Evaluating derivatives: principles and techniques of algorithmic differentiation
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
Society for Industrial and Applied Mathematics
2008
|
| Edition: | 2. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XXI, 438 S. graph. Darst. |
| ISBN: | 9780898716597 |
Staff View
MARC
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| 245 | 1 | 0 | |a Evaluating derivatives |b principles and techniques of algorithmic differentiation |c Andreas Griewank ; Andrea Walther |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Philadelphia |b Society for Industrial and Applied Mathematics |c 2008 | |
| 300 | |a XXI, 438 S. |b graph. Darst. | ||
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| 500 | |a Includes bibliographical references and index | ||
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Record in the Search Index
| _version_ | 1804138017595588608 |
|---|---|
| adam_text | Contents
Rules ix
Preface
xi
Prologue xv
Mathematical Symbols
xix
1
Introduction
1
1 Tangents and Gradients
13
2
A Framework for Evaluating Functions
15
2.1
The Lighthouse Example
...................... 16
2.2
Three-Part Evaluation Procedures
................. 18
2.3
Elemental Differentiability
...................... 23
2.4
Generalizations and Connections
.................. 25
2.5
Examples and Exercises
....................... 29
3
Fundamentals of Forward and Reverse
31
3.1
Forward Propagation of Tangents
.................. 32
3.2
Reverse Propagation of Gradients
.................. 37
3.3
Cheap Gradient Principle with Examples
............. 44
3.4
Approximate Error Analysis
..................... 50
3.5
Summary and Discussion
...................... 52
3.6
Examples and Exercises
....................... 56
4
Memory Issues and Complexity Bounds
61
4.1
Memory Allocation and Overwrites
................. 61
4.2
Recording
Prévalues on
the Tape
.................. 64
4.3
Selective Saves and Restores
..................... 70
4.4
A Temporal Complexity Model
................... 73
4.5
Complexity of Tangent Propagation
................ 80
4.6
Complexity of Gradient Propagation
................ 83
4.7
Examples and Exercises
....................... 88
Contents
5
Repeating and Extending Reverse
91
5.1
Adjoining Iterative Assignments
.................. 93
5.2
Adjoints
of
Adjoints
......................... 95
5.3
Tangents of
Adjoints
......................... 98
5.4
Complexity of Second-Order
Adjoints
............... 102
5.5
Examples and Exercises
....................... 105
6
Implementation and Software
107
6.1
Operator Overloading
........................ 110
6.2
Source Transformation
........................ 120
6.3
AD for Parallel Programs
...................... 129
6.4
Summary and Outlook
........................ 138
6.5
Examples and Exercises
....................... 139
II Jacobians and Hessians
143
7
Sparse Forward and Reverse
145
7.1
Quantifying Structure and Sparsity
................. 147
7.2
Sparse Derivative Propagation
................... 151
7.3
Sparse Second Derivatives
...................... 154
7.4
Examples and Exercises
....................... 159
8
Exploiting Sparsity by Compression
161
8.1
Curtis-Powell-Reid Seeding
..................... 164
8.2
Newsam-Ramsdell Seeding
..................... 168
8.3
Column Compression
......................... 171
8.4
Combined Column and Row Compression
............. 173
8.5
Second Derivatives, Two-Sided Compression
............ 176
8.6
Summary
............................... 181
8.7
Examples and Exercises
....................... 182
9
Going beyond Forward and Reverse
185
9.1
Jacobians as
Schur
Complements
.................. 188
9.2
Accumulation by Edge-Elimination
................. 193
9.3
Accumulation by Vertex-Elimination
................ 200
9.4
Face-Elimination on the Line-Graph
................ 204
9.5
NP-hardness via Ensemble Computation
.............. 207
9.6
Summary and Outlook
........................ 209
9.7
Examples and Exercises
....................... 209
10
Jacobian and Hessian Accumulation
211
10.1
Greedy and Other Heuristics
....................211
10.2
Local Preaccumulations
....................... 220
10.3
Scarcity and Vector Products
....................225
10.4
Hessians and Their Computational Graph
.............236
Contents
vii
10.5
Examples and Exercises
.......................241
11
Observations on Efficiency
245
11.1
Ramification of the Rank-One Example
..............245
11.2
Remarks on Partial Separability
..................248
