code.rst 19.3 KB
Newer Older
1

Martin Reinecke's avatar
Martin Reinecke committed
2
=============
3
4
Code Overview
=============
5

Martin Reinecke's avatar
Martin Reinecke committed
6
7
8
9

Executive summary
=================

10
11
The fundamental building blocks required for IFT computations are best
recognized from a large distance, ignoring all technical details.
12

13
14
From such a perspective,

Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
15
- IFT problems largely consist of the combination of several high dimensional
16
  *minimization* problems.
Philipp Arras's avatar
Philipp Arras committed
17
- Within NIFTy, *operators* are used to define the characteristic equations and
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
18
  properties of the problems.
Martin Reinecke's avatar
Martin Reinecke committed
19
20
- The equations are built mostly from the application of *linear operators*,
  but there may also be nonlinear functions involved.
21
- The unknowns in the equations represent either continuous physical *fields*,
Martin Reinecke's avatar
Martin Reinecke committed
22
23
24
25
  or they are simply individual measured *data points*.
- Discretized *fields* have geometrical information (like locations and volume
  elements) associated with every entry; this information is called the field's
  *domain*.
26
27
28
29
30
31
32
33
34
35
36
37
38

In the following sections, the concepts briefly presented here will be
discussed in more detail; this is done in reversed order of their introduction,
to avoid forward references.


Domains
=======


Abstract base class
-------------------

39
40
.. currentmodule:: nifty5.domains.domain

Philipp Arras's avatar
Philipp Arras committed
41
One of the fundamental building blocks of the NIFTy5 framework is the *domain*.
42
Its required capabilities are expressed by the abstract :py:class:`Domain` class.
43
A domain must be able to answer the following queries:
44
m
45
46
47
48
49
50
51
52
53
54
55

- its total number of data entries (pixels), which is accessible via the
  :attr:`~Domain.size` property
- the shape of the array that is supposed to hold these data entries
  (obtainable by means of the :attr:`~Domain.shape` property)
- equality comparison to another :class:`Domain` instance


Unstructured domains
--------------------

56
57
.. currentmodule:: nifty5.domains.unstructured_domain

58
59
60
61
62
63
64
65
66
67
68
Domains can be either *structured* (i.e. there is geometrical information
associated with them, like position in space and volume factors),
or *unstructured* (meaning that the data points have no associated manifold).

Unstructured domains can be described by instances of NIFTy's
:class:`UnstructuredDomain` class.


Structured domains
------------------

69
70
.. currentmodule:: nifty5.domains.structured_domain

71
72
73
74
75
In contrast to unstructured domains, these domains have an assigned geometry.
NIFTy requires them to provide the volume elements of their grid cells.
The additional methods are specified in the abstract class
:class:`StructuredDomain`:

Martin Reinecke's avatar
Martin Reinecke committed
76
- The properties :attr:`~StructuredDomain.scalar_dvol`,
77
78
79
80
81
82
83
84
85
86
87
88
89
  :attr:`~StructuredDomain.dvol`, and  :attr:`~StructuredDomain.total_volume`
  provide information about the domain's pixel volume(s) and its total volume.
- The property :attr:`~StructuredDomain.harmonic` specifies whether a domain
  is harmonic (i.e. describes a frequency space) or not
- Iff the domain is harmonic, the methods
  :meth:`~StructuredDomain.get_k_length_array`,
  :meth:`~StructuredDomain.get_unique_k_lengths`, and
  :meth:`~StructuredDomain.get_fft_smoothing_kernel_function` provide absolute
  distances of the individual grid cells from the origin and assist with
  Gaussian convolution.

NIFTy comes with several concrete subclasses of :class:`StructuredDomain`:

90
91
.. currentmodule:: nifty5.domains

Philipp Arras's avatar
Philipp Arras committed
92
- :class:`~rg_space.RGSpace` represents a regular Cartesian grid with an arbitrary
93
  number of dimensions, which is supposed to be periodic in each dimension.
Philipp Arras's avatar
Philipp Arras committed
94
95
- :class:`~hp_space.HPSpace` and :class:`~gl_space.GLSpace` describe pixelisations of the
  2-sphere; their counterpart in harmonic space is :class:`~lm_space.LMSpace`, which
96
  contains spherical harmonic coefficients.
Philipp Arras's avatar
Philipp Arras committed
97
- :class:`~power_space.PowerSpace` is used to describe one-dimensional power spectra.
98

