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Subfield · Mathematics

Numerical Analysis research

Numerical Analysis is a subfield of Mathematics. Science Explorer counts 130k research works in it since 1950. 10.0% of them reached the world's top 10% most cited for their field and year.

In short: study of algorithms that use numerical approximation for the problems of mathematical analysis.

Research works
130k
fractional, since 1950
In the world top 10%
13k
per year above
Top-10% rate
10.0%
share of its works in the world top 10%
Topics
5
Growth, 2018-21 → 2022-25
+8%
the tick is no change
Largest producer
China
34% of the 10 largest, 2022-now
Numerical Analysis

What is Numerical Analysis research about, and what is growing?

By the world's output in 2021-25, its most used keywords are Numerical Integration, Numerical Methods, Singular Perturbation, Differential Equations, Exponential Integrators, Optimization Software. Losing share: Optimization Software, Semidefinite Programming.

▲ gaining share of this field◆ holding its share▼ losing share

Size is the world's fractional works in 2021-25 in the topics tagged with each word; colour compares the word's share of this field with 2011-15. Each links to the topic it comes from most.

All 11 words, with their numbers
  1. Numerical Integration7,739◆ 0.98×2 topics
  2. Numerical Methods7,493◆ 1.12×1 topic
  3. Singular Perturbation7,493◆ 1.12×1 topic
  4. Differential Equations5,963◆ 0.97×1 topic
  5. Exponential Integrators5,963◆ 0.97×1 topic
  6. Optimization Software4,878▼ 0.85×1 topic
  7. Semidefinite Programming4,878▼ 0.85×1 topic
  8. Iterative Methods3,383◆ 1.06×1 topic
  9. Newton's Method3,383◆ 1.06×1 topic
  10. High-dimensional Integration1,776◆ 1.03×1 topic
  11. Quasi-Monte Carlo1,776◆ 1.03×1 topic

Which countries lead Numerical Analysis research?

On research quality in 2022-now, China, the United States and Germany rank highest in Numerical Analysis. By volume, China and the United States publish the most (4.1k and 1.8k works in 2022-now).

By research quality, 2022-now

  1. 1 China #2 all time 1.05× citation excellence · 1.03× journals · 100% breadth 12.2% top 10% · 3.7k works
  2. 2 United States #1 all time 1.05× citation excellence · 1.54× journals · 100% breadth 10.3% top 10% · 1.6k works
  3. 3 Germany #3 all time 1.08× citation excellence · 1.39× journals · 100% breadth 9.3% top 10% · 598 works
  4. 4 United Kingdom #5 all time 0.95× citation excellence · 1.44× journals · 100% breadth 10.7% top 10% · 274 works
  5. 5 France #4 all time 0.77× citation excellence · 1.25× journals · 100% breadth 6.8% top 10% · 476 works
  6. 6 India #14 all time 1.02× citation excellence · 0.45× journals · 100% breadth 11.0% top 10% · 1.6k works
  7. 7 Spain #7 all time 1.14× citation excellence · 0.99× journals · 100% breadth 9.9% top 10% · 327 works
  8. 8 Italy #12 all time 1.31× citation excellence · 0.68× journals · 100% breadth 13.1% top 10% · 427 works
  9. 9 Saudi Arabia #15 all time 1.60× citation excellence · 0.32× journals · 100% breadth 19.8% top 10% · 529 works
  10. 10 Japan #19 all time 0.64× citation excellence · 0.91× journals · 100% breadth 5.0% top 10% · 275 works

The track on each row is that row's top-10% rate. The ranking itself is the quality score, which compares the rate with what each place's own topics predict, adds its share of papers in the Nature Index journals, and counts output and breadth too.

Of 20 countries ranked in Numerical Analysis. Ranked on the quality score for 2022-now; the bracket is the all-time rank.