11.3
Advice on Problem Preparation
...................255
11.4
Examples and Exercises
.......................257
III Advances and Reversals
259
12
Reversal Schedules and Checkpointing
261
12.1
Reversal with Recording or Recalculation
.............262
12.2
Reversal of Call Trees
........................265
12.3
Reversals of Evolutions by Checkpointing
.............278
12.4
Parallel Reversal Schedules
.....................295
12.5
Examples and Exercises
.......................298
13
Taylor and Tensor Coefficients
299
13.1
Higher-Order
Derivative
Vectors
.................. 300
13.2
Taylor Polynomial Propagation
................... 303
13.3
Multivariate Tensor Evaluation
................... 311
13.4
Higher-Order Gradients and Jacobians
............... 317
13.5
Special Relations in Dynamical Systems
.............. 326
13.6
Summary and Outlook
........................ 332
13.7
Examples and Exercises
....................... 333
14
Differentiation without Differentiability
335
14.1
Another Look at the Lighthouse Example
.............337
14.2
Putting the Pieces Together
.....................341
14.3
One-Sided Laurent Expansions
...................350
14.4
Summary and Conclusion
......................362
14.5
Examples and Exercises
.......................363
15
Implicit and Iterative Differentiation
367
15.1
Results of the Implicit Function Theorem
............. 370
15.2
Iterations and the Newton Scenario
................. 375
15.3
Direct Derivative Recurrences
.................... 381
15.4
Adjoint Recurrences and Their Convergence
............ 386
15.5
Second-Order
Adjoints
........................ 390
15.6
Summary and Outlook
........................ 393
15.7
Examples and Exercises
....................... 394
Epilogue
397
List of Figures
399
viii Contents
List of Tables
403
Assumptions and Definitions
407
Propositions, Corollaries, and Lemmas
409
Bibliography
411
Index
433
Algorithmic, or automatic, differentiation (AD) is a growing area of theoretical research and
software development concerned with the accurate and efficient evaluation of derivatives for
function evaluations given as computer programs. The resulting derivative values are useful for all
scientific computations that are based on linear, quadratic, or higher-order approximations to
nonlinear scalar or vector functions. The field opens up an exciting opportunity to develop new
algorithms that reflect the true cost of accurate derivatives and to use them for improvements in
speed and reliability.
This second edition has been updated and expanded to cover recent developments in applications
and theory, including an elegant NP completeness argument by
Uwe Naumann
and a brief
introduction to scarcity, a generalization of sparsity. There is also added material on checkpointing
and iterative differentiation. To improve readability the more detailed analysis of memory and
complexity bounds has been relegated to separate, optional chapters. The book consists of three
parts: a stand-alone introduction to the fundamentals of AD and its software; a thorough treatment
of methods for sparse problems; and final chapters on program-reversal schedules, higher derivatives,
nonsmooth problems, and iterative processes. Each of the
15
chapters concludes
with examples and exercises.
Evaluating Derivatives will be valuable to designers of algorithms and software
for nonlinear computational problems. Current numerical software users should
gain the insight necessary to choose and deploy existing AD software tools
to the best advantage.
Andreas Griewank is head of the Department of Mathematics at
Humboldt-Universität zu
Berlin and a member of the
DFG
Research Center
Matheon,
Mathematics for Key Technologies. He is author of the first edition
of this book, published in
2000.
A former senior scientist at Argonne National
Laboratory, his main research interests are nonlinear optimization and scientific
computing.
Andrea Warther has been junior professor for the analysis and optimization
of computer models at
Technische Universität
Dresden since
2003.
Her main
research interests are scientific computing and nonlinear optimization.
For more information about SIAM books, journals,
conferences, memberships, or activities, contact:
Society for Industrial and Applied Mathematics
3600
Market Street, 6th Floor
Philadelphia, PA
19104-2688
USA
+1-215-382-9800 ·
Fax
+1-215-386-7999
siam@siam.org
·
www.siam.org
|
| adam_txt |
Contents
Rules ix
Preface
xi
Prologue xv
Mathematical Symbols
xix
1