Philipp Arras's avatar
Philipp Arras committed
99
100
101
102
Among these, :class:`~rg_space.RGSpace` can be harmonic or not (depending on
constructor arguments), :class:`~gl_space.GLSpace`, :class:`~hp_space.HPSpace`,
and :class:`~power_space.PowerSpace` are pure position domains (i.e.
nonharmonic), and :class:`~lm_space.LMSpace` is always harmonic.
103
104
105
106
107
108


Combinations of domains
=======================

The fundamental classes described above are often sufficient to specify the
109
110
domain of a field. In some cases, however, it will be necessary to define the
field on a product of elementary domains instead of a single one.
Philipp Arras's avatar
Philipp Arras committed
111
More sophisticated operators also require a set of several such fields.
112
113
Some examples are:

Philipp Arras's avatar
Philipp Arras committed
114
115
116
- sky emission depending on location and energy. This could be represented by a
  product of an :class:`~hp_space.HPSpace` (for location) with an
  :class:`~rg_space.RGSpace` (for energy).
Philipp Arras's avatar
Philipp Arras committed
117
- a polarized field, which could be modeled as a product of any structured
Martin Reinecke's avatar
Martin Reinecke committed
118
  domain (representing location) with a four-element
Philipp Arras's avatar
Philipp Arras committed
119
  :class:`~unstructured_domain.UnstructuredDomain` holding Stokes I, Q, U and V components.
Martin Reinecke's avatar
Martin Reinecke committed
120
121
122
- a model for the sky emission, which holds both the current realization
  (on a harmonic domain) and a few inferred model parameters (e.g. on an
  unstructured grid).
123

Philipp Arras's avatar
Philipp Arras committed
124
.. currentmodule:: nifty5
125

Philipp Arras's avatar
Philipp Arras committed
126
127
128
129
Consequently, NIFTy defines a class called :class:`~domain_tuple.DomainTuple`
holding a sequence of :class:`~domains.domain.Domain` objects, which is used to
specify full field domains. In principle, a :class:`~domain_tuple.DomainTuple`
can even be empty, which implies that the field living on it is a scalar.
Martin Reinecke's avatar
Martin Reinecke committed
130

Philipp Arras's avatar
Philipp Arras committed
131
132
133
134
135
136
137
138
A :class:`~domain_tuple.DomainTuple` supports iteration and indexing, and also
provides the properties :attr:`~domain_tuple.DomainTuple.shape`,
:attr:`~domain_tuple.DomainTuple.size` in analogy to the elementary
:class:`~domains.domain.Domain`.

An aggregation of several :class:`~domain_tuple.DomainTuple` s, each member
identified by a name, is described by the :class:`~multi_domain.MultiDomain`
class.
139
140
141
142

Fields
======

Martin Reinecke's avatar
Martin Reinecke committed
143
144
145
Fields on a single DomainTuple
------------------------------

Philipp Arras's avatar
Philipp Arras committed
146
A :class:`~field.Field` object consists of the following components:
147

Philipp Arras's avatar
Philipp Arras committed
148
- a domain in form of a :class:`~domain_tuple.DomainTuple` object
149
150
151
- a data type (e.g. numpy.float64)
- an array containing the actual values

Martin Reinecke's avatar
Martin Reinecke committed
152
Usually, the array is stored in the form of a ``numpy.ndarray``, but for very
Martin Reinecke's avatar
Martin Reinecke committed
153
154
155
156
157
158
159
160
161
162
163
164
165
166
resource-intensive tasks NIFTy also provides an alternative storage method to
be used with distributed memory processing.

Fields support a wide range of arithmetic operations, either involving two
fields with equal domains, or a field and a scalar.
Contractions (like summation, integration, minimum/maximum, computation of
statistical moments) can be carried out either over an entire field (producing
a scalar result) or over sub-domains (resulting in a field living on a smaller
domain). Scalar products of two fields can also be computed easily.

There is also a set of convenience functions to generate fields with constant
values or fields filled with random numbers according to a user-specified
distribution.