All countries ranked in Numerical Analysis

By volume, 2022-now

  1. 1 China 4.1k works
  2. 2 United States 1.8k works
  3. 3 India 1.7k works
  4. 4 Russia 1.3k works
  5. 5 Germany 667 works
  6. 6 Türkiye 602 works
  7. 7 Saudi Arabia 562 works
  8. 8 Iran 536 works
  9. 9 France 516 works
  10. 10 Italy 465 works

How concentrated that is

The same countries as shares of everything the list above accounts for. A node where two countries do two thirds of the work and one spread evenly across twelve read alike as a ranking and not at all alike here.

China: 33.6%United States: 14.4%India: 14.0%Russia: 10.7%6 others listed: 27.3%34%largest
China4,120 · 33.6%United States1,771 · 14.4%India1,711 · 14.0%Russia1,312 · 10.7%6 others listed3,348 · 27.3%

Shares of the rows listed above, not of the whole node.

Which institutions lead Numerical Analysis research?

Chinese Academy of Sciences (China) ranks first in Numerical Analysis on research quality in 2022-now, followed by Centre National de la Recherche Scientifique and National Academy of Sciences of Ukraine.

By research quality, 2022-now

  1. 1 Chinese Academy of Sciences China · Beijing · #8 all time 15.1% top 10% · 64 works
  2. 2 Centre National de la Recherche Scientifique France · Paris · #5 all time 5.9% top 10% · 200 works
  3. 3 National Academy of Sciences of Ukraine Ukraine · Kyiv 5.4% top 10% · 76 works
  4. 4 Lomonosov Moscow State University Russia · Moscow 4.9% top 10% · 115 works
  5. 5 Russian Academy of Sciences Russia · Moscow 2.1% top 10% · 370 works

The track on each row is that row's top-10% rate. The ranking itself is the quality score, which compares the rate with what each place's own topics predict, adds its share of papers in the Nature Index journals, and counts output and breadth too.

Of 5 institutions above the floor of 40 fractional works in Numerical Analysis.

Top 100 Numerical Analysis institutions

By volume, 2022-now

  1. 1 Russian Academy of SciencesRussia 392 works
  2. 2 Centre National de la Recherche ScientifiqueFrance 217 works
  3. 3 Lomonosov Moscow State UniversityRussia 120 works
  4. 4 National Academy of Sciences of UkraineUkraine 79 works
  5. 5 Chinese Academy of SciencesChina 72 works
  6. 6 University of MosulIraq 58 works
  7. 7 Islamic Azad University, TehranIran 58 works
  8. 8 Bulgarian Academy of SciencesBulgaria 58 works
  9. 9 Harbin Institute of TechnologyChina 56 works
  10. 10 Qassim UniversitySaudi Arabia 55 works

Who are the leading researchers in Numerical Analysis?

Counting only their work in Numerical Analysis, Chi‐Wang Shu, Mehdi Dehghan and Dumitru Bǎleanu have the most papers in the world top 10% for their subfield and year beyond the 10% expected.

  1. 1 Chi‐Wang Shu United States · #63,185 worldwide overall 43.8 of 87 papers here in the top 10%
  2. 2 Mehdi Dehghan Iran · #1,004,501 worldwide overall 79.4 of 161 papers here in the top 10%
  3. 3 Dumitru Bǎleanu Türkiye · #1,922,304 worldwide overall 109.2 of 242 papers here in the top 10%
  4. 4 Ian Turner Australia · #183,959 worldwide overall 24.9 of 56 papers here in the top 10%
  5. 5 Christodoulos A. Floudas United States · #11,996 worldwide overall 20.8 of 32 papers here in the top 10%
  6. 6 Christian Kanzow Germany · #27,094 worldwide overall 22.1 of 48 papers here in the top 10%
  7. 7 Zhi‐zhong Sun China · #69,017 worldwide overall 21.6 of 44 papers here in the top 10%
  8. 8 T. E. Simos Greece · #441,958 worldwide overall 39.0 of 137 papers here in the top 10%
  9. 9 Jorge Nocedal United States · #39,819 worldwide overall 20.2 of 35 papers here in the top 10%
  10. 10 Ignacio E. Grossmann United States · #1,649 worldwide overall 20.6 of 40 papers here in the top 10%

Ranked on their work in Numerical Analysis alone: how many more of their papers here reached the world top 10% for their subfield and year than the 10% expected, among 168,003 ranked researchers with a paper here. Each paper counts once, so many strong papers outrank one famous one, and a long record of ordinary papers does not climb. Identities come from OpenAlex author disambiguation, which occasionally merges or splits people.