Introduction
1
1 Tangents and Gradients
13
2
A Framework for Evaluating Functions
15
2.1
The Lighthouse Example
. 16
2.2
Three-Part Evaluation Procedures
. 18
2.3
Elemental Differentiability
. 23
2.4
Generalizations and Connections
. 25
2.5
Examples and Exercises
. 29
3
Fundamentals of Forward and Reverse
31
3.1
Forward Propagation of Tangents
. 32
3.2
Reverse Propagation of Gradients
. 37
3.3
Cheap Gradient Principle with Examples
. 44
3.4
Approximate Error Analysis
. 50
3.5
Summary and Discussion
. 52
3.6
Examples and Exercises
. 56
4
Memory Issues and Complexity Bounds
61
4.1
Memory Allocation and Overwrites
. 61
4.2
Recording
Prévalues on
the Tape
. 64
4.3
Selective Saves and Restores
. 70
4.4
A Temporal Complexity Model
. 73
4.5
Complexity of Tangent Propagation
. 80
4.6
Complexity of Gradient Propagation
. 83
4.7
Examples and Exercises
. 88
Contents
5
Repeating and Extending Reverse
91
5.1
Adjoining Iterative Assignments
. 93
5.2
Adjoints
of
Adjoints
. 95
5.3
Tangents of
Adjoints
. 98
5.4
Complexity of Second-Order
Adjoints
. 102
5.5
Examples and Exercises
. 105
6
Implementation and Software
107
6.1
Operator Overloading
. 110
6.2
Source Transformation
. 120
6.3
AD for Parallel Programs
. 129
6.4
Summary and Outlook
. 138
6.5
Examples and Exercises
. 139
II Jacobians and Hessians
143
7
Sparse Forward and Reverse
145
7.1
Quantifying Structure and Sparsity
. 147
7.2
Sparse Derivative Propagation
. 151
7.3
Sparse Second Derivatives
. 154
7.4
Examples and Exercises
. 159
8
Exploiting Sparsity by Compression
161
8.1
Curtis-Powell-Reid Seeding
. 164
8.2
Newsam-Ramsdell Seeding
. 168
8.3
Column Compression
. 171
8.4
Combined Column and Row Compression
. 173
8.5
Second Derivatives, Two-Sided Compression
. 176
8.6
Summary
. 181
8.7
Examples and Exercises
. 182
9
Going beyond Forward and Reverse
185
9.1
Jacobians as
Schur
Complements
. 188
9.2
Accumulation by Edge-Elimination
. 193
9.3
Accumulation by Vertex-Elimination
. 200
9.4
Face-Elimination on the Line-Graph
. 204
9.5
NP-hardness via Ensemble Computation
. 207
9.6
Summary and Outlook
. 209
9.7
Examples and Exercises
. 209
10
Jacobian and Hessian Accumulation
211
10.1
Greedy and Other Heuristics
.211
10.2
Local Preaccumulations
. 220
10.3
Scarcity and Vector Products
.225
10.4
Hessians and Their Computational Graph
.236
Contents
vii
10.5
Examples and Exercises
.241
11
Observations on Efficiency
245
11.1
Ramification of the Rank-One Example
.245
11.2
Remarks on Partial Separability
.248
11.3
Advice on Problem Preparation
.255
11.4
Examples and Exercises
.257
III Advances and Reversals
259
12
Reversal Schedules and Checkpointing
261
12.1
Reversal with Recording or Recalculation
.262
12.2
Reversal of Call Trees
.265
12.3
Reversals of Evolutions by Checkpointing
.278
12.4
Parallel Reversal Schedules
.295
12.5
Examples and Exercises
.298
13
Taylor and Tensor Coefficients
299
13.1
Higher-Order
Derivative
Vectors
. 300
13.2
Taylor Polynomial Propagation
. 303
13.3
Multivariate Tensor Evaluation
. 311
13.4
Higher-Order Gradients and Jacobians
. 317
13.5
Special Relations in Dynamical Systems
. 326
13.6
Summary and Outlook
. 332
13.7
Examples and Exercises
. 333
14
Differentiation without Differentiability
335
14.1
Another Look at the Lighthouse Example
.337
14.2
Putting the Pieces Together
.341
14.3
One-Sided Laurent Expansions
.350
14.4
Summary and Conclusion
.362
14.5
Examples and Exercises
.363
15
Implicit and Iterative Differentiation
367
15.1
Results of the Implicit Function Theorem
. 370
15.2
Iterations and the Newton Scenario
. 375
15.3
Direct Derivative Recurrences
. 381
15.4
Adjoint Recurrences and Their Convergence
. 386
15.5
Second-Order
Adjoints
. 390
15.6
Summary and Outlook
. 393
15.7
Examples and Exercises
. 394
Epilogue
397
List of Figures
399
viii Contents
List of Tables
403
Assumptions and Definitions
407
Propositions, Corollaries, and Lemmas
409
Bibliography
411
Index
433
Algorithmic, or automatic, differentiation (AD) is a growing area of theoretical research and
software development concerned with the accurate and efficient evaluation of derivatives for
function evaluations given as computer programs. The resulting derivative values are useful for all
scientific computations that are based on linear, quadratic, or higher-order approximations to
nonlinear scalar or vector functions. The field opens up an exciting opportunity to develop new
algorithms that reflect the true cost of accurate derivatives and to use them for improvements in
speed and reliability.