Martin Reinecke's avatar
Martin Reinecke committed
167
168
169
170
171
Like almost all NIFTy objects, fields are immutable: their value or any other
attribute cannot be modified after construction. To manipulate a field in ways
that are not covered by the provided standard operations, its data content must
be extracted first, then changed, and a new field has to be created from the
result.
Martin Reinecke's avatar
Martin Reinecke committed
172

173
Fields defined on a MultiDomain
Martin Reinecke's avatar
Martin Reinecke committed
174
-------------------------------
Martin Reinecke's avatar
Martin Reinecke committed
175

Philipp Arras's avatar
Philipp Arras committed
176
177
178
The :class:`~multi_field.MultiField` class can be seen as a dictionary of
individual :class:`~field.Field` s, each identified by a name, which is defined
on a :class:`~multi_domain.MultiDomain`.
Martin Reinecke's avatar
Martin Reinecke committed
179
180
181
182
183


Operators
=========

Philipp Arras's avatar
Philipp Arras committed
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
All transformations between different NIFTy fields are expressed in the form of
:class:`~operators.operator.Operator` objects. The interface of this class is
rather minimalistic: it has a property called
:attr:`~operators.operator.Operator.domain` which returns a
:class:`~domain_tuple.DomainTuple` or :class:`~multi_domain.MultiDomain` object
specifying the structure of the :class:`~field.Field` or
:class:`~multi_field.MultiField` it expects as input, another property
:attr:`~operators.operator.Operator.target` describing its output, and finally
an overloaded :attr:`~operators.operator.Operator.apply` method, which can take:

- a :class:`~field.Field`/:class:`~multi_field.MultiField` object, in which case
  it returns the transformed :class:`~field.Field`/:class:`~multi_field.MultiField`.
- a :class:`~linearization.Linearization` object, in which case it returns the
  transformed :class:`~linearization.Linearization`.

This is the interface that all objects derived from
:class:`~operators.operator.Operator` must implement. In addition,
:class:`~operators.operator.Operator` objects can be added/subtracted,
multiplied, chained (via the :attr:`__call__` method
or the `@` operator) and support point-wise application of functions like
:class:`exp()`, :class:`log()`, :class:`sqrt()`, :class:`conjugate()`.
Martin Reinecke's avatar
Martin Reinecke committed
205

206

Martin Reinecke's avatar
Martin Reinecke committed
207
208
209
Advanced operators
------------------

Philipp Arras's avatar
Philipp Arras committed
210
NIFTy provides a library of commonly employed operators which can be used for
Martin Reinecke's avatar
Martin Reinecke committed
211
212
213
specific inference problems. Currently these are:

- :class:`AmplitudeOperator`, which returns a smooth power spectrum.
Philipp Arras's avatar
Philipp Arras committed
214
215
216
217
- :class:`InverseGammaOperator`, which models point sources which are
  distributed according to a inverse-gamma distribution.
- :class:`CorrelatedField`, which models a diffuse log-normal field. It takes an
  amplitude operator to specify the correlation structure of the field.
Martin Reinecke's avatar
Martin Reinecke committed
218
219


220
221
222
Linear Operators
================

Philipp Arras's avatar
Philipp Arras committed
223
A linear operator (represented by NIFTy5's abstract :class:`operators.linear_operator.LinearOperator`
Martin Reinecke's avatar
Martin Reinecke committed
224
225
class) is derived from `Operator` and can be interpreted as an
(implicitly defined) matrix. Since its operation is linear, it can provide some
Philipp Arras's avatar
Philipp Arras committed
226
additional functionality which is not available for the more generic :class:`operators.operator.Operator`
Martin Reinecke's avatar
Martin Reinecke committed
227
class.
228

Martin Reinecke's avatar
Martin Reinecke committed
229

Martin Reinecke's avatar
Martin Reinecke committed
230
231
Linear Operator basics
----------------------
Martin Reinecke's avatar
Martin Reinecke committed
232

233
234
There are four basic ways of applying an operator :math:`A` to a field :math:`f`:

Martin Reinecke's avatar
Martin Reinecke committed
235
236
237
238
- direct application: :math:`A\cdot f`
- adjoint application: :math:`A^\dagger \cdot f`
- inverse application: :math:`A^{-1}\cdot f`
- adjoint inverse application: :math:`(A^\dagger)^{-1}\cdot f`
239

Martin Reinecke's avatar
Martin Reinecke committed
240
241
(Because of the linearity, inverse adjoint and adjoint inverse application
are equivalent.)
242