Top 100 Numerical Analysis researchers

What are the most cited papers in Numerical Analysis?

The most cited paper in Numerical Analysis is "Adam: A Method for Stochastic Optimization" (Mandi, Jayanta, Canoy, Rocsildes, Bucarey, Víctor et al., 2015), with 50,327 citations. Next come "Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces" (29,587) and "Deterministic Nonperiodic Flow" (19,616).

Most cited papers
#PaperReleasedCitations
1 Adam: A Method for Stochastic Optimization Mandi, Jayanta, Canoy, Rocsildes, Bucarey, Víctor et al. · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) Open access May 2015 50,327
2 Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces Rainer Storn, Kenneth V. Price · Journal of Global Optimization Dec 1997 29,587
3 Deterministic Nonperiodic Flow Edward N. Lorenz · Journal of the Atmospheric Sciences Open access Mar 1963 19,616
4 No free lunch theorems for optimization David H. Wolpert, William G. Macready · IEEE Transactions on Evolutionary Computation Apr 1997 14,318
5 On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming Andreas Wächter, Lorenz T. Biegler · Mathematical Programming 28 Apr 2005 9,722
All 99 most cited papers, with filters and sorting

Where is Numerical Analysis research done?

The largest centres of Numerical Analysis research in 2022-now are Moscow (Russia), Beijing (China), Shanghai (China) and Nanjing (China). Among places with at least 20 works in it, it is an unusually large share of all research in Lawton.

Largest cities, 2022-now

  1. 1 MoscowRussia 542 works
  2. 2 BeijingChina 459 works
  3. 3 ShanghaiChina 278 works
  4. 4 NanjingChina 200 works
  5. 5 Xi'anChina 157 works
  6. 6 ChangshaChina 154 works
  7. 7 ParisFrance 153 works
  8. 8 GuangzhouChina 153 works
  9. 9 RiyadhSaudi Arabia 151 works
  10. 10 TehranIran 145 works

Where it is the local speciality

  1. LawtonUS · 40.9 works286×
← less than its size predicts1×more →

Location quotient: how much more of its research is in Numerical Analysis than the world average.

See Numerical Analysis on the map

Numerical Analysis research by country

The best places for Numerical Analysis research inside each country, with that country's researchers and world standing.

Where is the best place to study Numerical Analysis?

Among universities, judged by research, Stanford University, Massachusetts Institute of Technology and ETH Zurich score highest, combining standing in the field's ranking, citation excellence, specialisation, size, growth and international reach. Research strength is one signal when choosing where to study; it does not measure teaching.

0%10%20%mean 15.32%fractional works in this field (log) →share of work in the world top 10% →Massachusetts Instit…Princeton UniversityStanford UniversityETH Zurich
above the meannear itbelow it

One dot per university in the first ten of the table below, at the default weights. The upper left is the interesting corner: small places doing unusually strong work.

