is always convex, even when the original problem is non-convex
the duality solution give an lower boundary for the original solution
when strong duality satisfy, the duality solution is the original solution
Krylov Space
refer
Going deeper into RF: understaing Q-learning and linear function approximation