Martin Reinecke's avatar
Martin Reinecke committed
243
244
These different actions of a linear operator ``Op`` on a field ``f`` can be
invoked in various ways:
245
246
247
248
249
250
251
252
253
254
255
256
257

- direct multiplication: ``Op(f)`` or ``Op.times(f)`` or ``Op.apply(f, Op.TIMES)``
- adjoint multiplication: ``Op.adjoint_times(f)`` or ``Op.apply(f, Op.ADJOINT_TIMES)``
- inverse multiplication: ``Op.inverse_times(f)`` or ``Op.apply(f, Op.INVERSE_TIMES)``
- adjoint inverse multiplication: ``Op.adjoint_inverse_times(f)`` or ``Op.apply(f, Op.ADJOINT_INVERSE_TIMES)``

Operator classes defined in NIFTy may implement an arbitrary subset of these
four operations. This subset can be queried using the
:attr:`~LinearOperator.capability` property.

If needed, the set of supported operations can be enhanced by iterative
inversion methods;
for example, an operator defining direct and adjoint multiplication could be
Martin Reinecke's avatar
Martin Reinecke committed
258
enhanced by this approach to support the complete set. This functionality is
259
260
261
262
263
264
265
266
267
268
269
270
271
272
provided by NIFTy's :class:`InversionEnabler` class, which is itself a linear
operator.

Direct multiplication and adjoint inverse multiplication transform a field
living on the operator's :attr:`~LinearOperator.domain` to one living on the operator's :attr:`~LinearOperator.target`, whereas adjoint multiplication
and inverse multiplication transform from :attr:`~LinearOperator.target` to :attr:`~LinearOperator.domain`.

Operators with identical domain and target can be derived from
:class:`EndomorphicOperator`; typical examples for this category are the :class:`ScalingOperator`, which simply multiplies its input by a scalar
value, and :class:`DiagonalOperator`, which multiplies every value of its input
field with potentially different values.

Further operator classes provided by NIFTy are

Martin Reinecke's avatar
Martin Reinecke committed
273
274
- :class:`HarmonicTransformOperator` for transforms from a harmonic domain to
  its counterpart in position space, and their adjoint
275
- :class:`PowerDistributor` for transforms from a :class:`PowerSpace` to
Martin Reinecke's avatar
Martin Reinecke committed
276
  an associated harmonic domain, and their adjoint
277
278
279
280
- :class:`GeometryRemover`, which transforms from structured domains to
  unstructured ones. This is typically needed when building instrument response
  operators.

Martin Reinecke's avatar
Martin Reinecke committed
281

Martin Reinecke's avatar
Martin Reinecke committed
282
283
Syntactic sugar
---------------
Martin Reinecke's avatar
Martin Reinecke committed
284

Martin Reinecke's avatar
Martin Reinecke committed
285
Nifty5 allows simple and intuitive construction of altered and combined
286
operators.
Philipp Arras's avatar
Philipp Arras committed
287
288
As an example, if ``A``, ``B`` and ``C`` are of type :class:`operators.linear_operator.LinearOperator`
and ``f1`` and ``f2`` are of type :class:`~field.Field`, writing::
289

Martin Reinecke's avatar
Martin Reinecke committed
290
    X = A(B.inverse(A.adjoint)) + C
291
292
293
294
295
296
297
298
299
300
301
    f2 = X(f1)

will perform the operation suggested intuitively by the notation, checking
domain compatibility while building the composed operator.
The combined operator infers its domain and target from its constituents,
as well as the set of operations it can support.
The properties :attr:`~LinearOperator.adjoint` and
:attr:`~LinearOperator.inverse` return a new operator which behaves as if it
were the original operator's adjoint or inverse, respectively.


Philipp Arras's avatar
Philipp Arras committed
302
303
Operators
=========
304

Philipp Arras's avatar
Philipp Arras committed
305
Operator classes (represented by NIFTy5's abstract :class:`operators.operator.Operator` class) are used to construct
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
306
the equations of a specific inference problem.
Philipp Arras's avatar
Philipp Arras committed
307
308
309
Most operators are defined via a position, which is a :class:`~multi_field.MultiField` object,
their value at this position, which is again a :class:`~multi_field.MultiField` object and a Jacobian derivative,
which is a :class:`operators.linear_operator.LinearOperator` and is needed for the minimization procedure.
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
310

Philipp Arras's avatar
Philipp Arras committed
311
Using the existing basic operator classes one can construct more complicated operators, as
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
312
NIFTy allows for easy and self-consinstent combination via point-wise multiplication,
Philipp Arras's avatar
Philipp Arras committed
313
addition and subtraction. The operator resulting from these operations then automatically
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
314
315
316
317
contains the correct Jacobians, positions and values.
Notably, :class:`Constant` and :class:`Variable` allow for an easy way to turn
inference of specific quantities on and off.