This second edition has been updated and expanded to cover recent developments in applications
and theory, including an elegant NP completeness argument by
Uwe Naumann
and a brief
introduction to scarcity, a generalization of sparsity. There is also added material on checkpointing
and iterative differentiation. To improve readability the more detailed analysis of memory and
complexity bounds has been relegated to separate, optional chapters. The book consists of three
parts: a stand-alone introduction to the fundamentals of AD and its software; a thorough treatment
of methods for sparse problems; and final chapters on program-reversal schedules, higher derivatives,
nonsmooth problems, and iterative processes. Each of the
15
chapters concludes
with examples and exercises.
Evaluating Derivatives will be valuable to designers of algorithms and software
for nonlinear computational problems. Current numerical software users should
gain the insight necessary to choose and deploy existing AD software tools
to the best advantage.
Andreas Griewank is head of the Department of Mathematics at
Humboldt-Universität zu
Berlin and a member of the
DFG
Research Center
Matheon,
Mathematics for Key Technologies. He is author of the first edition
of this book, published in
2000.
A former senior scientist at Argonne National
Laboratory, his main research interests are nonlinear optimization and scientific
computing.
Andrea Warther has been junior professor for the analysis and optimization
of computer models at
Technische Universität
Dresden since
2003.
Her main
research interests are scientific computing and nonlinear optimization.
For more information about SIAM books, journals,
conferences, memberships, or activities, contact:
Society for Industrial and Applied Mathematics
3600
Market Street, 6th Floor
Philadelphia, PA
19104-2688
USA
+1-215-382-9800 ·
Fax
+1-215-386-7999
siam@siam.org
·
www.siam.org |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Griewank, Andreas 1950-2021 Walther, Andrea 1970- |
| author_GND | (DE-588)112508162 (DE-588)121832546 |
| author_facet | Griewank, Andreas 1950-2021 Walther, Andrea 1970- |
| author_role | aut aut |
| author_sort | Griewank, Andreas 1950-2021 |
| author_variant | a g ag a w aw |
| building | Verbundindex |
| bvnumber | BV035069827 |
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| callnumber-search | QA304 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 910 |
| ctrlnum | (OCoLC)227574816 (DE-599)BVBBV035069827 |
| dewey-full | 515/.33 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.33 |
| dewey-search | 515/.33 |
| dewey-sort | 3515 233 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed. |
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| genre | (DE-588)4151278-9 Einführung gnd-content |
| genre_facet | Einführung |
| id | DE-604.BV035069827 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:03:48Z |
| indexdate | 2024-07-09T21:21:30Z |
| institution | BVB |
| isbn | 9780898716597 |
| language | English |
| lccn | 2008021064 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016738224 |
| oclc_num | 227574816 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-20 DE-706 DE-703 DE-29T DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-20 DE-706 DE-703 DE-29T DE-11 |
| physical | XXI, 438 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | marc |
| spelling | Griewank, Andreas 1950-2021 Verfasser (DE-588)112508162 aut Evaluating derivatives principles and techniques of algorithmic differentiation Andreas Griewank ; Andrea Walther 2. ed. Philadelphia Society for Industrial and Applied Mathematics 2008 XXI, 438 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Datenverarbeitung Differential calculus Data processing Automatische Differentiation (DE-588)4314524-3 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Automatische Differentiation (DE-588)4314524-3 s DE-604 Walther, Andrea 1970- Verfasser (DE-588)121832546 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016738224&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016738224&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Griewank, Andreas 1950-2021 Walther, Andrea 1970- Evaluating derivatives principles and techniques of algorithmic differentiation Datenverarbeitung Differential calculus Data processing Automatische Differentiation (DE-588)4314524-3 gnd |
| subject_GND | (DE-588)4314524-3 (DE-588)4151278-9 |
| title | Evaluating derivatives principles and techniques of algorithmic differentiation |
| title_auth | Evaluating derivatives principles and techniques of algorithmic differentiation |
| title_exact_search | Evaluating derivatives principles and techniques of algorithmic differentiation |
| title_exact_search_txtP | Evaluating derivatives principles and techniques of algorithmic differentiation |
| title_full | Evaluating derivatives principles and techniques of algorithmic differentiation Andreas Griewank ; Andrea Walther |
| title_fullStr | Evaluating derivatives principles and techniques of algorithmic differentiation Andreas Griewank ; Andrea Walther |
| title_full_unstemmed | Evaluating derivatives principles and techniques of algorithmic differentiation Andreas Griewank ; Andrea Walther |
| title_short | Evaluating derivatives |
| title_sort | evaluating derivatives principles and techniques of algorithmic differentiation |
| title_sub | principles and techniques of algorithmic differentiation |
| topic | Datenverarbeitung Differential calculus Data processing Automatische Differentiation (DE-588)4314524-3 gnd |
| topic_facet | Datenverarbeitung Differential calculus Data processing Automatische Differentiation Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016738224&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016738224&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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