Standingthis field's institution ranking, three quarters all time and a quarter 2022-now
50%
Citation excellenceshare of its work in the world top 10%, 2022-now and all time averaged
20%
Specialisationhow much of its work is in this field, counted up to 10× the world share
5%
Sizeits output in this field, on a log scale
15%
Growthoutput growth, counted from -20% to +20%
5%
Internationalshare of its work with a co-author abroad
5%
Each slider sets a weight; the share it gets of the score is beside it.
#UniversityScoreRank in field (all time)Top 10%SpecialisationWorksGrowth
1Stanford UniversityUnited States 65.7- (#1)12.7%0.9×23 -10.7%
2Massachusetts Institute of TechnologyUnited States 63.0- (#3)21.7%2.1×26 +18.9%
3ETH ZurichSwitzerland 56.5- (#4)10.0%1.8×21 +19.1%
4Princeton UniversityUnited States 54.5- (#2)16.2%1.4×10 -58.6%
5New York UniversityUnited States 51.7- (#6)17.0%0.9×17 +96.2%
6École Polytechnique Fédérale de LausanneSwitzerland 50.6- (#10)17.6%3.2×22 +21.4%
7National University of SingaporeSingapore 48.8- (#11)21.3%1.1×22 +18.5%
8University of California, BerkeleyUnited States 47.6- (#7)15.4%1.4×23 -14.8%
9Georgia Institute of TechnologyUnited States 44.8- (#16)8.7%3.3×35 +27.5%
10University of California San DiegoUnited States 44.8- (#17)12.6%1.9×32 +113.8%

Universities only. The score blends standing (50%), citation excellence (20%), size (15%), specialisation (5%), growth (5%), international reach (5%), 2022-now; growth compares 2018-21 with 2022-25 and counts up to +20%, since a jump from a small base is mostly noise. Standing is the score in this field's institution ranking, three quarters over all time and a quarter 2022-now; excellence averages the two windows too. Four years alone undercount fields such as AI, where many conference and arXiv papers from 2022 on carry no affiliation in the data.

Every institution in Numerical Analysis, country by country

Is Numerical Analysis research growing?

Output in 2022-2025 was 8% higher than in 2018-2021, peaking in 2024. The fastest-growing topics are Numerical methods for differential equations, Mathematical Approximation and Integration and Differential Equations and Numerical Methods.

still counting19801990200020102020
grewheldshrank

The same series as a ribbon: one cell per year, darker for more. The line above answers how much; this answers when.

Its share of all research

Growing output can still be a shrinking share: all of science grows. This is Numerical Analysis as a percentage of every work published that year, 0.090% in 2024.

still counting0.10%0.20%0.30%Numerical Analysis0.077% in 202519801990200020102020

Fractional works of Numerical Analysis over all fractional works that year. How the balance of science has shifted.

How much of it reaches the world top 10%

Year by year.

still counting5%10%15%Numerical Analysis10.2% in 2025200020102020

Of each year's works in it, the share that reached the world's 10% most cited for their field and year. A field is the world for its own topics, so there is no other line to hold it against.

Which topics inside it are moving

Growth and decline on one axis around a shared zero. Two lists side by side hide the thing that matters: whether the growth dwarfs the decline, or the other way round.

  1. Numerical methods for differential equations+14%
  2. Mathematical Approximation and Integration+11%
  3. Differential Equations and Numerical Methods+8%
  4. Advanced Optimization Algorithms Research+7%
  5. Iterative Methods for Nonlinear Equations-1%
← shrinkingno changegrowing →

Growth in output from 2018-21 to 2022-25. Every topic is a page of its own.

What does Numerical Analysis include?

Numerical methods for di…37kNumerical methods for differential equations: 37k fractional worksDifferential Equations a…36kDifferential Equations and Numerical Methods: 36k fractional worksAdvanced Optimization Al…30kAdvanced Optimization Algorithms Research: 30k fractional worksIterative Methods for No…17kIterative Methods for Nonlinear Equations: 17k fractional worksMathematical Approximati…10kMathematical Approximation and Integration: 10k fractional works

Area is fractional works in that topic. Every name below is a page of its own.

Share who leads Numerical Analysis

Who leads Numerical Analysis, on Science Explorer

A card with the countries and institutions that lead Numerical Analysis, for a post or a slide.

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1200 × 630 PNG, sized for social posts, drawn from the latest snapshot.

The leading institutions' badges5Each links to that institution's page and updates with every snapshot

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