Philipp Arras's avatar
Philipp Arras committed
318
The basic operator classes also allow for more complex operations on operators such as
319
the application of :class:`LinearOperators` or local non-linearities.
Philipp Arras's avatar
Philipp Arras committed
320
321
As an example one may consider the following combination of ``x``, which is an operator of type
:class:`Variable` and ``y``, which is an operator of type :class:`Constant`::
322

Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
323
	z = x*x + y
324

Philipp Arras's avatar
Philipp Arras committed
325
``z`` will then be an operator with the following properties::
326
327

	z.value = x.value*x.value + y.value
Martin Reinecke's avatar
Martin Reinecke committed
328
	z.position = Union(x.position, y.position)
329
330
331
	z.jacobian = 2*makeOp(x.value)


Philipp Arras's avatar
Philipp Arras committed
332
Basic operators
Philipp Arras's avatar
Philipp Arras committed
333
---------------
Philipp Arras's avatar
Philipp Arras committed
334
# FIXME All this is outdated!
335

Philipp Arras's avatar
Philipp Arras committed
336
Basic operator classes provided by NIFTy are
337

Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
338
- :class:`Constant` contains a constant value and has a zero valued Jacobian.
Philipp Arras's avatar
Philipp Arras committed
339
  Like other operators, it has a position, but its value does not depend on it.
340
- :class:`Variable` returns the position as its value, its derivative is one.
Philipp Arras's avatar
Philipp Arras committed
341
- :class:`LinearModel` applies a :class:`operators.linear_operator.LinearOperator` on the model.
342
- :class:`LocalModel` applies a non-linearity locally on the model.
Martin Reinecke's avatar
Martin Reinecke committed
343
	value and Jacobian are combined into corresponding :class:`MultiFields` and operators.
Martin Reinecke's avatar
tweaks    
Martin Reinecke committed
344

345

Philipp Arras's avatar
Philipp Arras committed
346
347
Advanced operators
------------------
348

Philipp Arras's avatar
Philipp Arras committed
349
NIFTy also provides a library of more sophisticated operators which are used for more
350
351
specific inference problems. Currently these are:

Philipp Arras's avatar
Philipp Arras committed
352
353
354
- :class:`AmplitudeOperator`, which returns a smooth power spectrum.
- :class:`InverseGammaOperator`, which models point sources which follow a inverse gamma distribution.
- :class:`CorrelatedField`, which models a diffuse log-normal field. It takes an amplitude operator
355
356
357
	to specify the correlation structure of the field.


358
359
.. _minimization:

Martin Reinecke's avatar
Martin Reinecke committed
360

361
362
363
Minimization
============

Martin Reinecke's avatar
Martin Reinecke committed
364
365
366
367
368
369
Most problems in IFT are solved by (possibly nested) minimizations of
high-dimensional functions, which are often nonlinear.


Energy functionals
------------------
370

Philipp Arras's avatar
Philipp Arras committed
371
In NIFTy5 such functions are represented by objects of type :class:`Energy`.
Martin Reinecke's avatar
Martin Reinecke committed
372
373
These hold the prescription how to calculate the function's
:attr:`~Energy.value`, :attr:`~Energy.gradient` and
Martin Reinecke's avatar
Martin Reinecke committed
374
(optionally) :attr:`~Energy.metric` at any given :attr:`~Energy.position`
Martin Reinecke's avatar
Martin Reinecke committed
375
in parameter space.
Martin Reinecke's avatar
Martin Reinecke committed
376
Function values are floating-point scalars, gradients have the form of fields
Martin Reinecke's avatar
Martin Reinecke committed
377
living on the energy's position domain, and metrics are represented by
Martin Reinecke's avatar
Martin Reinecke committed
378
linear operator objects.
379

Martin Reinecke's avatar
Martin Reinecke committed
380
381
Energies are classes that typically have to be provided by the user when
tackling new IFT problems.
Philipp Arras's avatar
Philipp Arras committed
382
Some examples of concrete energy classes delivered with NIFTy5 are
Martin Reinecke's avatar
Martin Reinecke committed
383
:class:`QuadraticEnergy` (with position-independent metric, mainly used with
Philipp Arras's avatar
Philipp Arras committed
384
conjugate gradient minimization) and :class:`~nifty5.library.WienerFilterEnergy`.
385

Martin Reinecke's avatar
Martin Reinecke committed
386
387
388

Iteration control
-----------------
Martin Reinecke's avatar
Martin Reinecke committed
389
390

Iterative minimization of an energy reqires some means of
Martin Reinecke's avatar
Martin Reinecke committed
391
checking the quality of the current solution estimate and stopping once
Martin Reinecke's avatar
Martin Reinecke committed
392
393
394
it is sufficiently accurate. In case of numerical problems, the iteration needs
to be terminated as well, returning a suitable error description.

Philipp Arras's avatar
Philipp Arras committed
395
In NIFTy5, this functionality is encapsulated in the abstract
Martin Reinecke's avatar
Martin Reinecke committed
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
:class:`IterationController` class, which is provided with the initial energy
object before starting the minimization, and is updated with the improved
energy after every iteration. Based on this information, it can either continue
the minimization or return the current estimate indicating convergence or
failure.

Sensible stopping criteria can vary significantly with the problem being
solved; NIFTy provides one concrete sub-class of :class:`IterationController`
called :class:`GradientNormController`, which should be appropriate in many
circumstances, but users have complete freedom to implement custom sub-classes
for their specific applications.


Minimization algorithms
-----------------------

Martin Reinecke's avatar
Martin Reinecke committed
412
413
414
415
416
417
418
419
420
421
422
423
424
All minimization algorithms in NIFTy inherit from the abstract
:class:`Minimizer` class, which presents a minimalistic interface consisting
only of a :meth:`~Minimizer.__call__` method taking an :class:`Energy` object
and optionally a preconditioning operator, and returning the energy at the
discovered minimum and a status code.

For energies with a quadratic form (i.e. which
can be expressed by means of a :class:`QuadraticEnergy` object), an obvious
choice of algorithm is the :class:`ConjugateGradient` minimizer.

A similar algorithm suited for nonlinear problems is provided by
:class:`NonlinearCG`.

Martin Reinecke's avatar
Martin Reinecke committed
425
Many minimizers for nonlinear problems can be characterized as
Martin Reinecke's avatar
Martin Reinecke committed
426
427
428
429
430
431
432
433
434

- first deciding on a direction for the next step
- then finding a suitable step length along this direction, resulting in the
  next energy estimate.

This family of algorithms is encapsulated in NIFTy's :class:`DescentMinimizer`
class, which currently has three concrete implementations:
:class:`SteepestDescent`, :class:`VL_BFGS`, and :class:`RelaxedNewton`.
Of these algorithms, only :class:`RelaxedNewton` requires the energy object to
Martin Reinecke's avatar
Martin Reinecke committed
435
provide a :attr:`~Energy.metric` property, the others only need energy
Martin Reinecke's avatar
Martin Reinecke committed
436
437
values and gradients.

Martin Reinecke's avatar
Martin Reinecke committed
438
439
440
441
442
The flexibility of NIFTy's design allows using externally provided
minimizers. With only small effort, adapters for two SciPy minimizers were
written; they are available under the names :class:`NewtonCG` and
:class:`L_BFGS_B`.

Martin Reinecke's avatar
Martin Reinecke committed
443
444
445
446

Application to operator inversion
---------------------------------

Martin Reinecke's avatar
Martin Reinecke committed
447
448
449
450
451
452
453
454
455
456
It is important to realize that the machinery presented here cannot only be
used for minimizing IFT Hamiltonians, but also for the numerical inversion of
linear operators, if the desired application mode is not directly available.
A classical example is the information propagator

:math:`D = \left(R^\dagger N^{-1} R + S^{-1}\right)^{-1}`,

which must be applied when calculating a Wiener filter. Only its inverse
application is straightforward; to use it in forward direction, we make use
of NIFTy's :class:`InversionEnabler` class, which internally performs a
Martin Reinecke's avatar
Martin Reinecke committed
457
minimization of a :class:`QuadraticEnergy` by means of the
Martin Reinecke's avatar
Martin Reinecke committed
458
:class:`ConjugateGradient` algorithm.